Author’s note
With the exception of this note, the entire document is machine-written. I have carefully supervised its preparation to ensure that the proofs are readable and that previous literature is appropriately cited. I welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.
The author was partially supported by NSF grant DMS-2450004.
Introduction
Background
The statistics of spacings between consecutive energy levels have been studied since Wigner proposed to model the spectra of heavy nuclei by the eigenvalues of large random matrices [78 , 80 , 81 ] . A basic feature of such spectra is the rarity of very close neighboring levels, which is known as level repulsion . The origin of level repulsion goes back to an observation of von Neumann and Wigner [76 ] . Real symmetric matrices with a repeated eigenvalue form a set of codimension two, and complex Hermitian matrices with a repeated eigenvalue form a set of codimension three. Heuristically, the probability that a given spacing of a random matrix is smaller than s s s s β + 1 s^{\beta+1} s β + 1 β = 1 \beta=1 β = 1 β = 2 \beta=2 β = 2 s β s^\beta s β [36 , 37 ] . The same exponent appears in Wigner’s surmise for the spacing distribution [80 ] and in the factor ∏ i < j ∣ λ i − λ j ∣ β \prod_{i<j}|\lambda_i-\lambda_j|^\beta ∏ i < j ∣ λ i − λ j ∣ β [20 , 21 , 56 ] .
The real symmetric case corresponds to systems with time-reversal symmetry. For such systems, Bohigas, Giannoni, and Schmit conjectured that quantum spectra whose classical dynamics is chaotic share the local statistics of the Gaussian orthogonal ensemble (GOE) [8 ] , in contrast with the Poisson statistics, without repulsion, expected for integrable systems [6 ] . This picture is supported by experimental data (see, for example, [38 , 66 ] for nuclear spectra and microwave billiards).
This paper concerns the smallest spacings in a spectrum of N N N s β s^\beta s β N − 1 / ( β + 1 ) N^{-1/(\beta+1)} N − 1/ ( β + 1 ) s β d s s^\beta\,\mathrm{d}s s β d s [35 , 63 ] , and numerical simulations agree with it [49 , 63 ] . (We emphasize that this Poisson process describes the rare small spacings. Its intensity vanishes at zero because of level repulsion, and it should not be confused with the Poisson level statistics of integrable systems.) Smallest gaps also appear in numerical linear algebra. For example, the rate at which the Toda flow, which is closely related to the QR algorithm, diagonalizes a symmetric matrix is governed by its smallest eigenvalue gap [3 , 17 , 18 ] , and numerical experiments suggest that the resulting running times are universal for Wigner matrices, including matrices with discrete entries [59 ] .
We call real symmetric or complex Hermitian random matrices with independent entries on and above the diagonal Wigner matrices . The class considered in this paper is specified in Definition 1.1 below. In mathematics, the local eigenvalue statistics of Wigner matrices are now known to be universal, in the sense that under mild conditions on the entry distributions they agree asymptotically with those of the Gaussian ensembles. This was established through the work of many authors. We mention [25 , 26 , 27 , 28 , 41 , 70 ] for the averaged bulk statistics, [22 , 69 ] for the distribution of a single gap, and [12 , 46 , 47 ] for statistics at a fixed energy, and we refer to the book [23 ] for an account of the dynamical approach. Matrices with discrete entries are natural in this context. Wigner’s first paper on the semicircle law concerns real symmetric matrices whose off-diagonal entries are ± 1 \pm1 ± 1 [79 ] , and the centered and rescaled adjacency matrix of a dense Erdős–Rényi graph has independent Bernoulli entries above the diagonal.
Universality of a single gap does not determine the smallest gap among N N N [75 ] . Building on Soshnikov’s analysis of extreme spacings in determinantal point processes [65 ] , Ben Arous and Bourgade [3 ] obtained the joint limit of the smallest gaps of the CUE and the GUE together with their locations, and Figalli and Guionnet [34 ] carried these results over to several-matrix models. Determinantal structure is also behind the results of Shi and Jiang [62 ] for complex Ginibre matrices and those of Charlier [14 ] for the two-dimensional Coulomb gas at β = 2 \beta= 2 β = 2
For the Gaussian orthogonal ensemble (GOE), the correlation functions are Pfaffians, and the smallest gaps were determined by Feng, Tian, and Wei [32 ] . Feng, Li, and Yao [33 ] gave a detailed treatment of the Gaussian symplectic ensemble, together with a sketch of a GOE version that also records the locations of the gaps, and the smallest gaps of real Gaussian matrices were studied in [52 ] . Feng and Wei [30 ] treated the circular β \beta β β \beta β [3 ] determined the leading order, and Feng and Wei [31 ] determined the fluctuations.
For Wigner matrices, Bourgade [10 ] proved universality of the smallest gaps for both real symmetric and complex Hermitian generalized Wigner matrices under smoothness assumptions on the entry distributions. Bourgade also proved universality of the largest gaps in the complex Hermitian case under a smoothness assumption. Landon, Lopatto, and Marcinek [48 ] proved a comparison theorem that removes this smoothness assumption. In both works, the explicit largest-gap limit laws were stated for complex Hermitian matrices because the corresponding limiting distribution for the GOE was not known at the time; their comparison arguments also apply to real symmetric matrices. Whether the smallest gaps are universal without smoothness assumptions was raised in [10 ] . Recently, Zhang [82 ] proved universality of the smallest gaps for complex Hermitian Wigner matrices whose entries have real and imaginary parts with sufficiently many well-separated atoms (at least ten suffice), through a comparison argument that matches many moments of the entries. In [82 Conjecture 1.6] , he conjectured the analogous result for real symmetric matrices, noting that it does not seem accessible by his methods, and he further conjectured that two atoms suffice.
For real symmetric matrices with discrete entries, the strongest results available were lower bounds. Nguyen, Tao, and Vu [57 ] proved tail bounds for gaps that imply, in the normalization of this paper, that the smallest gap is at least N − 2 − o ( 1 ) N^{-2-o(1)} N − 2 − o ( 1 ) [13 , 51 , 64 ] . The absence of repeated eigenvalues was proved in [13 , 53 , 71 ] . These lower bounds are smaller than the true order N − 3 / 2 N^{-3/2} N − 3/2 N 1 / 2 N^{1/2} N 1/2
Main results
We now state our results. We begin with the definition of the matrices we consider.
Definition 1.1. Let X X X Z Z Z E X = E Z = 0 \mathbb{E}X=\mathbb{E}Z=0 E X = E Z = 0 E X 2 = 1 \mathbb{E}X^2=1 E X 2 = 1 E Z 2 > 0 \mathbb{E}Z^2>0 E Z 2 > 0 c 0 , C 0 > 0 c_0,C_0>0 c 0 , C 0 > 0
P ( ∣ X ∣ > y ) + P ( ∣ Z ∣ > y ) ≤ C 0 e − c 0 y 2 for all y ≥ 0. (1) \mathbb{P}(|X|>y)+\mathbb{P}(|Z|>y)\le C_0e^{-c_0y^2}\qquad\text{for all }y\ge0. \tag*{(1)} P ( ∣ X ∣ > y ) + P ( ∣ Z ∣ > y ) ≤ C 0 e − c 0 y 2 for all y ≥ 0. (1) For every integer N ≥ 2 N\ge2 N ≥ 2 ( X a b ) 1 ≤ a < b ≤ N (X_{ab})_{1\le a<b\le N} ( X ab ) 1 ≤ a < b ≤ N X X X ( Z a ) 1 ≤ a ≤ N (Z_a)_{1\le a\le N} ( Z a ) 1 ≤ a ≤ N Z Z Z X a b X_{ab} X ab N × N N\times N N × N H = H N H=H_N H = H N
H a b = H b a = N − 1 / 2 X a b ( 1 ≤ a < b ≤ N ) , H a a = N − 1 / 2 Z a ( 1 ≤ a ≤ N ) , (2) H_{ab}=H_{ba}=N^{-1/2}X_{ab}\quad(1\le a<b\le N),\qquad H_{aa}=N^{-1/2}Z_a\quad(1\le a\le N), \tag*{(2)} H ab = H ba = N − 1/2 X ab ( 1 ≤ a < b ≤ N ) , H aa = N − 1/2 Z a ( 1 ≤ a ≤ N ) , (2) the real symmetric Wigner matrix associated with X X X
No assumption on the symmetry, smoothness, or support of the laws of X X X Z Z Z X X X E X 3 \mathbb{E}X^3 E X 3 X X X Z Z Z X X X
For a matrix H H H λ 1 ≤ ⋯ ≤ λ N \lambda_1\le\cdots\le\lambda_N λ 1 ≤ ⋯ ≤ λ N n n n [ n ] = { 1 , … , n } [n]=\{1,\ldots,n\} [ n ] = { 1 , … , n } i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] g i = λ i + 1 − λ i g_i=\lambda_{i+1}-\lambda_i g i = λ i + 1 − λ i i i i
ρ s c ( E ) = 1 2 π max { 4 − E 2 , 0 } , c ( E ) = π 2 6 ρ s c ( E ) 3 ( E ∈ R ) (3) \rho_{\mathrm{sc}}(E)=\frac{1}{2\pi}\sqrt{\max\{4-E^2,0\}},\qquad c(E)=\frac{\pi^2}{6}\rho_{\mathrm{sc}}(E)^3\qquad(E\in\mathbb{R}) \tag*{(3)} ρ sc ( E ) = 2 π 1 max { 4 − E 2 , 0 } , c ( E ) = 6 π 2 ρ sc ( E ) 3 ( E ∈ R ) (3) denote the semicircle density and the function that determines the intensity in our main result. We identify a locally finite point configuration on [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty) \times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) vague topology , which we define as the coarsest topology for which the maps μ ↦ ∫ f d μ \mu\mapsto\int f\,\mathrm{d}\mu μ ↦ ∫ f d μ f f f [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty) \times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) + ∞ +\infty + ∞
Theorem 1.2. Let H H H Definition 1.1 . Then the point processes
Ξ N = ∑ i ∈ [ N − 1 ] : λ i ∈ ( − 2 , 2 ) δ ( N 3 / 2 g i , λ i ) (4) \Xi_N = \sum_{i\in[N-1]:\lambda_i\in(-2,2)} \delta_{(N^{3/2}g_i,\lambda_i)} \tag*{(4)} Ξ N = i ∈ [ N − 1 ] : λ i ∈ ( − 2 , 2 ) ∑ δ ( N 3/2 g i , λ i ) (4) on [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty) \times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 )
ν ( d x d E ) = c ( E ) x d x d E . (5) \nu(\mathrm{d}x\,\mathrm{d}E) = c(E)x\,\mathrm{d}x\,\mathrm{d}E. \tag*{(5)} ν ( d x d E ) = c ( E ) x d x d E . (5) Consequently, for every nonempty interval I I I ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) x ≥ 0 x \ge0 x ≥ 0
lim N → ∞ P ( N 3 / 2 min i ∈ [ N − 1 ] : λ i ∈ I g i > x ) = exp ( − π 2 x 2 12 ∫ I ρ s c ( E ) 3 d E ) . (6) \lim_{N\to\infty} \mathbb{P}\left(N^{3/2}\min_{i\in[N-1]:\lambda_i\in I} g_i > x\right) = \exp\left(-\frac{\pi^2x^2}{12}\int_I \rho_{\mathrm{sc}}(E)^3\,\mathrm{d}E\right). \tag*{(6)} N → ∞ lim P ( N 3/2 i ∈ [ N − 1 ] : λ i ∈ I min g i > x ) = exp ( − 12 π 2 x 2 ∫ I ρ sc ( E ) 3 d E ) . (6) Theorem 1.2 identifies the smallest gaps in the bulk of the spectrum for every matrix as in Definition 1.1 , including Bernoulli matrices. It settles [82 Conjecture 1.6] , together with the stronger version with two atoms, for the matrices of Definition 1.1 . The conjecture concerns real generalized Wigner matrices whose entries have tails at most c − 1 e − x c c^{-1}e^{-xc} c − 1 e − x c E Z 2 > 0 \mathbb{E}Z^2 > 0 E Z 2 > 0 [32 , 33 ] and with the one obtained under smoothness assumptions in [10 ] . For every interval I I I ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) k k k I I I k k k + ∞ +\infty + ∞ k k k k k k I I I [10 Corollary 1.3] , which gives the limiting law of the k k k k k k N 3 / 2 ( 2 − 1 ∫ I c ( E ) d E ) 1 / 2 g i N^{3/2}(2^{-1}\int_I c(E)\,\mathrm{d}E)^{1/2}g_i N 3/2 ( 2 − 1 ∫ I c ( E ) d E ) 1/2 g i i i i λ i ∈ I \lambda_i \in I λ i ∈ I 2 x 2 k − 1 e − x 2 / ( k − 1 ) ! 2x^{2k-1}e^{-x^2}/(k-1)! 2 x 2 k − 1 e − x 2 / ( k − 1 )! ( 0 , ∞ ) (0,\infty) ( 0 , ∞ )
We have not attempted to optimize the assumptions. The sub-Gaussian condition (1.1) is used in several places, for instance through the results of [57 ] . The condition E Z 2 > 0 \mathbb{E}Z^2 > 0 E Z 2 > 0 [10 , 29 ] , which assume that all entry variances are bounded below by a multiple of N − 1 N^{-1} N − 1 [57 ] use this property.
Our second result is a law of large numbers for the number of gaps at scales between the smallest gaps and the typical spacing. It is the main input to the proof of Theorem 1.2 and may be of independent interest. We let E i = ( λ i + λ i + 1 ) / 2 E_i=(\lambda_i+\lambda_{i+1})/2 E i = ( λ i + λ i + 1 ) /2 i i i ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) R \mathbb{R} R [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) [ 0 , ∞ ) × R [0,\infty)\times\mathbb{R} [ 0 , ∞ ) × R
Theorem 1.3. Let H H H Definition 1.1 , and let α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) φ : ( − 2 , 2 ) → R \varphi:(-2,2)\to\mathbb{R} φ : ( − 2 , 2 ) → R f : [ 0 , ∞ ) → R f:[0,\infty)\to\mathbb{R} f : [ 0 , ∞ ) → R
N 2 α − 1 ∑ i = 1 N − 1 φ ( E i ) f ( N 1 + α g i ) ⟶ ∫ − 2 2 φ ( E ) c ( E ) d E ⋅ ∫ 0 ∞ y f ( y ) d y (7) N^{2\alpha-1}\sum_{i=1}^{N-1}\varphi(E_i)f\left(N^{1+\alpha}g_i\right)\longrightarrow\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E\cdot\int_{0}^{\infty}y f(y)\,\mathrm{d}y \tag*{(7)} N 2 α − 1 i = 1 ∑ N − 1 φ ( E i ) f ( N 1 + α g i ) ⟶ ∫ − 2 2 φ ( E ) c ( E ) d E ⋅ ∫ 0 ∞ y f ( y ) d y (7) in probability as N → ∞ N\to\infty N → ∞
We call a compact interval contained in ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) bulk interval . In words, Theorem 1.3 states that for every bulk interval I I I y > 0 y>0 y > 0 y N − 1 − α yN^{-1-\alpha} y N − 1 − α I I I N 1 − 2 α ( y 2 / 2 ) ∫ I c ( E ) d E N^{1-2\alpha}(y^2/2)\int_I c(E)\,\mathrm{d}E N 1 − 2 α ( y 2 /2 ) ∫ I c ( E ) d E Theorem 1.2 , this number tends to zero in probability for α > 1 / 2 \alpha>1/2 α > 1/2 α = 1 / 2 \alpha=1/2 α = 1/2 α ≥ 1 / 2 \alpha\ge1/2 α ≥ 1/2 α < 1 / 4 \alpha<1/4 α < 1/4 Proposition 3.6 ) follows from the Pfaffian formulas for its correlation functions. We prove it in Appendix § 12 , where it serves as the main input to the base case of our proof.
Our methods also apply to complex Hermitian matrices, as follows.
Definition 1.4. Let Z Z Z E Z = 0 \mathbb{E}Z=0 E Z = 0 E Z 2 > 0 \mathbb{E}Z^2>0 E Z 2 > 0 X C X_{\mathrm{C}} X C E X C = 0 \mathbb{E}X_{\mathrm{C}}=0 E X C = 0 E ∣ X C ∣ 2 = 1 \mathbb{E}|X_{\mathrm{C}}|^2=1 E ∣ X C ∣ 2 = 1 E X C 2 = 0 \mathbb{E}X_{\mathrm{C}}^2=0 E X C 2 = 0 c 0 , C 0 > 0 c_0,C_0>0 c 0 , C 0 > 0 P ( ∣ X C ∣ > y ) + P ( ∣ Z ∣ > y ) ≤ C 0 e − c 0 y 2 \mathbb{P}(|X_{\mathrm{C}}|>y)+\mathbb{P}(|Z|>y)\le C_0e^{-c_0y^2} P ( ∣ X C ∣ > y ) + P ( ∣ Z ∣ > y ) ≤ C 0 e − c 0 y 2 y ≥ 0 y\ge0 y ≥ 0 N ≥ 2 N\ge2 N ≥ 2 ( X C , a b ) 1 ≤ a < b ≤ N (X_{\mathrm{C},ab})_{1\le a<b\le N} ( X C , ab ) 1 ≤ a < b ≤ N X C X_{\mathrm{C}} X C ( Z a ) 1 ≤ a ≤ N (Z_a)_{1\le a\le N} ( Z a ) 1 ≤ a ≤ N Z Z Z X C , a b X_{\mathrm{C},ab} X C , ab N × N N\times N N × N H H H H a b = N − 1 / 2 X C , a b H_{ab}=N^{-1/2}X_{\mathrm{C},ab} H ab = N − 1/2 X C , ab H b a = H a b ‾ H_{ba}=\overline{H_{ab}} H ba = H ab a < b a<b a < b H a a = N − 1 / 2 Z a H_{aa}=N^{-1/2}Z_a H aa = N − 1/2 Z a complex Hermitian Wigner matrix associated with X C X_{\mathrm{C}} X C Z Z Z
The condition E X C 2 = 0 \mathbb{E}X_{\mathrm{C}}^2=0 E X C 2 = 0 X C X_{\mathrm{C}} X C X C X_{\mathrm{C}} X C X C X_{\mathrm{C}} X C 2 − 1 / 2 X 2^{-1/2}X 2 − 1/2 X X X X Definition 1.1 .
Theorem 1.5. Let H H H λ 1 ≤ ⋯ ≤ λ N \lambda_1\le\cdots\le\lambda_N λ 1 ≤ ⋯ ≤ λ N g i = λ i + 1 − λ i g_i=\lambda_{i+1}-\lambda_i g i = λ i + 1 − λ i c 2 ( E ) = ( π 2 / 3 ) ρ s c ( E ) 4 c_2(E)=(\pi^2/3)\rho_{\mathrm{sc}}(E)^4 c 2 ( E ) = ( π 2 /3 ) ρ sc ( E ) 4
∑ i ∈ [ N − 1 ] : λ i ∈ ( − 2 , 2 ) δ ( N 4 / 3 g i , λ i ) (8) \sum_{i\in[N-1]:\lambda_i\in(-2,2)}\delta_{\left(N^{4/3}g_i,\lambda_i\right)} \tag*{(8)} i ∈ [ N − 1 ] : λ i ∈ ( − 2 , 2 ) ∑ δ ( N 4/3 g i , λ i ) (8) on [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty) \times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) c 2 ( E ) x 2 d x d E c_2(\mathrm{E})x^2\,\mathrm{d}x\,\mathrm{dE} c 2 ( E ) x 2 d x dE I I I ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) x ≥ 0 x \ge0 x ≥ 0
lim N → ∞ P ( N 4 / 3 min i ∈ [ N − 1 ] : λ i ∈ I g i > x ) = exp ( − π 2 x 3 9 ∫ I ρ s c ( E ) 4 d E ) . (9) \lim_{N\to\infty}\mathbb{P}\left(N^{4/3}\min_{i\in[N-1]:\lambda_i\in I}g_i>x\right)=\exp\left(-\frac{\pi^2x^3}{9}\int_I\rho_{\mathrm{sc}}(\mathrm{E})^4\,\mathrm{dE}\right). \tag*{(9)} N → ∞ lim P ( N 4/3 i ∈ [ N − 1 ] : λ i ∈ I min g i > x ) = exp ( − 9 π 2 x 3 ∫ I ρ sc ( E ) 4 dE ) . (9) The limit agrees with the one for the GUE [3 ] . Theorem 1.5 removes the smoothness assumption of [10 ] and the assumption on the number of atoms of [82 ] for complex Hermitian Wigner matrices with identically distributed entries above the diagonal. We give the proof of Theorem 1.5 in less detail than that of Theorem 1.2. In Section 9, we sketch how the arguments of § 6 and 6 and Appendix § 12 adapt to complex Hermitian matrices; the resampling argument is not needed in this case. We give the details of the two estimates that are new. The first bounds, for κ ∈ ( 0 , 1 / 8 ) \kappa\in(0,1/8) κ ∈ ( 0 , 1/8 ) N − 1 − κ N^{-1-\kappa} N − 1 − κ N 1 − 8 κ + o ( 1 ) N^{1-8\kappa+o(1)} N 1 − 8 κ + o ( 1 ) N 1 − 3 α ′ N^{1-3\alpha'} N 1 − 3 α ′ N − 1 − α ′ N^{-1-\alpha'} N − 1 − α ′ α ′ < 8 κ / 3 \alpha' < 8\kappa/3 α ′ < 8 κ /3 3 × 3 3 \times3 3 × 3
Methods and relation to previous work
We give an overview of the proofs in Section 2. Here we describe how their ingredients relate to the literature.
In [10 ] , the smoothness of the entry distributions is used through the reverse heat flow [25 ] , which represents the matrix, up to a small error in total variation, as a matrix with a Gaussian component. Dyson Brownian motion [19 ] must run for a time at least N − 1 / 2 + 2 ε N^{-1/2+2\varepsilon} N − 1/2 + 2 ε ε > 0 \varepsilon> 0 ε > 0 N − 3 / 2 N^{-3/2} N − 3/2 N − 1 / 4 + ε N^{-1/4+\varepsilon} N − 1/4 + ε X a b X_{ab} X ab N − 1 / 4 + ε N^{-1/4+\varepsilon} N − 1/4 + ε
The alternative is a comparison argument, which replaces the entries one at a time. As explained in § 3.1 , for real symmetric matrices a single entry moves a gap by an amount comparable to the smallest gaps, so the standard perturbative comparison estimate does not give a small error at this scale. Zhang gives a different reason for restricting his result to the complex Hermitian case, which is consistent with ours. The error in his Green function comparison, which matches a fixed number of moments, is small only for spectral parameters at distance much larger than N − 3 / 2 N^{-3/2} N − 3/2 [82 (11)] and the paragraph after it), while the smallest gaps of real symmetric matrices are of order N − 3 / 2 N^{-3/2} N − 3/2 N − 3 / 2 N^{-3/2} N − 3/2
Instead, we use the fact that resampling a small random set of entries does not change the law of the matrix. Resampling a few random entries of a Wigner matrix, and the heuristic that each resampled entry moves an eigenvalue by one step of a random walk, were used by Bordenave, Lugosi, and Zhivotovskiy to study the noise sensitivity of the top eigenvector [9 ] ; see also [16 ] for a related dynamics. We use resampling instead to smooth the law of the gaps while preserving the law of the matrix. The relevant random walk acts on the traceless part of the 2 × 2 2 \times2 2 × 2 R 2 \mathbb{R}^{2} R 2 [76 ] .
The analysis of this random walk requires a local limit theorem that resolves the law of the walk at the scale of a single step, which for discrete entries requires excluding arithmetic structure in the eigenvectors. For this, we use net and decoupling arguments developed for the invertibility of random matrices and the structure of their eigenvectors by Rudelson and Vershynin [60 ] , Vershynin [74 ] , Nguyen, Tao, and Vu [57 ] , and Tao and Vu [71 ] . The local limit theorem itself is proved by Fourier analysis with smooth test functions, as in the classical theorems of Shepp and Stone [61 , 67 ] , and the passage from local limits to Poisson convergence uses the Bonferroni inequalities, as in the factorial moment method used for extreme gaps in [3 , 65 ] . We also use the tail bound of Nguyen, Tao, and Vu for three consecutive eigenvalues [57 Corollary 2.5] at several points.
The resampling argument requires the number of gaps at a scale slightly larger than N − 3 / 2 N^{-3/2} N − 3/2 Theorem 1.3 . Its proof uses a comparison argument along the Ornstein–Uhlenbeck flow. Our comparison estimate follows the continuity estimate of Bourgade and Yau [11 Lemma A.1] (see also [23 ] , Chapter 15), which descends from the Lindeberg method of Chatterjee [15 ] , the four moment theorem of Tao and Vu [70 ] , and the Green function comparison of Erdős, Yau, and Yin [28 ] ; the expansion in a single entry is the cumulant expansion of Khorunzhy, Khoruzhenko, and Pastur [44 ] , in the form of [54 Proposition 3.1] .
Comparison arguments for statistics that involve all eigenvalues appear in [48 ] , which uses regularizations of eigenvalues through the Helffer–Sjöstrand formula following Knowles and Yin [45 ] , and in [82 ] , which uses resolvent expansions with spectral parameters at distance less than N − 1 N^{-1} N − 1 [4 , 24 , 29 ] , and we bound their derivatives by Cauchy’s estimates. We are not aware of an earlier use of this device in comparison arguments.
The proof of Theorem 1.3 is an induction on α \alpha α α < 1 / 8 \alpha< 1/8 α < 1/8 [10 Corollary 3.3] between Dyson Brownian motion started from H H H § 12 , which relies on the Pfaffian formulas for the GOE correlation functions [1 , 56 , 72 ] in the form given in [33 ] .
To go beyond α = 1 / 8 \alpha= 1/8 α = 1/8 H H H [41 ] and Erdős, Péché, Ramírez, Schlein, and Yau [25 ] (see also [26 ] and [23, Chapter 12]); the regime of Dyson Brownian motion run for times much smaller than $N^{-1}$, which is the one relevant here, was studied by von Soosten and Warzel [77 ] . Our perturbation lemma for entrywise small perturbations of diagonal matrices is a variant of the quadratic residual bounds of Mathias {citep}[55 ] and Li and Li {citep}[50 ] , with entrywise bounds in place of operator norms. The computation of gap counts at a smaller scale through independent $2 \times 2$ Gaussian blocks makes rigorous, in this regime, the $2 \times 2$ heuristic behind Wigner’s surmise. The induction on the scale is in the spirit of the multiscale arguments for local laws (see {citep}[4 Section 5] ) and of the induction on the size of index sets in {citep}[5 ] `.
Notation
Constants c , C > 0 c, C > 0 c , C > 0 X X X Z Z Z X C X_{\mathrm C} X C Z Z Z N N N c ( E ) c(E) c ( E ) ε \varepsilon ε C k C_k C k
We say that an event holds with overwhelming probability if for every D > 0 D > 0 D > 0 C D > 0 C_D > 0 C D > 0 C D N − D C_DN^{-D} C D N − D N N N o ( 1 ) o(1) o ( 1 ) N → ∞ N \to\infty N → ∞ o P ( 1 ) o_{\mathrm P}(1) o P ( 1 ) O P ( 1 ) O_{\mathrm P}(1) O P ( 1 )
For x , y ∈ R n x,y \in\mathbb{R}^n x , y ∈ R n ∣ x ∣ \lvert x\rvert ∣ x ∣ x x x x ⋅ y x \cdot y x ⋅ y v ∈ R N v \in\mathbb{R}^N v ∈ R N ∥ v ∥ 2 \lVert v\rVert_2 ∥ v ∥ 2 ∥ v ∥ ∞ \lVert v\rVert_\infty ∥ v ∥ ∞
For a matrix A A A ∥ A ∥ \lVert A\rVert ∥ A ∥ A T A^{T} A T A ∗ A^* A ∗ N × N N \times N N × N A A A K ⊂ [ N ] K \subset[N] K ⊂ [ N ] A K K = ( A j k ) j , k ∈ K A_{KK} = (A_{jk})_{j,k \in K} A K K = ( A j k ) j , k ∈ K Id n \operatorname{Id}_n Id n n × n n \times n n × n e a e_a e a a a a R N \mathbb{R}^N R N
For a real symmetric or complex Hermitian N × N N \times N N × N A A A λ 1 ( A ) ≤ ⋯ ≤ λ N ( A ) \lambda_1(A) \le\cdots\le\lambda_N(A) λ 1 ( A ) ≤ ⋯ ≤ λ N ( A ) H H H λ i = λ i ( H ) \lambda_i = \lambda_i(H) λ i = λ i ( H ) u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N H u i = λ i u i Hu_i = \lambda_i u_i H u i = λ i u i u i ( a ) = e a T u i u_i(a) = e_a^T u_i u i ( a ) = e a T u i a a a u i u_i u i H H H H H H
Overview of the proof
This section explains the proofs of Theorem 1.2 and Theorem 1.3 , and, in § 3.6 , the changes needed for Theorem 1.5 . It is written to be read on its own. The precise statements and proofs occupy Sections 3 to 8 and Appendix § 12 ; a roadmap is given in § 3.7 .
Throughout this section, except in § 3.6 , H H H Definition 1.1 . It is real symmetric, its entries above the diagonal are N − 1 / 2 X a b N^{-1/2}X_{ab} N − 1/2 X ab X a b X_{ab} X ab X X X λ 1 ≤ ⋯ ≤ λ N \lambda_1 \le\cdots\le\lambda_N λ 1 ≤ ⋯ ≤ λ N H H H u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] g i = λ i + 1 − λ i g_i=\lambda_{i+1}-\lambda_i g i = λ i + 1 − λ i i i i E i = ( λ i + λ i + 1 ) / 2 E_i=(\lambda_i+\lambda_{i+1})/2 E i = ( λ i + λ i + 1 ) /2 c ( E ) = ( π 2 / 6 ) ρ s c ( E ) 3 c(E)=(\pi^2/6)\rho_{\mathrm{sc}}(E)^3 c ( E ) = ( π 2 /6 ) ρ sc ( E ) 3 ρ s c \rho_{\mathrm{sc}} ρ sc ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) D > 0 D>0 D > 0 C D N − D C_DN^{-D} C D N − D ε > 0 \varepsilon>0 ε > 0 ε \varepsilon ε o ( 1 ) o(1) o ( 1 )
The main difficulty
We set d = N − 3 / 2 d=N^{-3/2} d = N − 3/2 d d d critical scale . Theorem 1.2 states that the smallest gaps of H H H d d d I I I x > 0 x>0 x > 0 i i i g i ≤ x d g_i\le xd g i ≤ x d λ i ∈ I \lambda_i\in I λ i ∈ I
∫ I ∫ 0 x c ( E ) y d y d E = x 2 2 ∫ I c ( E ) d E . (10) \int_I\int_0^x c(E)y\,\mathrm{d}y\,\mathrm{d}E=\frac{x^2}{2}\int_I c(E)\,\mathrm{d}E. \tag*{(10)} ∫ I ∫ 0 x c ( E ) y d y d E = 2 x 2 ∫ I c ( E ) d E . (10) The usual route to universality of local eigenvalue statistics compares H H H [15 , 28 , 70 ] . For the smallest gaps of real symmetric matrices, the standard perturbative comparison estimate does not give a small error at this scale. First-order perturbation theory suggests that changing the entries H a b H_{ab} H ab H b a H_{ba} H ba h h h λ i \lambda_i λ i 2 h u i ( a ) u i ( b ) 2hu_i(a)u_i(b) 2 h u i ( a ) u i ( b ) N − 1 / 2 N^{-1/2} N − 1/2 max i , a ∣ u i ( a ) ∣ ≤ N − 1 / 2 + ε \max_{i,a}|u_i(a)| \le N^{-1/2+\varepsilon} max i , a ∣ u i ( a ) ∣ ≤ N − 1/2 + ε ε > 0 \varepsilon> 0 ε > 0 [4 , 29 ] . So a single entry moves an eigenvalue, and therefore a gap, by about N − 3 / 2 = d N^{-3/2}=d N − 3/2 = d d d d
In [10 ] , this obstacle was avoided by assuming that the entry distributions are smooth, which allows one to replace H H H d d d N − 4 / 3 N^{-4/3} N − 4/3 N − 3 / 2 N^{-3/2} N − 3/2 § 3.6 .
Our argument has two parts. At the scale d d d H H H § 3.2 to § 3.4 . This argument requires the number of gaps at a scale slightly larger than d d d N − 1 − α N^{-1-\alpha} N − 1 − α α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) d d d N − 1 N^{-1} N − 1 N − 3 / 2 N^{-3/2} N − 3/2 § 3.5 , we prove a law of large numbers for the number of gaps at such scales, using a comparison argument together with an induction on α \alpha α
Four steps of the proof are new: the resampling argument, together with the reduction of its local limit theorem to the exclusion of eigenvectors whose coordinates take few values, which replaces the use of smooth entry distributions in [10 ] ; the holomorphic bound on derivatives in the comparison argument, which replaces bounds obtained from eigenvalue perturbation formulas or from resolvent regularizations of eigenvalues; the induction on α \alpha α 2 × 2 2 \times2 2 × 2 α < 1 / 8 \alpha< 1/8 α < 1/8 E X 3 ≠ 0 \mathbb{E}X^3 \ne0 E X 3 = 0 3 × 3 3 \times3 3 × 3
Resampling
Let τ = N 1 / 100 \tau= N^{1/100} τ = N 1/100 m = 2 ⌊ τ / 8 ⌋ m = 2\lfloor\tau/8\rfloor m = 2 ⌊ τ /8 ⌋ m = τ / 4 + O ( 1 ) m = \tau/4 + O(1) m = τ /4 + O ( 1 ) ( a 1 , b 1 ) , … , ( a m , b m ) (a_1,b_1),\ldots,(a_m,b_m) ( a 1 , b 1 ) , … , ( a m , b m ) w i t h $ 1 ≤ a < b ≤ N with \$1 \le a < b \le N w i t h $1 ≤ a < b ≤ N Y 1 , … , Y m Y_1,\ldots,Y_m Y 1 , … , Y m X X X H H H ( a 1 , b 1 ) , … , ( a m , b m ) (a_1,b_1),\ldots,(a_m,b_m) ( a 1 , b 1 ) , … , ( a m , b m ) selected positions. Let H ~ \widetilde{H} H H H H l = 1 , … , m l=1,\ldots,m l = 1 , … , m ( a l , b l ) (a_l,b_l) ( a l , b l ) ( b l , a l ) (b_l,a_l) ( b l , a l ) N − 1 / 2 Y l N^{-1/2}Y_l N − 1/2 Y l H ~ \widetilde{H} H H H H H ~ \widetilde{H} H H H H
Heuristically, one replacement moves a gap by an amount of order d d d m m m m ≈ τ / 4 m \approx\tau/4 m ≈ τ /4 d d d 2 × 2 2 \times2 2 × 2 h 1 , h 2 h_1,h_2 h 1 , h 2 h 12 h_{12} h 12
2 ∣ ( h 1 − h 2 2 , h 12 ) ∣ . (11) 2\left|\left(\frac{h_1-h_2}{2},h_{12}\right)\right|. \tag*{(11)} 2 ( 2 h 1 − h 2 , h 12 ) . (11) Fix i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] g ~ i \widetilde{g}_i g i i i i ( i + 1 ) (i+1) ( i + 1 ) H ~ \widetilde{H} H a < b a<b a < b D a b ( i ) ∈ R 2 D_{ab}^{(i)} \in\mathbb{R}^2 D ab ( i ) ∈ R 2
D a b ( i ) = N ( u i ( a ) u i ( b ) − u i + 1 ( a ) u i + 1 ( b ) , u i ( a ) u i + 1 ( b ) + u i ( b ) u i + 1 ( a ) ) , (12) D_{ab}^{(i)}=N\left(u_i(a)u_i(b)-u_{i+1}(a)u_{i+1}(b),u_i(a)u_{i+1}(b)+u_i(b)u_{i+1}(a)\right), \tag*{(12)} D ab ( i ) = N ( u i ( a ) u i ( b ) − u i + 1 ( a ) u i + 1 ( b ) , u i ( a ) u i + 1 ( b ) + u i ( b ) u i + 1 ( a ) ) , (12) and we set
z i = ( − g i 2 d , 0 ) , S i = ∑ l = 1 m ( Y l − X a l b l ) D a l b l ( i ) . (13) z_i=\left(-\frac{g_i}{2d},0\right),\qquad S_i=\sum_{l=1}^{m}(Y_l-X_{a_lb_l})D_{a_lb_l}^{(i)}. \tag*{(13)} z i = ( − 2 d g i , 0 ) , S i = l = 1 ∑ m ( Y l − X a l b l ) D a l b l ( i ) . (13) We apply (11) to the compression of H ~ \widetilde{H} H u i u_i u i u i + 1 u_{i+1} u i + 1 u i T H ~ u i u_i^{T}\widetilde{H}u_i u i T H u i u i + 1 T H ~ u i + 1 u_{i+1}^{T}\widetilde{H}u_{i+1} u i + 1 T H u i + 1 u i T H ~ u i + 1 u_i^{T}\widetilde{H}u_{i+1} u i T H u i + 1 1 − O ( τ 2 N − 2 ) 1-O(\tau^2N^{-2}) 1 − O ( τ 2 N − 2 ) 2 d ∣ z i + S i ∣ 2d|z_i+S_i| 2 d ∣ z i + S i ∣ u i u_i u i u i + 1 u_{i+1} u i + 1 § 3.3 , we therefore obtain the approximation g ~ i ≈ 2 d ∣ z i + S i ∣ \widetilde{g}_i \approx2d|z_i+S_i| g i ≈ 2 d ∣ z i + S i ∣ S i S_i S i linearized increment and 2 d ∣ z i + S i ∣ 2d|z_i+S_i| 2 d ∣ z i + S i ∣ linearized gap of the pair ( i , i + 1 ) (i,i+1) ( i , i + 1 )
Conditionally on H H H S i S_i S i m m m ( Y − X a b ) D a b ( i ) (Y-X_{ab})D_{ab}^{(i)} ( Y − X ab ) D ab ( i ) ( a , b ) (a,b) ( a , b ) a < b a<b a < b Y Y Y X X X X a b X_{ab} X ab N ε N^\varepsilon N ε ∣ ( Y − X a b ) D a b ( i ) ∣ ≤ 2 N 2 ε ( ∣ Y ∣ + N ε ) |(Y-X_{ab})D_{ab}^{(i)}| \le2N^{2\varepsilon}(|Y|+N^\varepsilon) ∣ ( Y − X ab ) D ab ( i ) ∣ ≤ 2 N 2 ε ( ∣ Y ∣ + N ε ) 4 I d 2 + o ( 1 ) 4I_{d_2}+o(1) 4 I d 2 + o ( 1 ) ( Y − X a b ) 2 (Y-X_{ab})^2 ( Y − X ab ) 2 2 2 2 u i u_i u i u i + 1 u_{i+1} u i + 1 D a b ( i ) ( D a b ( i ) ) T D_{ab}^{(i)}(D_{ab}^{(i)})^T D ab ( i ) ( D ab ( i ) ) T a < b a<b a < b 2 I d 2 2I_{d_2} 2 I d 2 X a b X_{ab} X ab ∥ X ˚ ∥ ≤ N 1 / 2 + ε \|\mathring{X}\| \le N^{1/2+\varepsilon} ∥ X ˚ ∥ ≤ N 1/2 + ε X ˚ \mathring{X} X ˚ X a b 2 − 1 X_{ab}^2-1 X ab 2 − 1
So S i S_i S i 4 m I d 2 ≈ τ I d 2 4mI_{d_2}\approx\tau I_{d_2} 4 m I d 2 ≈ τ I d 2 y ∈ R 2 y\in\mathbb{R}^2 y ∈ R 2 ( 2 π τ ) − 1 e − ∣ y ∣ 2 / ( 2 τ ) (2\pi\tau)^{-1}e^{-|y|^2/(2\tau)} ( 2 π τ ) − 1 e − ∣ y ∣ 2 / ( 2 τ ) S i S_i S i ψ : R 2 → R \psi:\mathbb{R}^2\to\mathbb{R} ψ : R 2 → R
τ E [ ψ ( z i + S i ) ∣ H ] = 1 2 π τ exp ( − ∣ z i ∣ 2 2 τ ) ∫ R 2 ψ ( y ) d y + o ( 1 ) . (14) \tau\,\mathbb{E}\left[\psi(z_i+S_i)\mid H\right]=\frac{1}{2\pi\tau}\exp\left(-\frac{|z_i|^2}{2\tau}\right)\int_{\mathbb{R}^2}\psi(y)\,\mathrm{d}y+o(1). \tag*{(14)} τ E [ ψ ( z i + S i ) ∣ H ] = 2 π τ 1 exp ( − 2 τ ∣ z i ∣ 2 ) ∫ R 2 ψ ( y ) d y + o ( 1 ) . (14) We set ℓ = d τ 1 / 2 = N − 299 / 200 \ell=d\tau^{1/2}=N^{-299/200} ℓ = d τ 1/2 = N − 299/200 x > 0 x>0 x > 0
P ( g ~ i ≤ x d ∣ H ) ≈ P ( ∣ z i + S i ∣ ≤ x 2 ∣ H ) ≈ π x 2 4 ⋅ 1 2 π τ exp ( − ∣ z i ∣ 2 2 τ ) = x 2 8 τ exp ( − g i 2 8 ℓ 2 ) . (15) \mathbb{P}(\widetilde g_i\le xd\mid H)\approx\mathbb{P}\left(|z_i+S_i|\le\frac{x}{2}\mid H\right)\approx\frac{\pi x^2}{4}\cdot\frac{1}{2\pi\tau}\exp\left(-\frac{|z_i|^2}{2\tau}\right)=\frac{x^2}{8\tau}\exp\left(-\frac{g_i^2}{8\ell^2}\right). \tag*{(15)} P ( g i ≤ x d ∣ H ) ≈ P ( ∣ z i + S i ∣ ≤ 2 x ∣ H ) ≈ 4 π x 2 ⋅ 2 π τ 1 exp ( − 2 τ ∣ z i ∣ 2 ) = 8 τ x 2 exp ( − 8 ℓ 2 g i 2 ) . (15) This mechanism is illustrated in Figure 2.1 . A gap g i g_i g i ℓ \ell ℓ τ − 1 \tau^{-1} τ − 1 H ~ \widetilde H H d d d g i g_i g i ℓ \ell ℓ
Figure 2.1. The linearized picture of a pair ( i , i + 1 ) (i,i+1) ( i , i + 1 ) z i = ( − g i / ( 2 d ) , 0 ) z_i=(-g_i/(2d),0) z i = ( − g i / ( 2 d ) , 0 ) g i g_i g i H H H S i S_i S i m ≈ τ / 4 m\approx\tau/4 m ≈ τ /4 4 I d 2 4\mathrm{Id}_2 4 Id 2 τ I d 2 \tau\mathrm{Id}_2 τ Id 2 τ 1 / 2 \tau^{1/2} τ 1/2 2 τ 1 / 2 2\tau^{1/2} 2 τ 1/2 o ( d ) o(d) o ( d ) H ~ \widetilde H H x d xd x d z i + S i z_i+S_i z i + S i x / 2 x/2 x /2 ( x 2 / ( 8 τ ) ) exp ( − g i 2 / ( 8 ℓ 2 ) ) (x^2/(8\tau))\exp(-g_i^2/(8\ell^2)) ( x 2 / ( 8 τ )) exp ( − g i 2 / ( 8 ℓ 2 ))
To sum (2.6) over i i i H H H ℓ \ell ℓ α = 99 / 200 \alpha=99/200 α = 99/200 N − 1 − α = ℓ N^{-1-\alpha}=\ell N − 1 − α = ℓ N 1 − 2 α = τ N^{1-2\alpha}=\tau N 1 − 2 α = τ Theorem 1.3 with this value of α \alpha α H H H ℓ \ell ℓ Theorem 1.3 concerns continuous test functions with compact support. To apply it to the indicator function of a bulk interval I I I y ↦ e − y 2 / 8 y\mapsto e^{-y^2/8} y ↦ e − y 2 /8 Theorem 1.3 (see Proposition 4.1 ). It states that for every bulk interval I I I C I > 0 C_I > 0 C I > 0
∑ 1 ≤ i < j ≤ N : ( λ i + λ j ) / 2 ∈ I ( 1 + ( λ j − λ i ) 2 ℓ 2 ) − 61 ≤ C I τ . (16) \sum_{\substack{1 \le i < j \le N:\\(\lambda_i+\lambda_j)/2\in I}} \left(1+\frac{(\lambda_j-\lambda_i)^2}{\ell^2}\right)^{-61} \le C_I\tau. \tag*{(16)} 1 ≤ i < j ≤ N : ( λ i + λ j ) /2 ∈ I ∑ ( 1 + ℓ 2 ( λ j − λ i ) 2 ) − 61 ≤ C I τ . (16) We obtain
1 τ ∑ i : E i ∈ I exp ( − g ~ i 2 8 ℓ 2 ) ⟶ 4 ∫ I c ( E ) d E (17) \frac{1}{\tau}\sum_{i:E_i\in I}\exp\left(-\frac{\widetilde g_i^2}{8\ell^2}\right) \longrightarrow4\int_I c(E)\,\mathrm{d}E \tag*{(17)} τ 1 i : E i ∈ I ∑ exp ( − 8 ℓ 2 g i 2 ) ⟶ 4 ∫ I c ( E ) d E (17) in probability, where we used that ∫ 0 ∞ y e − y 2 / 8 d y = 4 \int_0^\infty ye^{-y^2/8}\,\mathrm{d}y=4 ∫ 0 ∞ y e − y 2 /8 d y = 4 x 2 / 8 x^2/8 x 2 /8 τ \tau τ
Summing (2.6) over all pairs requires care. The local limit theorem (2.5) has an error o ( τ − 1 ) o(\tau^{-1}) o ( τ − 1 ) g i ≤ N − 7 / 5 g_i\le N^{-7/5} g i ≤ N − 7/5 § 3.3 ), and there are about N 1 / 5 N^{1/5} N 1/5 τ \tau τ g i ≤ R ℓ g_i\le R\ell g i ≤ R ℓ R R R O R ( τ ) O_R(\tau) O R ( τ ) g i ≥ ℓ g_i\ge\ell g i ≥ ℓ ∣ z i + S i ∣ ≤ x / 2 \lvert z_i+S_i\rvert\le x/2 ∣ z i + S i ∣ ≤ x /2 C τ − 1 e − c g i / ℓ C\tau^{-1}e^{-cg_i/\ell} C τ − 1 e − c g i / ℓ Proposition 4.4 ), and these bounds are summable against (2.7), with a sum that tends to zero as R → ∞ R\to\infty R → ∞
To obtain a Poisson limit, we need the joint behavior of S i 1 , … , S i k S_{i_1},\ldots,S_{i_k} S i 1 , … , S i k i 1 < ⋯ < i k i_1<\cdots<i_k i 1 < ⋯ < i k i j + 1 ≥ i j + 2 i_{j+1}\ge i_j+2 i j + 1 ≥ i j + 2 j j j ( i j , i j + 1 ) (i_j,i_j+1) ( i j , i j + 1 ) § 3.3 that, with probability tending to one, the pairs with small gaps are disjoint. The vectors S i 1 , … , S i k S_{i_1},\ldots,S_{i_k} S i 1 , … , S i k Y 1 , … , Y m Y_1,\ldots,Y_m Y 1 , … , Y m 4 m I d 2 k 4mI_{d_{2k}} 4 m I d 2 k ( S i 1 , … , S i k ) (S_{i_1},\ldots,S_{i_k}) ( S i 1 , … , S i k ) 2 ∣ z i + S i ∣ 2\lvert z_i+S_i\rvert 2 ∣ z i + S i ∣ H ~ \widetilde H H H ~ \widetilde H H H H H H ~ \widetilde H H H H H
The choice of τ \tau τ τ \tau τ τ \tau τ τ = N 1 / 100 \tau=N^{1/100} τ = N 1/100
Comparing actual and linearized gaps
We now explain why the approximation g ~ i ≈ 2 d ∣ z i + S i ∣ \widetilde g_i\approx2d\lvert z_i+S_i\rvert g i ≈ 2 d ∣ z i + S i ∣ o ( d ) o(d) o ( d ) i i i Δ = H ~ − H \Delta=\widetilde H-H Δ = H − H U U U u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N W = U T Δ U W=U^{\mathsf T}\Delta U W = U T Δ U H H H Δ \Delta Δ 2 m 2m 2 m 2 N − 1 / 2 + ε 2N^{-1/2+\varepsilon} 2 N − 1/2 + ε
max i , j ∣ W i j ∣ ≤ ∑ a < b ∣ Δ a b ∣ ∣ u i ( a ) u j ( b ) + u i ( b ) u j ( a ) ∣ ≤ 4 m N − 3 / 2 + 3 ε ≤ w N , (18) \max_{i,j}|W_{ij}| \le\sum_{a<b}|\Delta_{ab}|\,|u_i(a)u_j(b)+u_i(b)u_j(a)| \le4mN^{-3/2+3\varepsilon} \le w_N, \tag*{(18)} i , j max ∣ W ij ∣ ≤ a < b ∑ ∣ Δ ab ∣ ∣ u i ( a ) u j ( b ) + u i ( b ) u j ( a ) ∣ ≤ 4 m N − 3/2 + 3 ε ≤ w N , (18) w N = τ N − 3 / 2 + 3 ε , w_N=\tau N^{-3/2+3\varepsilon}, w N = τ N − 3/2 + 3 ε , where we used that 4 m ≤ τ 4m\le\tau 4 m ≤ τ N − 1 N^{-1} N − 1 d d d
Let s = N − 7 / 5 s=N^{-7/5} s = N − 7/5 K ⊂ [ N ] \mathcal{K}\subset[N] K ⊂ [ N ] λ j + 1 − λ j ≤ s \lambda_{j+1}-\lambda_j\le s λ j + 1 − λ j ≤ s j , j + 1 ∈ K j,j+1\in\mathcal{K} j , j + 1 ∈ K [57 ] , the expected number of indices i ∈ [ N − 2 ] i\in[N-2] i ∈ [ N − 2 ] λ i + 2 − λ i ≤ r \lambda_{i+2}-\lambda_i\le r λ i + 2 − λ i ≤ r O ( N ( N r ) 3 ) O(N(Nr)^3) O ( N ( N r ) 3 ) r ≥ N − 2 r\ge N^{-2} r ≥ N − 2 r = 4 s r=4s r = 4 s λ i + 2 − λ i > 4 s \lambda_{i+2}-\lambda_i>4s λ i + 2 − λ i > 4 s i i i 1 − O ( N ( N s ) 3 ) = 1 − O ( N − 1 / 5 ) 1-O(N(Ns)^3)=1-O(N^{-1/5}) 1 − O ( N ( N s ) 3 ) = 1 − O ( N − 1/5 )
For each cluster K \mathcal{K} K H + Δ H+\Delta H + Δ ∣ K ∣ |\mathcal{K}| ∣ K ∣ λ j \lambda_j λ j j ∈ K j\in\mathcal{K} j ∈ K ( Λ + W ) K K (\Lambda+W)_{\mathcal{K}\mathcal{K}} ( Λ + W ) KK Λ = diag ( λ 1 , … , λ N ) \Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_N) Λ = diag ( λ 1 , … , λ N )
2 ∣ K ∣ ( max i , j ∣ W i j ∣ ) 2 sup w ∑ j ∉ K 1 ∣ λ j − w ∣ ≤ 4 w N 2 L N = δ N 2 , (19) 2|\mathcal{K}|\left(\max_{i,j}|W_{ij}|\right)^2\sup_w\sum_{j\notin\mathcal{K}}\frac{1}{|\lambda_j-w|}\le4w_N^2\mathcal{L}_N=\frac{\delta_N}{2}, \tag*{(19)} 2∣ K ∣ ( i , j max ∣ W ij ∣ ) 2 w sup j ∈ / K ∑ ∣ λ j − w ∣ 1 ≤ 4 w N 2 L N = 2 δ N , (19) L N = 2 s − 1 log N , δ N = 8 w N 2 L N . \mathcal{L}_N=2s^{-1}\log N,\qquad\delta_N=8w_N^2\mathcal{L}_N. L N = 2 s − 1 log N , δ N = 8 w N 2 L N . Here the supremum is over the complex numbers w w w [ λ min K , λ max K ] [\lambda_{\min\mathcal{K}},\lambda_{\max\mathcal{K}}] [ λ m i n K , λ m a x K ] s / 4 s/4 s /4 L N \mathcal{L}_N L N λ j \lambda_j λ j j ∉ K j\notin\mathcal{K} j ∈ / K 3 s / 4 3s/4 3 s /4 w w w 4 s 4s 4 s λ j + 2 − λ j > 4 s \lambda_{j+2}-\lambda_j>4s λ j + 2 − λ j > 4 s j j j τ \tau τ s s s d d d
δ N d = 16 N − 2 / 25 + 6 ε log N , (20) \frac{\delta_N}{d}=16N^{-2/25+6\varepsilon}\log N, \tag*{(20)} d δ N = 16 N − 2/25 + 6 ε log N , (20) which tends to zero when ε < 1 / 75 \varepsilon<1/75 ε < 1/75 ε = 1 / 100 \varepsilon=1/100 ε = 1/100
For a two-element cluster K = { i , i + 1 } \mathcal{K}=\{i,i+1\} K = { i , i + 1 } ( Λ + W ) K K (\Lambda+W)_{\mathcal{K}\mathcal{K}} ( Λ + W ) KK § 3.2 , and by (11) its eigenvalue gap equals 2 d ∣ z i + S i ∣ 2d|z_i+S_i| 2 d ∣ z i + S i ∣ g ~ i = 2 d ∣ z i + S i ∣ + o ( d ) \widetilde{g}_i=2d|z_i+S_i|+o(d) g i = 2 d ∣ z i + S i ∣ + o ( d ) H ~ \widetilde{H} H s / 2 s/2 s /2 Δ \Delta Δ § 3.5 .
A local limit theorem at the scale of a single step
The local limit theorem (14) is needed at unit scale, which is the size of a single increment ( Y − X a b ) D a b ( i ) (Y-X_{ab})D_{ab}^{(i)} ( Y − X ab ) D ab ( i ) X X X X X X ± 1 \pm1 ± 1 Y − X a b ∈ { − 2 , 0 , 2 } Y-X_{ab}\in\{-2,0,2\} Y − X ab ∈ { − 2 , 0 , 2 }
Φ H ( ξ ) = E [ exp ( i ξ ⋅ ( Y − X a b ) D a b ( i ) ) | H ] , ξ ∈ R 2 , (21) \Phi_H(\xi)=\mathbb{E}\left[\exp\left(i\xi\cdot(Y-X_{ab})D_{ab}^{(i)}\right)\middle|H\right],\qquad\xi\in\mathbb{R}^2, \tag*{(21)} Φ H ( ξ ) = E [ exp ( i ξ ⋅ ( Y − X ab ) D ab ( i ) ) H ] , ξ ∈ R 2 , (21) for ξ \xi ξ ( a , b ) (a,b) ( a , b ) Y Y Y X X X ψ \psi ψ (14) . The first step is a symmetrization, which removes the old entries X a b X_{ab} X ab H H H X X X Y ′ Y' Y ′ Y Y Y ( a , b ) (a,b) ( a , b ) z ↦ ∣ z ∣ 2 z\mapsto|z|^2 z ↦ ∣ z ∣ 2
1 − ∣ Φ H ( ξ ) ∣ 2 ≥ ( N 2 ) − 1 ∑ a < b E [ 1 − cos ( ( Y − Y ′ ) ξ ⋅ D a b ( i ) ) ] . (22) 1-\left|\Phi_H(\xi)\right|^2\geq\binom{N}{2}^{-1}\sum_{a<b}\mathbb{E}\left[1-\cos\left((Y-Y')\xi\cdot D_{ab}^{(i)}\right)\right]. \tag*{(22)} 1 − ∣ Φ H ( ξ ) ∣ 2 ≥ ( 2 N ) − 1 a < b ∑ E [ 1 − cos ( ( Y − Y ′ ) ξ ⋅ D ab ( i ) ) ] . (22) For b ∈ [ N ] b\in[N] b ∈ [ N ] r b = N 1 / 2 ( u i ( b ) , u i + 1 ( b ) ) ∈ R 2 r_b=N^{1/2}(u_i(b),u_{i+1}(b))\in\mathbb{R}^2 r b = N 1/2 ( u i ( b ) , u i + 1 ( b )) ∈ R 2 ξ = ( ξ ( 1 ) , ξ ( 2 ) ) ∈ R 2 \xi=(\xi^{(1)},\xi^{(2)})\in\mathbb{R}^2 ξ = ( ξ ( 1 ) , ξ ( 2 ) ) ∈ R 2 Γ ( ξ ) \Gamma(\xi) Γ ( ξ ) 2 × 2 2\times2 2 × 2 ( ξ ( 1 ) , ξ ( 2 ) ) (\xi^{(1)},\xi^{(2)}) ( ξ ( 1 ) , ξ ( 2 ) ) ( ξ ( 2 ) , − ξ ( 1 ) ) (\xi^{(2)},-\xi^{(1)}) ( ξ ( 2 ) , − ξ ( 1 ) ) (12) gives ξ ⋅ D a b ( i ) = r a T Γ ( ξ ) r b \xi\cdot D_{ab}^{(i)}=r_a^{T}\Gamma(\xi)r_b ξ ⋅ D ab ( i ) = r a T Γ ( ξ ) r b (22) is small. Since Var X = 1 \operatorname{Var}X=1 Var X = 1 ∣ Y − Y ′ ∣ |Y-Y'| ∣ Y − Y ′ ∣ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) Y − Y ′ Y-Y' Y − Y ′ ( Y − Y ′ ) r a T Γ ( ξ ) r b (Y-Y')r_a^{T}\Gamma(\xi)r_b ( Y − Y ′ ) r a T Γ ( ξ ) r b 2 π Z 2\pi\mathbb{Z} 2 π Z ( a , b ) (a,b) ( a , b ) a ′ , a ′ ′ a',a'' a ′ , a ′′ 2 × 2 2\times2 2 × 2 Q Q Q r a ′ T r_{a'}^{T} r a ′ T r a ′ ′ T r_{a''}^{T} r a ′′ T 2 π Z 2\pi\mathbb{Z} 2 π Z b b b n b ∈ Z 2 n_b\in\mathbb{Z}^2 n b ∈ Z 2
( Y − Y ′ ) Q Γ ( ξ ) r b = 2 π n b + (small error) , so that r b ≈ 2 π Y − Y ′ Γ ( ξ ) − 1 Q − 1 n b . (23) (Y-Y')Q\Gamma(\xi)r_b=2\pi n_b+\text{(small error)},\qquad\text{so that}\qquad r_b\approx\frac{2\pi}{Y-Y'}\Gamma(\xi)^{-1}Q^{-1}n_b. \tag*{(23)} ( Y − Y ′ ) Q Γ ( ξ ) r b = 2 π n b + (small error) , so that r b ≈ Y − Y ′ 2 π Γ ( ξ ) − 1 Q − 1 n b . (23) When N − c ≤ ∣ ξ ∣ ≤ N c N^{-c}\leq|\xi|\leq N^c N − c ≤ ∣ ξ ∣ ≤ N c c > 0 c>0 c > 0 n b n_b n b N C c N^{Cc} N C c N 1 / 2 u i N^{1/2}u_i N 1/2 u i v ↦ ( N − 1 ∑ b v ( b ) 2 ) 1 / 2 v\mapsto(N^{-1}\sum_bv(b)^2)^{1/2} v ↦ ( N − 1 ∑ b v ( b ) 2 ) 1/2 N C c N^{Cc} N C c k k k 2 k 2k 2 k a ′ , a ′ ′ a',a'' a ′ , a ′′ H H H Lemma 8.5 ). The proof replaces the eigenvector and its eigenvalue by nearby elements of a deterministic net of approximate eigenvectors and approximate eigenvalues, and it combines a union bound over this net with anticoncentration.
For a given vector v v v E E E [ N ] [N] [ N ] N / 2 N/2 N /2 ∣ v ∣ |v| ∣ v ∣ a a a N 1 / 2 ( ( H − E ) v ) ( a ) N^{1/2}((H-E)v)(a) N 1/2 (( H − E ) v ) ( a ) X a b v ( b ) X_{ab}v(b) X ab v ( b ) b b b X a ~ b X_{\tilde{a}b} X a ~ b a ~ \tilde{a} a ~ b b b a a a ( H − E ) v (H-E)v ( H − E ) v N / 4 N/4 N /4 N N N [57 , 60 , 74 ] .
Gap counts at the scales N − 1 − α N^{-1-\alpha} N − 1 − α
It remains to explain Theorem 1.3 , which supplies (17) . For α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) ℓ α = N − 1 − α \ell_\alpha= N^{-1-\alpha} ℓ α = N − 1 − α M α = N 1 − 2 α M_\alpha= N^{1-2\alpha} M α = N 1 − 2 α Theorem 1.3 states that for every y > 0 y > 0 y > 0 I I I g i ≤ y ℓ α g_i \le y\ell_\alpha g i ≤ y ℓ α E i ∈ I E_i \in I E i ∈ I M α ( y 2 / 2 ) ∫ I c ( E ) d E M_\alpha(y^2/2) \int_I c(E)\,\mathrm{d}E M α ( y 2 /2 ) ∫ I c ( E ) d E ℓ α \ell_\alpha ℓ α N − 3 / 2 = o ( ℓ α ) N^{-3/2} = o(\ell_\alpha) N − 3/2 = o ( ℓ α )
H t = e − t / 2 H + ( 1 − e − t ) 1 / 2 G , t ∈ [ 0 , 1 ] , (24) H_t = e^{-t/2}H + (1-e^{-t})^{1/2}G,\qquad t \in[0,1], \tag*{(24)} H t = e − t /2 H + ( 1 − e − t ) 1/2 G , t ∈ [ 0 , 1 ] , (24) where G G G H H H 1 / N 1/N 1/ N 2 / N 2/N 2/ N
We work with smoothed pair statistics. Let p ≥ 2 p \ge2 p ≥ 2 f p ( w ) = ( 1 + w 2 ) − p f_p(w) = (1+w^2)^{-p} f p ( w ) = ( 1 + w 2 ) − p w ∈ C ∖ { i , − i } w \in\mathbb{C} \setminus\{i,-i\} w ∈ C ∖ { i , − i } φ \varphi φ ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) φ N \varphi_N φ N φ \varphi φ N − 1 / 10 N^{-1/10} N − 1/10 φ N \varphi_N φ N φ N \varphi_N φ N ω p > 0 \omega_p > 0 ω p > 0 p p p N × N N \times N N × N A A A w 1 , … , w N {w_1,\ldots,w_N} w 1 , … , w N ∣ Im w j ∣ < ω p ℓ α |\operatorname{Im} w_j| < \omega_p\ell_\alpha ∣ Im w j ∣ < ω p ℓ α j j j θ > 0 \theta> 0 θ > 0
X α ( A ) = ∑ 1 ≤ i < j ≤ N φ N ( w i + w j 2 ) f p ( w j − w i ℓ α ) , F ( A ) = exp ( − θ X α ( A ) M α ) . (25) \mathcal{X}_\alpha(A) = \sum_{1 \le i < j \le N} \varphi_N\left(\frac{w_i+w_j}{2}\right) f_p\left(\frac{w_j-w_i}{\ell_\alpha}\right),\qquad F(A) = \exp\left(-\frac{\theta\mathcal{X}_\alpha(A)}{M_\alpha}\right). \tag*{(25)} X α ( A ) = 1 ≤ i < j ≤ N ∑ φ N ( 2 w i + w j ) f p ( ℓ α w j − w i ) , F ( A ) = exp ( − M α θ X α ( A ) ) . (25) The sum does not depend on the order of the eigenvalues, since f p f_p f p X α ( A ) \mathcal{X}_\alpha(A) X α ( A ) pair statistic of A A A § 6 , it is denoted X α ( A ; φ N , f p ) \mathcal{X}_\alpha(A;\varphi_N,f_p) X α ( A ; φ N , f p ) A A A ℓ α \ell_\alpha ℓ α
X α ( H ) M α ⟶ ∫ 0 ∞ y f p ( y ) d y ⋅ ∫ − 2 2 φ ( E ) c ( E ) d E (26) \frac{\mathcal{X}_\alpha(H)}{M_\alpha} \longrightarrow\int_0^\infty y f_p(y)\,dy \cdot\int_{-2}^{2} \varphi(E)c(E)\,dE \tag*{(26)} M α X α ( H ) ⟶ ∫ 0 ∞ y f p ( y ) d y ⋅ ∫ − 2 2 φ ( E ) c ( E ) d E (26) in probability, for all sufficiently large p p p φ \varphi φ
For p ≥ 2 p \ge2 p ≥ 2 ( i , j ) (i,j) ( i , j ) j ≥ i + 2 j \ge i+2 j ≥ i + 2 ℓ α \ell_\alpha ℓ α o ( M α ) o(M_\alpha) o ( M α ) X α ( H ) \mathcal{X}_\alpha(H) X α ( H ) f p f_p f p [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) φ \varphi φ (26) implies Theorem 1.3 at the scale ℓ α \ell_\alpha ℓ α § 8.4 .
Base case. For α < 1 / 8 \alpha< 1/8 α < 1/8 c ∗ ∈ ( 0 , 1 / 2 ) c_* \in(0,1/2) c ∗ ∈ ( 0 , 1/2 ) N − 1 + ε ≤ t < 1 N^{-1+\varepsilon} \le t < 1 N − 1 + ε ≤ t < 1 H t H_t H t G G G λ i + 1 ( H t ) − λ i ( H t ) \lambda_{i+1}(H_t)-\lambda_i(H_t) λ i + 1 ( H t ) − λ i ( H t ) λ i + 1 ( G ) − λ i ( G ) \lambda_{i+1}(G)-\lambda_i(G) λ i + 1 ( G ) − λ i ( G ) N ε / ( N 2 t ) N^\varepsilon/(N^2t) N ε / ( N 2 t ) i i i c ∗ N ≤ i ≤ ( 1 − c ∗ ) N c_*N \le i \le(1-c_*)N c ∗ N ≤ i ≤ ( 1 − c ∗ ) N Proposition 3.5 ). We take t = T t=T t = T T = N − 3 / 4 − α T=N^{-3/4-\alpha} T = N − 3/4 − α N − 5 / 4 + α + ε N^{-5/4+\alpha+\varepsilon} N − 5/4 + α + ε o ( ℓ α ) o(\ell_\alpha) o ( ℓ α ) α < 1 / 8 \alpha< 1/8 α < 1/8 H T H_T H T H H H § 12 from the Pfaffian formulas for the GOE correlation functions. It then remains to compare H T H_T H T H H H
Comparison. We compare the Laplace transforms E F ( H t ) \mathbb{E}F(H_t) E F ( H t ) E F ( H ) \mathbb{E}F(H) E F ( H ) F F F ( λ i ( A ) − λ j ( A ) ) − 1 (\lambda_i(A)-\lambda_j(A))^{-1} ( λ i ( A ) − λ j ( A ) ) − 1
We avoid this through holomorphy. For a < b a<b a < b V a b = e a e b T + e b e a T V^{ab}=e_a e_b^{T}+e_b e_a^{T} V ab = e a e b T + e b e a T A A A ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3 A A A N − 1 / 2 + ε N^{-1/2+\varepsilon} N − 1/2 + ε N − 1 N^{-1} N − 1 N ε N^\varepsilon N ε A A A [4 , 29 ] , these conditions hold for H t H_t H t Proposition 3.2 ). We show that there exists c = c p > 0 c=c_p>0 c = c p > 0 N N N ζ \zeta ζ ∣ ζ ∣ ≤ c N − α − 4 ε |\zeta|\le cN^{-\alpha-4\varepsilon} ∣ ζ ∣ ≤ c N − α − 4 ε w w w A + ζ V a b A+\zeta V^{ab} A + ζ V ab ∣ Im w ∣ < ω p ℓ α |\operatorname{Im}w|<\omega_p\ell_\alpha ∣ Im w ∣ < ω p ℓ α Lemma 5.2 ). Hence F ( A + ζ V a b ) F(A+\zeta V^{ab}) F ( A + ζ V ab ) X α \mathcal{X}_\alpha X α F ( A + ζ V a b ) F(A+\zeta V^{ab}) F ( A + ζ V ab ) ζ \zeta ζ Lemma 5.5 ). Since the imaginary parts of the eigenvalues are much smaller than N − 1 / 5 N^{-1/5} N − 1/5 N − 1 / 10 N^{-1/10} N − 1/10 φ N \varphi_N φ N ω p \omega_p ω p (25) has nonnegative real part (Lemmas 5.1 and 5.5 ). Therefore ∣ F ( A + ζ V a b ) ∣ ≤ 1 |F(A+\zeta V^{ab})|\le1 ∣ F ( A + ζ V ab ) ∣ ≤ 1 ζ \zeta ζ ∣ ζ ∣ ≤ c N − α − 4 ε / 2 |\zeta|\le cN^{-\alpha-4\varepsilon}/2 ∣ ζ ∣ ≤ c N − α − 4 ε /2 n ≥ 1 n\ge1 n ≥ 1
∣ d n d ζ n F ( A + ζ V a b ) ∣ ≤ n ! ( c N − α − 4 ε 2 ) − n . (27) \left|\frac{\mathrm{d}^n}{\mathrm{d}\zeta^n}F(A+\zeta V^{ab})\right|
\le n!\left(\frac{cN^{-\alpha-4\varepsilon}}{2}\right)^{-n}. \tag*{(27)} d ζ n d n F ( A + ζ V ab ) ≤ n ! ( 2 c N − α − 4 ε ) − n . (27) No information about gaps enters this bound. With (27) , a Taylor expansion in one entry at a time along the interpolation (24) , in the style of the continuity estimate of Bourgade and Yau [11 ] , gives for every D > 0 D > 0 D > 0
∣ E F ( H t ) − E F ( H ) ∣ ≤ C t N 1 / 2 + 3 α + 12 ε + C D N − D . (28) \left|\mathbb{E}F(H_t)-\mathbb{E}F(H)\right| \le CtN^{1/2+3\alpha+12\varepsilon}+C_DN^{-D}. \tag*{(28)} ∣ E F ( H t ) − E F ( H ) ∣ ≤ C t N 1/2 + 3 α + 12 ε + C D N − D . (28) The factor N 1 / 2 + 3 α + 12 ε N^{1/2+3\alpha+12\varepsilon} N 1/2 + 3 α + 12 ε N − 3 / 2 ⋅ N 3 ( α + 4 ε ) N^{-3/2}\cdot N^{3(\alpha+4\varepsilon)} N − 3/2 ⋅ N 3 ( α + 4 ε ) N 2 N^2 N 2 E X 3 = 0 \mathbb{E}X^3=0 E X 3 = 0 t = T t=T t = T (28) is C N − 1 / 4 + 2 α + 12 ε + C D N − D CN^{-1/4+2\alpha+12\varepsilon}+C_DN^{-D} C N − 1/4 + 2 α + 12 ε + C D N − D α < 1 / 8 \alpha<1/8 α < 1/8 θ > 0 \theta>0 θ > 0 (26) for H T H_T H T (26) for H H H α < 1 / 8 \alpha<1/8 α < 1/8
Induction on α \alpha α Let α \alpha α α ′ \alpha' α ′ 1 / 10 ≤ α < α ′ < 1 / 2 1/10\le\alpha<\alpha'<1/2 1/10 ≤ α < α ′ < 1/2 (26) holds. To prove (26) with α ′ \alpha' α ′ α \alpha α H t H_t H t ℓ α \ell_\alpha ℓ α ℓ α ′ \ell_{\alpha'} ℓ α ′ t α = N ℓ α 2 = N − 1 − 2 α t_\alpha=N\ell_\alpha^2=N^{-1-2\alpha} t α = N ℓ α 2 = N − 1 − 2 α U U U Λ \Lambda Λ § 3.3 , and let W ′ = ( 1 − e − t α ) 1 / 2 U T G U W'=(1-e^{-t_\alpha})^{1/2}U^{T}GU W ′ = ( 1 − e − t α ) 1/2 U T G U
U T H t α U = e − t α / 2 Λ + W ′ , (29) U^{T}H_{t_\alpha}U=e^{-t_\alpha/2}\Lambda+W', \tag*{(29)} U T H t α U = e − t α /2 Λ + W ′ , (29) so H t α H_{t_\alpha} H t α e − t α / 2 Λ + W ′ e^{-t_\alpha/2}\Lambda+W' e − t α /2 Λ + W ′ W ′ W' W ′ e − t α / 2 Λ e^{-t_\alpha/2}\Lambda e − t α /2 Λ W W W D y D_y D y G G G H H H W ′ W' W ′ ( 1 − e − t α ) 1 / 2 (1-e^{-t_\alpha})^{1/2} ( 1 − e − t α ) 1/2 ℓ α \ell_\alpha ℓ α
We group the eigenvalues of H H H § 3.3 , with the threshold q = N − 1 − κ q=N^{-1-\kappa} q = N − 1 − κ s s s κ = α − α ′ / 6 \kappa=\alpha-\alpha'/6 κ = α − α ′ /6 (19) with e − t α / 2 Λ e^{-t_\alpha/2}\Lambda e − t α /2 Λ W ′ W' W ′ q q q Λ \Lambda Λ W W W s s s W ′ W' W ′ N ε ℓ α N^\varepsilon\ell_\alpha N ε ℓ α K = { i , i + 1 } \mathcal{K}=\{i,i+1\} K = { i , i + 1 } H t α H_{t_\alpha} H t α ( e − t α / 2 Λ + W ′ ) K K (e^{-t_\alpha/2}\Lambda+W')_{\mathcal{K}\mathcal{K}} ( e − t α /2 Λ + W ′ ) KK N C ε ℓ α 2 / q N^{C\varepsilon}\ell_\alpha^2/q N C ε ℓ α 2 / q (11) , the gap of this block equals
∣ ( e − t α / 2 g i + W i + 1 , i + 1 ′ − W i i ′ , 2 W i , i + 1 ′ ) ∣ . (30) \left|(e^{-t_\alpha/2}g_i+W'_{i+1,i+1}-W'_{ii},\,2W'_{i,i+1})\right|. \tag*{(30)} ( e − t α /2 g i + W i + 1 , i + 1 ′ − W ii ′ , 2 W i , i + 1 ′ ) . (30) Here W i + 1 , i + 1 ′ − W i i ′ W'_{i+1,i+1}-W'_{ii} W i + 1 , i + 1 ′ − W ii ′ 2 W i , i + 1 ′ 2W'_{i,i+1} 2 W i , i + 1 ′ 4 ( 1 − e − t α ) / N = 4 ℓ α 2 ( 1 + o ( 1 ) ) 4(1-e^{-t_\alpha})/N=4\ell_\alpha^2(1+o(1)) 4 ( 1 − e − t α ) / N = 4 ℓ α 2 ( 1 + o ( 1 )) H H H ℓ α ′ \ell_{\alpha'} ℓ α ′ H t α H_{t_\alpha} H t α ℓ α \ell_\alpha ℓ α H H H ℓ α \ell_\alpha ℓ α (26) with α ′ \alpha' α ′ α \alpha α H t α H_{t_\alpha} H t α (28) , with α ′ \alpha' α ′ α \alpha α t = t α t=t_\alpha t = t α H H H t α N 1 / 2 + 3 α ′ = N − 1 / 2 − 2 α + 3 α ′ t_\alpha N^{1/2+3\alpha'}=N^{-1/2-2\alpha+3\alpha'} t α N 1/2 + 3 α ′ = N − 1/2 − 2 α + 3 α ′
Two further conditions are needed. We have
ℓ α 2 q = N − 1 − 2 α + κ , N ( N q ) 3 = N 1 − 3 κ . (31) \frac{\ell_\alpha^2}{q}=N^{-1-2\alpha+\kappa},\qquad N(Nq)^3=N^{1-3\kappa}. \tag*{(31)} q ℓ α 2 = N − 1 − 2 α + κ , N ( N q ) 3 = N 1 − 3 κ . (31) The error in the Schur complement estimate, which is of order ℓ α 2 / q \ell_\alpha^2/q ℓ α 2 / q N C ε N^{C\varepsilon} N C ε o ( ℓ α ′ ) o(\ell_{\alpha'}) o ( ℓ α ′ ) κ < 2 α − α ′ \kappa<2\alpha-\alpha' κ < 2 α − α ′ O ( N ( N q ) 3 ) O(N(Nq)^3) O ( N ( N q ) 3 ) o ( M α ′ ) o(M_{\alpha'}) o ( M α ′ ) 3 κ > 2 α ′ 3\kappa>2\alpha' 3 κ > 2 α ′ κ = α − α ′ / 6 \kappa=\alpha-\alpha'/6 κ = α − α ′ /6 α ′ < 6 α / 5 \alpha'<6\alpha/5 α ′ < 6 α /5
1 10 ≤ α < α ′ < min { 6 α 5 , 1 6 + 2 α 3 } . (32) \frac{1}{10}\leq\alpha<\alpha'<\min\left\{\frac{6\alpha}{5},\frac{1}{6}+\frac{2\alpha}{3}\right\}. \tag*{(32)} 10 1 ≤ α < α ′ < min { 5 6 α , 6 1 + 3 2 α } . (32) The bound p ≥ 61 p\geq61 p ≥ 61 (26) is proved, and the lower bound α ≥ 1 / 10 \alpha\geq1/10 α ≥ 1/10 (230) in the proof of Proposition 6.7 , where 2 p ( α ′ − κ ) > 1 + 2 α ′ 2p(\alpha'-\kappa)>1+2\alpha' 2 p ( α ′ − κ ) > 1 + 2 α ′ α ′ − κ > α / 6 ≥ 1 / 60 \alpha'-\kappa>\alpha/6\geq1/60 α ′ − κ > α /6 ≥ 1/60 α = 1 / 10 \alpha=1/10 α = 1/10 α < 1 / 2 \alpha<1/2 α < 1/2 Figure 2.2 .
Figure 2.2. The induction on α \alpha α ( α , α ′ ) (\alpha,\alpha') ( α , α ′ ) (32) . The staircase shows the first steps of the sequence α ( n + 1 ) = α ( n ) + min { α ( n ) , 1 / 2 − α ( n ) } / 10 \alpha^{(n+1)}=\alpha^{(n)}+\min\{\alpha^{(n)},1/2-\alpha^{(n)}\}/10 α ( n + 1 ) = α ( n ) + min { α ( n ) , 1/2 − α ( n ) } /10 α ( 0 ) = 1 / 10 \alpha^{(0)}=1/10 α ( 0 ) = 1/10 1 / 2 1/2 1/2 α < 1 / 8 \alpha<1/8 α < 1/8
Hermitian matrices
In this section only, H H H λ 1 ≤ ⋯ ≤ λ N \lambda_1\leq\cdots\leq\lambda_N λ 1 ≤ ⋯ ≤ λ N g i = λ i + 1 − λ i g_i=\lambda_{i+1}-\lambda_i g i = λ i + 1 − λ i G G G (24) is taken from the Gaussian unitary ensemble (GUE), normalized so that E ∣ G a b ∣ 2 = 1 / N \mathbb{E}|G_{ab}|^2=1/N E ∣ G ab ∣ 2 = 1/ N E G a b 2 = 0 \mathbb{E}G_{ab}^2=0 E G ab 2 = 0 a < b a<b a < b 1 / N 1/N 1/ N N − 4 / 3 N^{-4/3} N − 4/3 N − 3 / 2 N^{-3/2} N − 3/2 N − 4 / 3 N^{-4/3} N − 4/3 α = 1 / 3 \alpha= 1/3 α = 1/3 (28) at this scale has the right side
C t N 1 / 2 ( N ⋅ N − 4 / 3 ) − 3 N C ε = C t N 3 / 2 + C ε (33) CtN^{1/2}\left(N \cdot N^{-4/3}\right)^{-3}N^{C\varepsilon}=CtN^{3/2+C\varepsilon} \tag*{(33)} C t N 1/2 ( N ⋅ N − 4/3 ) − 3 N C ε = C t N 3/2 + C ε (33) when the third moments of the entries do not vanish. So the Gaussian component must be taken at a time t = N − 1 − 2 α 1 t=N^{-1-2\alpha_1} t = N − 1 − 2 α 1 α 1 > 1 / 4 \alpha_1>1/4 α 1 > 1/4 α 1 < 1 / 3 \alpha_1<1/3 α 1 < 1/3 H t H_t H t N − 1 − α 1 N^{-1-\alpha_1} N − 1 − α 1
An induction on α \alpha α § 3.5 , with Gaussian vectors in R 3 \mathbb{R}^3 R 3 R 2 \mathbb{R}^2 R 2 (30) , gives the Hermitian analogue of (26) . With c 2 ( E ) c_2(E) c 2 ( E ) Theorem 1.5 , it states that X α ( H ) / N 1 − 3 α \mathcal{X}_\alpha(H)/N^{1-3\alpha} X α ( H ) / N 1 − 3 α ∫ 0 ∞ y 2 f p ( y ) d y ⋅ ∫ − 2 2 φ ( E ) c 2 ( E ) d E \int_0^\infty y^2 f_p(y)\,\mathrm{d}y\cdot\int_{-2}^2\varphi(E)c_2(E)\,\mathrm{d}E ∫ 0 ∞ y 2 f p ( y ) d y ⋅ ∫ − 2 2 φ ( E ) c 2 ( E ) d E 1 / 8 1/8 1/8 α 1 = 8 / 25 \alpha_1=8/25 α 1 = 8/25 t = N − 1 − 2 α 1 t=N^{-1-2\alpha_1} t = N − 1 − 2 α 1 2 × 2 2\times2 2 × 2 H t H_t H t N − 4 / 3 N^{-4/3} N − 4/3
This is where nonzero third moments cause difficulty. We again group the eigenvalues into clusters, at a threshold q = N − 1 − κ q=N^{-1-\kappa} q = N − 1 − κ κ > 0 \kappa>0 κ > 0 i i i λ i + 2 − λ i ≤ q \lambda_{i+2}-\lambda_i\le q λ i + 2 − λ i ≤ q λ i \lambda_i λ i N ( N q ) 2 + 6 = N 1 − 8 κ N(Nq)^{2+6}=N^{1-8\kappa} N ( N q ) 2 + 6 = N 1 − 8 κ κ > 1 / 8 \kappa>1/8 κ > 1/8 N − 1 − κ N^{-1-\kappa} N − 1 − κ H H H κ < 1 / 8 \kappa<1/8 κ < 1/8
Instead of excluding triples, we show that they rarely produce small gaps. For κ < 1 / 8 \kappa<1/8 κ < 1/8 i i i λ i + 2 − λ i ≤ N − 1 − κ \lambda_{i+2}-\lambda_i\le N^{-1-\kappa} λ i + 2 − λ i ≤ N − 1 − κ λ i \lambda_i λ i N 1 − 8 κ + C ε N^{1-8\kappa+C\varepsilon} N 1 − 8 κ + C ε t = N − 1 − 2 α 1 t=N^{-1-2\alpha_1} t = N − 1 − 2 α 1 N − 4 / 3 N^{-4/3} N − 4/3 N C ε ( N − 4 / 3 / N − 1 − α 1 ) 3 = N − 1 + 3 α 1 + C ε N^{C\varepsilon}(N^{-4/3}/N^{-1-\alpha_1})^3=N^{-1+3\alpha_1+C\varepsilon} N C ε ( N − 4/3 / N − 1 − α 1 ) 3 = N − 1 + 3 α 1 + C ε
N 1 − 8 κ + C ε ⋅ N − 1 + 3 α 1 + C ε = N 3 α 1 − 8 κ + 2 C ε , (34) N^{1-8\kappa+C\varepsilon}\cdot N^{-1+3\alpha_1+C\varepsilon}=N^{3\alpha_1-8\kappa+2C\varepsilon}, \tag*{(34)} N 1 − 8 κ + C ε ⋅ N − 1 + 3 α 1 + C ε = N 3 α 1 − 8 κ + 2 C ε , (34) which tends to zero when κ > 3 α 1 / 8 \kappa>3\alpha_1/8 κ > 3 α 1 /8 κ < 1 / 8 \kappa<1/8 κ < 1/8 α 1 < 1 / 3 \alpha_1<1/3 α 1 < 1/3 N − 1 − κ N^{-1-\kappa} N − 1 − κ κ > 1 / 15 \kappa>1/15 κ > 1/15 α 1 = 8 / 25 \alpha_1=8/25 α 1 = 8/25
Organization of the paper
Section 3 collects estimates on eigenvalues and eigenvectors from the literature, a perturbation lemma for matrices that are entrywise small perturbations of diagonal matrices, and an estimate for Gaussian kernels. In Section 4, we state three propositions (the gap counts at the scale ℓ \ell ℓ α \alpha α
Preliminaries
In this section, we collect the facts about eigenvalues and eigenvectors that are used in the rest of the paper. § 4.1 contains regularity and rigidity estimates for H H H H t H_t H t G G G H H H H t H_t H t § 4.2 contains a deterministic lemma on the eigenvalues of a diagonal matrix perturbed by a Hermitian matrix with small entries, which we apply in Section 6, Section 7, and Section 9. § 4.3 contains an elementary Gaussian estimate used in Section 6 and Section 9.
We first fix some notation for the rest of the paper. We write σ Z 2 = E Z 2 \sigma_Z^2 = \mathbb{E}Z^2 σ Z 2 = E Z 2 Definition 1.1 , and we set X a a = Z a X_{aa} = Z_a X aa = Z a a ∈ [ N ] a \in[N] a ∈ [ N ] X a b = N 1 / 2 H a b X_{ab} = N^{1/2}H_{ab} X ab = N 1/2 H ab a ≤ b a \le b a ≤ b N × N N \times N N × N
V a b = e a e b T + e b e a T ( 1 ≤ a < b ≤ N ) , V a a = e a e a T ( a ∈ [ N ] ) , (35) V^{ab} = e_a e_b^T + e_b e_a^T \quad(1 \le a < b \le N), \qquad V^{aa} = e_a e_a^T \quad(a \in[N]), \tag*{(35)} V ab = e a e b T + e b e a T ( 1 ≤ a < b ≤ N ) , V aa = e a e a T ( a ∈ [ N ]) , (35) so that every real symmetric N × N N \times N N × N A A A A = ∑ a ≤ b A a b V a b A = \sum_{a \le b} A_{ab} V^{ab} A = ∑ a ≤ b A ab V ab G G G H H H N − 1 N^{-1} N − 1 2 N − 1 2N^{-1} 2 N − 1
H t = e − t / 2 H + ( 1 − e − t ) 1 / 2 G , t ∈ [ 0 , 1 ] . (36) H_t = e^{-t/2}H + (1-e^{-t})^{1/2}G, \qquad t \in[0,1]. \tag*{(36)} H t = e − t /2 H + ( 1 − e − t ) 1/2 G , t ∈ [ 0 , 1 ] . (36) For i ∈ [ N ] i \in[N] i ∈ [ N ] γ i \gamma_i γ i ∫ − 2 γ i ρ s c ( E ) d E = i / N \int_{-2}^{\gamma_i} \rho_{\mathrm{sc}}(E)\,\mathrm{d}E = i/N ∫ − 2 γ i ρ sc ( E ) d E = i / N uniformly in the parameter if the constant C D C_D C D
Estimates for eigenvalues and eigenvectors
We begin with the set of matrices on which our deterministic arguments take place.
Definition 3.1. Let ε > 0 \varepsilon> 0 ε > 0 G ε \mathcal{G}_\varepsilon G ε N × N N \times N N × N A A A
(i) ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3
(ii) Every unit eigenvector v ∈ R N v \in\mathbb{R}^N v ∈ R N A A A ∥ v ∥ ∞ ≤ N − 1 / 2 + ε \lVert v\rVert_\infty\le N^{-1/2+\varepsilon} ∥ v ∥ ∞ ≤ N − 1/2 + ε
(iii) For all E ∈ R E \in\mathbb{R} E ∈ R # { i ∈ [ N ] : λ i ( A ) ∈ [ E , E + N − 1 ] } ≤ N ε \#\{i \in[N] : \lambda_i(A) \in[E,E+N^{-1}]\} \le N^\varepsilon # { i ∈ [ N ] : λ i ( A ) ∈ [ E , E + N − 1 ]} ≤ N ε
We call the elements of G ε \mathcal{G}_\varepsilon G ε regular matrices .
Each of the three conditions becomes weaker as ε \varepsilon ε G ε ⊂ G ε ′ \mathcal{G}_\varepsilon\subset\mathcal{G}_{\varepsilon'} G ε ⊂ G ε ′ ε < ε ′ \varepsilon< \varepsilon' ε < ε ′ [43 ] , one checks that G ε \mathcal{G}_\varepsilon G ε
Proposition 3.2. The following statements hold.
(a) For all ε > 0 \varepsilon> 0 ε > 0 D > 0 D > 0 D > 0 C ε , D > 0 C_{\varepsilon,D} > 0 C ε , D > 0 P ( H t ∉ G ε ) ≤ C ε , D N − D \mathbb{P}(H_t \notin\mathcal{G}_\varepsilon) \le C_{\varepsilon,D}N^{-D} P ( H t ∈ / G ε ) ≤ C ε , D N − D N ≥ 2 N \ge2 N ≥ 2 t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ]
(b) For all c ∗ ∈ ( 0 , 1 / 2 ) c_* \in(0,1/2) c ∗ ∈ ( 0 , 1/2 ) ε > 0 \varepsilon> 0 ε > 0 D > 0 D > 0 D > 0 C > 0 C > 0 C > 0 N ≥ 2 N \ge2 N ≥ 2 t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ]
P ( max c ∗ N ≤ i ≤ ( 1 − c ∗ ) N ∣ λ i ( H t ) − γ i ∣ > N − 1 + ε ) ≤ C N − D . (37) \mathbb{P}\left(\max_{c_*N\le i\le(1-c_*)N}|\lambda_i(H_t)-\gamma_i|>N^{-1+\varepsilon}\right)\le CN^{-D}. \tag*{(37)} P ( c ∗ N ≤ i ≤ ( 1 − c ∗ ) N max ∣ λ i ( H t ) − γ i ∣ > N − 1 + ε ) ≤ C N − D . (37) (c) For every integer n ≥ 1 n \ge1 n ≥ 1 C n > 0 C_n > 0 C n > 0 E ∥ H t ∥ n ≤ C n \mathbb{E}\|H_t\|^n \le C_n E ∥ H t ∥ n ≤ C n N ≥ 2 N \ge2 N ≥ 2 t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ]
The same statements hold with H t H_t H t G G G
The next lemma is an edge-inclusive rigidity estimate with subpolynomial precision, due to Erdős, Yau, and Yin [29 ] . It is used only for the coupling, in the proof of Proposition 3.5 and of its Hermitian analogue in Section 9.
Lemma 3.3. There exists A 0 > 0 A_0 > 0 A 0 > 0 t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ]
∣ λ i ( H t ) − γ i ∣ ≤ e A 0 ( log log N ) 2 N − 2 / 3 min ( i , N + 1 − i ) − 1 / 3 for all i ∈ [ N ] . (38) |\lambda_i(H_t)-\gamma_i|\le e^{A_0(\log\log N)^2}N^{-2/3}\min(i,N+1-i)^{-1/3}\quad\text{for all }i\in[N]. \tag*{(38)} ∣ λ i ( H t ) − γ i ∣ ≤ e A 0 ( l o g l o g N ) 2 N − 2/3 min ( i , N + 1 − i ) − 1/3 for all i ∈ [ N ] . (38) The same statement holds with H t H_t H t G G G
Proof of Proposition 3.2. We deduce the proposition from the results of Benaych-Georges and Knowles [4 ] . Their Definition 2.2 concerns a sequence A = A ( N ) A=A^{(N)} A = A ( N ) N × N N\times N N × N
(1) The entries A a b A_{ab} A ab a ≤ b a\le b a ≤ b
(2) We have E A a b = 0 \mathbb{E}A_{ab}=0 E A ab = 0 a , b a,b a , b E ∣ A a b ∣ 2 = N − 1 \mathbb{E}|A_{ab}|^2=N^{-1} E ∣ A ab ∣ 2 = N − 1 a ≠ b a\ne b a = b E ∣ A a a ∣ 2 ≤ C N − 1 \mathbb{E}|A_{aa}|^2\le CN^{-1} E ∣ A aa ∣ 2 ≤ C N − 1 a a a
(3) For every integer n ≥ 1 n\ge1 n ≥ 1 C n > 0 C_n>0 C n > 0 E ∣ N 1 / 2 A a b ∣ n ≤ C n \mathbb{E}|N^{1/2}A_{ab}|^n\le C_n E ∣ N 1/2 A ab ∣ n ≤ C n N N N a a a b b b
We first check that (1)–(3) hold for ( H t N ) N ≥ 2 (H_{t_N})_{N\ge2} ( H t N ) N ≥ 2 ( t N ) (t_N) ( t N ) [ 0 , 1 ] [0,1] [ 0 , 1 ] H t H_t H t H H H G G G a ≤ b a\le b a ≤ b
N 1 / 2 ( H t ) a b = e − t / 2 X a b + ( 1 − e − t ) 1 / 2 N 1 / 2 G a b , (39) N^{1/2}(H_t)_{ab}=e^{-t/2}X_{ab}+(1-e^{-t})^{1/2}N^{1/2}G_{ab}, \tag*{(39)} N 1/2 ( H t ) ab = e − t /2 X ab + ( 1 − e − t ) 1/2 N 1/2 G ab , (39) and therefore
E ( H t ) a b 2 = 1 N ( a ≠ b ) , E ( H t ) a a 2 = e − t σ Z 2 + 2 ( 1 − e − t ) N ≤ max ( σ Z 2 , 2 ) N . \mathbb{E}(H_t)_{ab}^{2}=\frac{1}{N}\quad(a\ne b),\qquad\mathbb{E}(H_t)_{aa}^{2}=\frac{e^{-t}\sigma_Z^2+2(1-e^{-t})}{N}\le\frac{\max(\sigma_Z^2,2)}{N}. E ( H t ) ab 2 = N 1 ( a = b ) , E ( H t ) aa 2 = N e − t σ Z 2 + 2 ( 1 − e − t ) ≤ N max ( σ Z 2 , 2 ) . Using (3.5), Minkowski’s inequality, and the sub-Gaussian tails of X X X Z Z Z N 1 / 2 G a b N^{1/2}G_{ab} N 1/2 G ab N 1 / 2 ( H t ) a b N^{1/2}(H_t)_{ab} N 1/2 ( H t ) ab t t t N N N a a a b b b G G G [4 Example 2.3] .
Next, let A A A ε ∈ ( 0 , 1 / 4 ] \varepsilon\in(0,1/4] ε ∈ ( 0 , 1/4 ] c ∗ ∈ ( 0 , 1 / 2 ) c_*\in(0,1/2) c ∗ ∈ ( 0 , 1/2 ) D > 0 D>0 D > 0 N 0 N_0 N 0 N ≥ N 0 N\ge N_0 N ≥ N 0
P ( A ∉ G ε ) + P ( max c ∗ N ≤ i ≤ ( 1 − c ∗ ) N ∣ λ i ( A ) − γ i ∣ > N − 1 + ε ) ≤ N − D . (40) \mathbb{P}(A\notin\mathcal{G}_\varepsilon)+\mathbb{P}\left(\max_{c_*N\le i\le(1-c_*)N}\left|\lambda_i(A)-\gamma_i\right|>N^{-1+\varepsilon}\right)\le N^{-D}. \tag*{(40)} P ( A ∈ / G ε ) + P ( c ∗ N ≤ i ≤ ( 1 − c ∗ ) N max ∣ λ i ( A ) − γ i ∣ > N − 1 + ε ) ≤ N − D . (40) We first show that the claim implies (a) and (b). The events in (a) and (b) decrease as ε \varepsilon ε ε ≤ 1 / 4 \varepsilon\le1/4 ε ≤ 1/4 ε \varepsilon ε c ∗ c_* c ∗ D D D N N N t N ∈ [ 0 , 1 ] t_N\in[0,1] t N ∈ [ 0 , 1 ] (40) with A = H t N A=H_{t_N} A = H t N N − D N^{-D} N − D N N N t t t t N = 0 t_N=0 t N = 0 N N N ( H t N ) (H_{t_N}) ( H t N ) A = G A=G A = G G G G
We now prove the claim, and we begin with the norm and the diagonal resolvent entries. By [4 Theorem G.1] , we have ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3 w = E + i η w=E+\mathrm{i}\eta w = E + i η η > 0 \eta>0 η > 0
m s c ( w ) = ∫ − 2 2 ρ s c ( E ′ ) E ′ − w d E ′ , r ( w ) = ( Im m s c ( w ) N η ) 1 / 2 + 1 N η m_{\mathrm{sc}}(w)=\int_{-2}^{2}\frac{\rho_{\mathrm{sc}}(E')}{E'-w}\,\mathrm{d}E',\qquad r(w)=\left(\frac{\operatorname{Im}m_{\mathrm{sc}}(w)}{N\eta}\right)^{1/2}+\frac{1}{N\eta} m sc ( w ) = ∫ − 2 2 E ′ − w ρ sc ( E ′ ) d E ′ , r ( w ) = ( N η Im m sc ( w ) ) 1/2 + N η 1 denote the Stieltjes transform of the semicircle law and the error parameter of [4 (2.16)] , and recall that ∣ m s c ( w ) ∣ ≤ 1 \lvert m_{\mathrm{sc}}(w)\rvert\le1 ∣ m sc ( w )∣ ≤ 1
W ε = { E + i η : ∣ E ∣ ≤ 4 ε , N − 1 + ε / 4 ≤ η ≤ 4 ε } . \mathcal{W}_\varepsilon=\left\{E+\mathrm{i}\eta:\lvert E\rvert\le\frac{4}{\varepsilon},\ N^{-1+\varepsilon/4}\le\eta\le\frac{4}{\varepsilon}\right\}. W ε = { E + i η : ∣ E ∣ ≤ ε 4 , N − 1 + ε /4 ≤ η ≤ ε 4 } . The local semicircle law [4 Theorem 2.6] , in the form [4 (2.17)] that holds simultaneously for all spectral parameters, applied with the spectral domain W ε \mathcal{W}_\varepsilon W ε
∣ ( ( A − w ) − 1 ) a a − m s c ( w ) ∣ ≤ N ε / 4 r ( w ) for all a ∈ [ N ] and w ∈ W ε . \left|\left((A-w)^{-1}\right)_{aa}-m_{\mathrm{sc}}(w)\right|\le N^{\varepsilon/4}r(w)\qquad\text{for all }a\in[N]\text{ and }w\in\mathcal{W}_\varepsilon. ( ( A − w ) − 1 ) aa − m sc ( w ) ≤ N ε /4 r ( w ) for all a ∈ [ N ] and w ∈ W ε . We have r ≤ 2 r\le2 r ≤ 2 W ε \mathcal{W}_\varepsilon W ε Im m s c ≤ 1 \operatorname{Im}m_{\mathrm{sc}}\le1 Im m sc ≤ 1 N η ≥ 1 N\eta\ge1 N η ≥ 1
max a ∈ [ N ] sup w ∈ W ε ∣ ( ( A − w ) − 1 ) a a ∣ ≤ 3 N ε / 4 . (41) \max_{a\in[N]}\sup_{w\in\mathcal{W}_\varepsilon}\left|\left((A-w)^{-1}\right)_{aa}\right|\le3N^{\varepsilon/4}. \tag*{(41)} a ∈ [ N ] max w ∈ W ε sup ( ( A − w ) − 1 ) aa ≤ 3 N ε /4 . (41) Let U \mathcal{U} U ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3 Definition 3.1 hold on U \mathcal{U} U N ≥ N 0 ( ε ) N\ge N_0(\varepsilon) N ≥ N 0 ( ε ) [4 Theorem 2.10] ). We write them out because condition (ii) concerns every unit eigenvector, including the eigenvectors of multiple eigenvalues.
We first derive (ii). For an eigenvalue λ \lambda λ A A A Π λ \Pi_\lambda Π λ
( A − w ) − 1 = ∑ λ ′ Π λ ′ λ ′ − w , w ∈ C ∖ R , (A-w)^{-1}=\sum_{\lambda'}\frac{\Pi_{\lambda'}}{\lambda'-w},\qquad w\in\mathbb{C}\setminus\mathbb{R}, ( A − w ) − 1 = λ ′ ∑ λ ′ − w Π λ ′ , w ∈ C ∖ R , where the sum runs over the distinct eigenvalues of A A A v v v A A A λ \lambda λ a ∈ [ N ] a\in[N] a ∈ [ N ] η > 0 \eta>0 η > 0
∣ v ( a ) ∣ 2 = ∣ ( Π λ e a ) ⋅ v ∣ 2 ≤ ( Π λ ) a a ≤ η ∑ λ ′ η ( Π λ ′ ) a a ( λ ′ − λ ) 2 + η 2 = η Im ( ( A − λ − i η ) − 1 ) a a . (42) |v(a)|^2=|(\Pi_\lambda e_a)\cdot v|^2\leq(\Pi_\lambda)_{aa}\leq\eta\sum_{\lambda'}\frac{\eta(\Pi_{\lambda'})_{aa}}{(\lambda'-\lambda)^2+\eta^2}=\eta\operatorname{Im}\left((A-\lambda-i\eta)^{-1}\right)_{aa}. \tag*{(42)} ∣ v ( a ) ∣ 2 = ∣ ( Π λ e a ) ⋅ v ∣ 2 ≤ ( Π λ ) aa ≤ η λ ′ ∑ ( λ ′ − λ ) 2 + η 2 η ( Π λ ′ ) aa = η Im ( ( A − λ − i η ) − 1 ) aa . (42) Here the first equality follows from v = Π λ v v=\Pi_\lambda v v = Π λ v ∥ Π λ e a ∥ 2 = ( Π λ ) a a \|\Pi_\lambda e_a\|^2=(\Pi_\lambda)_{aa} ∥ Π λ e a ∥ 2 = ( Π λ ) aa ( Π λ ′ ) a a ≥ 0 (\Pi_{\lambda'})_{aa}\geq0 ( Π λ ′ ) aa ≥ 0 λ ′ \lambda' λ ′
We now use (42) on U \mathcal{U} U ∣ λ ∣ ≤ 3 ≤ 4 / ε |\lambda|\leq3\leq4/\varepsilon ∣ λ ∣ ≤ 3 ≤ 4/ ε λ + i N − 1 + ε / 4 ∈ W ε \lambda+iN^{-1+\varepsilon/4}\in\mathcal{W}_\varepsilon λ + i N − 1 + ε /4 ∈ W ε (42) with η = N − 1 + ε / 4 \eta=N^{-1+\varepsilon/4} η = N − 1 + ε /4
∣ v ( a ) ∣ 2 ≤ 3 N − 1 + ε / 2 ≤ N − 1 + 2 ε |v(a)|^2\leq3N^{-1+\varepsilon/2}\leq N^{-1+2\varepsilon} ∣ v ( a ) ∣ 2 ≤ 3 N − 1 + ε /2 ≤ N − 1 + 2 ε for N ≥ N 0 ( ε ) N\geq N_0(\varepsilon) N ≥ N 0 ( ε )
Next, we derive (iii). Fix E ∈ R E\in\mathbb{R} E ∈ R U \mathcal{U} U [ E , E + N − 1 ] [E,E+N^{-1}] [ E , E + N − 1 ] [ − 3 , 3 ] [-3,3] [ − 3 , 3 ] E ′ = E + ( 2 N ) − 1 E'=E+(2N)^{-1} E ′ = E + ( 2 N ) − 1 ∣ E ′ ∣ ≤ 4 |E'|\leq4 ∣ E ′ ∣ ≤ 4 η = N − 1 + ε / 4 \eta=N^{-1+\varepsilon/4} η = N − 1 + ε /4 E ′ + i η ∈ W ε E'+i\eta\in\mathcal{W}_\varepsilon E ′ + i η ∈ W ε i i i λ i ( A ) ∈ [ E , E + N − 1 ] \lambda_i(A)\in[E,E+N^{-1}] λ i ( A ) ∈ [ E , E + N − 1 ]
∣ λ i ( A ) − E ′ ∣ ≤ ( 2 N ) − 1 ≤ η , and hence η ( λ i ( A ) − E ′ ) 2 + η 2 ≥ 1 2 η . |\lambda_i(A)-E'|\leq(2N)^{-1}\leq\eta,\qquad\text{and hence}\qquad\frac{\eta}{(\lambda_i(A)-E')^2+\eta^2}\geq\frac{1}{2\eta}. ∣ λ i ( A ) − E ′ ∣ ≤ ( 2 N ) − 1 ≤ η , and hence ( λ i ( A ) − E ′ ) 2 + η 2 η ≥ 2 η 1 . Summing over these i i i
# { i : λ i ( A ) ∈ [ E , E + N − 1 ] } ≤ 2 η ∑ i = 1 N η ( λ i ( A ) − E ′ ) 2 + η 2 = 2 η ∑ a = 1 N Im ( ( A − E ′ − i η ) − 1 ) a a ≤ 6 N ε / 2 , \begin{aligned}
\#\{i:\lambda_i(A)\in[E,E+N^{-1}]\}\leq2\eta\sum_{i=1}^{N}\frac{\eta}{(\lambda_i(A)-E')^2+\eta^2} \\
&=2\eta\sum_{a=1}^{N}\operatorname{Im}\left((A-E'-i\eta)^{-1}\right)_{aa}\leq6N^{\varepsilon/2},
\end{aligned} # { i : λ i ( A ) ∈ [ E , E + N − 1 ]} ≤ 2 η i = 1 ∑ N ( λ i ( A ) − E ′ ) 2 + η 2 η = 2 η a = 1 ∑ N Im ( ( A − E ′ − i η ) − 1 ) aa ≤ 6 N ε /2 , where we used (3.7) in the last step. The right side is bounded by N ε N^\varepsilon N ε N ≥ N 0 ( ε ) N\geq N_0(\varepsilon) N ≥ N 0 ( ε )
As ∥ A ∥ ≤ 3 \|A\|\leq3 ∥ A ∥ ≤ 3 U \mathcal{U} U P ( U c ) ≤ N − D / 2 \mathbb{P}(\mathcal{U}^{c})\leq N^{-D/2} P ( U c ) ≤ N − D /2 N ≥ N 0 N\geq N_0 N ≥ N 0
We now prove the rigidity estimate in (40) . The eigenvalues in [4 ] are labeled in decreasing order, and the typical location γ i B K \gamma_i^{\mathrm{BK}} γ i BK i i i
N ∫ γ i B K 2 ρ s c ( E ) d E = i − 1 2 , N ∫ − 2 γ i ′ ρ s c ( E ) d E = i − 1 2 . N\int_{\gamma_i^{\mathrm{BK}}}^{2}\rho_{\mathrm{sc}}(E)\,\mathrm{d}E=i-\frac{1}{2},\qquad N\int_{-2}^{\gamma_i'}\rho_{\mathrm{sc}}(E)\,\mathrm{d}E=i-\frac{1}{2}. N ∫ γ i BK 2 ρ sc ( E ) d E = i − 2 1 , N ∫ − 2 γ i ′ ρ sc ( E ) d E = i − 2 1 . The second relation holds for γ i ′ = γ N + 1 − i B K \gamma_i'=\gamma_{N+1-i}^{\mathrm{BK}} γ i ′ = γ N + 1 − i BK [4 Theorem 2.9] and a union bound over i i i
∣ λ i ( A ) − γ i ′ ∣ ≤ N ε / 2 N − 2 / 3 min ( i , N + 1 − i ) − 1 / 3 for all i ∈ [ N ] . \lvert\lambda_i(A)-\gamma_i'\rvert\leq N^{\varepsilon/2}N^{-2/3}\min(i,N+1-i)^{-1/3}\qquad\text{for all }i\in[N]. ∣ λ i ( A ) − γ i ′ ∣ ≤ N ε /2 N − 2/3 min ( i , N + 1 − i ) − 1/3 for all i ∈ [ N ] . Let c ∗ N ≤ i ≤ ( 1 − c ∗ ) N c_*N\leq i\leq(1-c_*)N c ∗ N ≤ i ≤ ( 1 − c ∗ ) N N ≥ 2 / c ∗ N\geq2/c_* N ≥ 2/ c ∗ i / N i/N i / N ( i − 1 / 2 ) / N (i-1/2)/N ( i − 1/2 ) / N [ c ∗ / 2 , 1 − c ∗ / 2 ] [c_*/2,1-c_*/2] [ c ∗ /2 , 1 − c ∗ /2 ] ρ s c \rho_{\mathrm{sc}} ρ sc ∣ γ i − γ i ′ ∣ ≤ C N − 1 \lvert\gamma_i-\gamma_i'\rvert\leq CN^{-1} ∣ γ i − γ i ′ ∣ ≤ C N − 1 min ( i , N + 1 − i ) ≥ c ∗ N \min(i,N+1-i)\geq c_*N min ( i , N + 1 − i ) ≥ c ∗ N
∣ λ i ( A ) − γ i ∣ ≤ ∣ λ i ( A ) − γ i ′ ∣ + ∣ γ i ′ − γ i ∣ ≤ N ε / 2 c ∗ − 1 / 3 N − 1 + C N − 1 ≤ N − 1 + ε \lvert\lambda_i(A)-\gamma_i\rvert\leq\lvert\lambda_i(A)-\gamma_i'\rvert+\lvert\gamma_i'-\gamma_i\rvert\leq N^{\varepsilon/2}c_*^{-1/3}N^{-1}+CN^{-1}\leq N^{-1+\varepsilon} ∣ λ i ( A ) − γ i ∣ ≤ ∣ λ i ( A ) − γ i ′ ∣ + ∣ γ i ′ − γ i ∣ ≤ N ε /2 c ∗ − 1/3 N − 1 + C N − 1 ≤ N − 1 + ε for all such i i i N ≥ N 0 N\geq N_0 N ≥ N 0
Finally, we prove (c). By [4 Theorem H.1] , if an N × N N\times N N × N A A A E exp ( c ′ N ∣ A a b ∣ 2 ) ≤ 1 / c ′ \mathbb{E}\exp(c'N\lvert A_{ab}\rvert^2)\leq1/c' E exp ( c ′ N ∣ A ab ∣ 2 ) ≤ 1/ c ′ a , b a,b a , b c ′ > 0 c'>0 c ′ > 0
P ( ∥ A ∥ ≥ x ) ≤ exp ( − ( c ( x 2 − C ) N ) ) for all x > 0 , \mathbb{P}(\lVert A\rVert\geq x)\leq\exp\bigl(-(c(x^2-C)N)\bigr)\qquad\text{for all }x>0, P (∥ A ∥ ≥ x ) ≤ exp ( − ( c ( x 2 − C ) N ) ) for all x > 0 , where c , C > 0 c,C>0 c , C > 0 c ′ c' c ′ (39) , we have
N ∣ ( H t ) a b ∣ 2 ≤ 2 X a b 2 + 2 N G a b 2 , N\lvert(H_t)_{ab}\rvert^2\leq2X_{ab}^2+2NG_{ab}^2, N ∣( H t ) ab ∣ 2 ≤ 2 X ab 2 + 2 N G ab 2 , so the independence of H H H G G G X X X Z Z Z N 1 / 2 G a b N^{1/2}G_{ab} N 1/2 G ab H t H_t H t c ′ > 0 c'>0 c ′ > 0 t t t G G G x 0 = ( 2 C ) 1 / 2 x_0=(2C)^{1/2} x 0 = ( 2 C ) 1/2
E ∥ H t ∥ n = ∫ 0 ∞ n x n − 1 P ( ∥ H t ∥ ≥ x ) d x ≤ x 0 n + ∫ x 0 ∞ n x n − 1 e − c x 2 N / 2 d x ≤ C n , \mathbb{E}\lVert H_t\rVert^n=\int_0^\infty nx^{n-1}\mathbb{P}(\lVert H_t\rVert\geq x)\,\mathrm{d}x\leq x_0^n+\int_{x_0}^\infty nx^{n-1}e^{-cx^2N/2}\,\mathrm{d}x\leq Cn, E ∥ H t ∥ n = ∫ 0 ∞ n x n − 1 P (∥ H t ∥ ≥ x ) d x ≤ x 0 n + ∫ x 0 ∞ n x n − 1 e − c x 2 N /2 d x ≤ C n , where we used that x 2 − C ≥ x 2 / 2 x^2-C\geq x^2/2 x 2 − C ≥ x 2 /2 x ≥ x 0 x\geq x_0 x ≥ x 0 G G G
Proof of Lemma 3.3. We use [29 Theorem 2.2] . It concerns an N × N N\times N N × N A A A σ a b 2 = E A a b 2 \sigma_{ab}^2=\mathbb{E}A_{ab}^2 σ ab 2 = E A ab 2
The variances in each column sum to one, which is [29 (2.2)] .
(2) There exist constants 0 < C − ≤ C + 0 < C_{-} \le C_{+} 0 < C − ≤ C + C − ≤ N σ a b 2 C_{-} \le N\sigma_{ab}^{2} C − ≤ N σ ab 2 N σ a b 2 ≤ C + N\sigma_{ab}^{2} \le C_{+} N σ ab 2 ≤ C + a , b ∈ [ N ] a,b \in[N] a , b ∈ [ N ] [29 Example 1] , this implies the assumption [29 (2.3)] . Further, it implies [29, (2.4)} directly.
(3) The variables A a b / σ a b A_{ab}/\sigma_{ab} A ab / σ ab [29 (2.17)] .
The classical locations in [29 ] coincide with our γ i \gamma_i γ i A 1 > 1 A_1 > 1 A 1 > 1 L L L [29 (2.18)] . Under the three conditions above, [29 (2.25)] with L = A 1 log log N L = A_1 \log\log N L = A 1 log log N
P ( ∃ i ∈ [ N ] : ∣ λ i ( A ) − γ i ∣ ≥ e A 1 ( log log N ) 2 N − 2 / 3 min ( i , N + 1 − i ) − 1 / 3 ) ≤ C exp ( − c e c 1 A 1 ( log log N ) 2 ) (43) \mathbb{P}\left(\exists i \in[N] : \lvert\lambda_i(A)-\gamma_i\rvert\ge e^{A_1(\log\log N)^2}N^{-2/3}\min(i,N+1-i)^{-1/3}\right)
\le C\exp\left(-c e^{c_1A_1(\log\log N)^2}\right) \tag*{(43)} P ( ∃ i ∈ [ N ] : ∣ λ i ( A ) − γ i ∣ ≥ e A 1 ( l o g l o g N ) 2 N − 2/3 min ( i , N + 1 − i ) − 1/3 ) ≤ C exp ( − c e c 1 A 1 ( l o g l o g N ) 2 ) (43) for N ≥ N 0 N \ge N_0 N ≥ N 0 c 1 ∈ ( 0 , 1 ) c_1 \in(0,1) c 1 ∈ ( 0 , 1 ) A 1 A_1 A 1 c c c c 1 c_1 c 1 C C C N 0 N_0 N 0 C ± C_{\pm} C ± [29 (2.17)] . Here we used that ( log N ) L = e A 1 ( log log N ) 2 (\log N)^L = e^{A_1(\log\log N)^2} ( log N ) L = e A 1 ( l o g l o g N ) 2 L L L [29 (2.18)] holds for N N N e c 1 A 1 ( log log N ) 2 e^{c_1A_1(\log\log N)^2} e c 1 A 1 ( l o g l o g N ) 2 log N \log N log N (43) is bounded by C D N − D C_DN^{-D} C D N − D D > 0 D > 0 D > 0
The matrix H t H_t H t N − 1 N^{-1} N − 1
σ t 2 = e − t σ Z 2 + 2 ( 1 − e − t ) , r N , t = 1 + σ t 2 − 1 N . \sigma_t^2=e^{-t}\sigma_Z^2+2(1-e^{-t}),\qquad r_{N,t}=1+\frac{\sigma_t^2-1}{N}. σ t 2 = e − t σ Z 2 + 2 ( 1 − e − t ) , r N , t = 1 + N σ t 2 − 1 . so that σ t 2 \sigma_t^2 σ t 2 N 1 / 2 ( H t ) a a N^{1/2}(H_t)_{aa} N 1/2 ( H t ) aa Proposition 3.2 , we have
∑ a = 1 N E ( H t ) a b 2 = N − 1 N + σ t 2 N = r N , t for all b ∈ [ N ] , \sum_{a=1}^{N}\mathbb{E}(H_t)_{ab}^{2}=\frac{N-1}{N}+\frac{\sigma_t^2}{N}=r_{N,t}\qquad\text{for all }b\in[N], a = 1 ∑ N E ( H t ) ab 2 = N N − 1 + N σ t 2 = r N , t for all b ∈ [ N ] , so the matrix A = r N , t − 1 / 2 H t A=r_{N,t}^{-1/2}H_t A = r N , t − 1/2 H t
Recall that σ Z 2 > 0 \sigma_Z^2>0 σ Z 2 > 0 N ≥ 2 N \ge2 N ≥ 2
0 < min ( σ Z 2 , 2 ) ≤ σ t 2 ≤ max ( σ Z 2 , 2 ) , 1 2 ≤ r N , t ≤ 1 + max ( σ Z 2 , 2 ) . 0<\min(\sigma_Z^2,2)\le\sigma_t^2\le\max(\sigma_Z^2,2),\qquad\frac{1}{2}\le r_{N,t}\le1+\max(\sigma_Z^2,2). 0 < min ( σ Z 2 , 2 ) ≤ σ t 2 ≤ max ( σ Z 2 , 2 ) , 2 1 ≤ r N , t ≤ 1 + max ( σ Z 2 , 2 ) . The variances of the entries of A A A
N σ a b 2 = r N , t − 1 ( a ≠ b ) , N σ a a 2 = σ t 2 r N , t − 1 , N\sigma_{ab}^{2}=r_{N,t}^{-1}\quad(a\ne b),\qquad N\sigma_{aa}^{2}=\sigma_t^2r_{N,t}^{-1}, N σ ab 2 = r N , t − 1 ( a = b ) , N σ aa 2 = σ t 2 r N , t − 1 , so we obtain the second condition in the form
min ( σ Z 2 , 1 ) 1 + max ( σ Z 2 , 2 ) ≤ N σ a b 2 ≤ 2 max ( σ Z 2 , 2 ) for all a , b ∈ [ N ] . \frac{\min(\sigma_Z^2,1)}{1+\max(\sigma_Z^2,2)}\le N\sigma_{ab}^{2}\le2\max(\sigma_Z^2,2)\qquad\text{for all }a,b\in[N]. 1 + max ( σ Z 2 , 2 ) min ( σ Z 2 , 1 ) ≤ N σ ab 2 ≤ 2 max ( σ Z 2 , 2 ) for all a , b ∈ [ N ] . We next check the third condition. By (3.5), we have
A a b σ a b = e − t / 2 X a b + ( 1 − e − t ) 1 / 2 N 1 / 2 G a b ( a ≠ b ) , A a a σ a a = e − t / 2 Z a + ( 1 − e − t ) 1 / 2 N 1 / 2 G a a σ t . \frac{A_{ab}}{\sigma_{ab}}=e^{-t/2}X_{ab}+(1-e^{-t})^{1/2}N^{1/2}G_{ab}\quad(a\ne b),\qquad\frac{A_{aa}}{\sigma_{aa}}=\frac{e^{-t/2}Z_a+(1-e^{-t})^{1/2}N^{1/2}G_{aa}}{\sigma_t}. σ ab A ab = e − t /2 X ab + ( 1 − e − t ) 1/2 N 1/2 G ab ( a = b ) , σ aa A aa = σ t e − t /2 Z a + ( 1 − e − t ) 1/2 N 1/2 G aa . These variables have sub-Gaussian tails with constants independent of t t t N N N a a a b b b σ t 2 ≥ min ( σ Z 2 , 2 ) \sigma_t^2\ge\min(\sigma_Z^2,2) σ t 2 ≥ min ( σ Z 2 , 2 ) [29 (2.17)] with constants independent of t t t A = r N , t − 1 / 2 H t A=r_{N,t}^{-1/2}H_t A = r N , t − 1/2 H t t t t
We now return to H t H_t H t N ≥ N 0 N\ge N_0 N ≥ N 0 ∣ λ i ( A ) ∣ ≤ 3 |\lambda_i(A)|\le3 ∣ λ i ( A ) ∣ ≤ 3 i i i λ i ( H t ) = r N , t 1 / 2 λ i ( A ) \lambda_i(H_t)=r_{N,t}^{1/2}\lambda_i(A) λ i ( H t ) = r N , t 1/2 λ i ( A )
∣ λ i ( H t ) − λ i ( A ) ∣ = ∣ r N , t 1 / 2 − 1 ∣ ∣ λ i ( A ) ∣ ≤ 3 ∣ r N , t − 1 ∣ ≤ C N − 1 ≤ C N − 2 / 3 min ( i , N + 1 − i ) − 1 / 3 , i ∈ [ N ] , \begin{aligned}
\left|\lambda_i(H_t)-\lambda_i(A)\right|=\left|r_{N,t}^{1/2}-1\right|\left|\lambda_i(A)\right|\le3\left|r_{N,t}-1\right| \\
&\le CN^{-1}\le CN^{-2/3}\min(i,N+1-i)^{-1/3},\qquad i\in[N],
\end{aligned} ∣ λ i ( H t ) − λ i ( A ) ∣ = r N , t 1/2 − 1 ∣ λ i ( A ) ∣ ≤ 3 ∣ r N , t − 1 ∣ ≤ C N − 1 ≤ C N − 2/3 min ( i , N + 1 − i ) − 1/3 , i ∈ [ N ] , where we used ∣ r N , t 1 / 2 − 1 ∣ ≤ ∣ r N , t − 1 ∣ |r_{N,t}^{1/2}-1|\le|r_{N,t}-1| ∣ r N , t 1/2 − 1∣ ≤ ∣ r N , t − 1∣ N N N i ∈ [ N ] i\in[N] i ∈ [ N ]
∣ λ i ( H t ) − γ i ∣ ≤ ( e A 1 ( log log N ) 2 + C ) N − 2 / 3 min ( i , N + 1 − i ) − 1 / 3 ≤ e ( A 1 + 1 ) ( log log N ) 2 N − 2 / 3 min ( i , N + 1 − i ) − 1 / 3 . \begin{aligned}
|\lambda_i(H_t)-\gamma_i|\le\left(e^{A_1(\log\log N)^2}+C\right)N^{-2/3}\min(i,N+1-i)^{-1/3} \\
&\le e^{(A_1+1)(\log\log N)^2}N^{-2/3}\min(i,N+1-i)^{-1/3}.
\end{aligned} ∣ λ i ( H t ) − γ i ∣ ≤ ( e A 1 ( l o g l o g N ) 2 + C ) N − 2/3 min ( i , N + 1 − i ) − 1/3 ≤ e ( A 1 + 1 ) ( l o g l o g N ) 2 N − 2/3 min ( i , N + 1 − i ) − 1/3 . This is (3.4) with A 0 = A 1 + 1 A_0=A_1+1 A 0 = A 1 + 1 G G G σ t 2 \sigma_t^2 σ t 2 2 2 2
We next record the tail bound of Nguyen, Tao, and Vu [57 ] for the distance between eigenvalues whose labels differ by two.
Proposition 3.4. For every C 1 > 0 C_1>0 C 1 > 0 C > 0 C>0 C > 0
sup 1 ≤ i ≤ N − 2 P ( λ i + 2 − λ i ≤ δ / N ) ≤ C δ 3 for all δ ≥ N − C 1 . (44) \sup_{1\le i\le N-2}\mathbb{P}\left(\lambda_{i+2}-\lambda_i\le\delta/N\right)\le C\delta^3\qquad\text{for all }\delta\ge N^{-C_1}. \tag*{(44)} 1 ≤ i ≤ N − 2 sup P ( λ i + 2 − λ i ≤ δ / N ) ≤ C δ 3 for all δ ≥ N − C 1 . (44) Further, for r > 0 r>0 r > 0 N 3 ( r ) \mathcal{N}_3(r) N 3 ( r ) i ∈ [ N − 2 ] i\in[N-2] i ∈ [ N − 2 ] λ i + 2 − λ i ≤ r \lambda_{i+2}-\lambda_i\le r λ i + 2 − λ i ≤ r E N 3 ( r ) ≤ C N ( N r ) 3 \mathbb{E}\mathcal{N}_3(r)\le CN(Nr)^3 E N 3 ( r ) ≤ C N ( N r ) 3 r ≥ N − 1 − C 1 r\ge N^{-1-C_1} r ≥ N − 1 − C 1
Proof. This is [57 Corollary 2.5] with l = 2 l=2 l = 2 c l c_l c l [57 Section 1] , whose strictly upper-triangular entries are independent and identically distributed sub-Gaussian random variables with mean zero and variance one, and whose diagonal entries are independent sub-Gaussian random variables with mean zero and variance at most N 1 − o ( 1 ) N^{1-o(1)} N 1 − o ( 1 ) N 1 / 2 H N^{1/2}H N 1/2 H X a b X_{ab} X ab Z a Z_a Z a Z a Z_a Z a σ Z 2 \sigma_Z^2 σ Z 2
For every C 1 > 0 C_1>0 C 1 > 0 [57 Corollary 2.5] gives C > 0 C>0 C > 0
sup 1 ≤ i ≤ N − 2 P ( λ i + 2 ( N 1 / 2 H ) − λ i ( N 1 / 2 H ) ≤ δ N − 1 / 2 ) ≤ C δ 3 for all δ ≥ N − C 1 . \sup_{1\le i\le N-2}\mathbb{P}\left(\lambda_{i+2}(N^{1/2}H)-\lambda_i(N^{1/2}H)\le\delta N^{-1/2}\right)\le C\delta^3 \qquad\text{for all }\delta\ge N^{-C_1}. 1 ≤ i ≤ N − 2 sup P ( λ i + 2 ( N 1/2 H ) − λ i ( N 1/2 H ) ≤ δ N − 1/2 ) ≤ C δ 3 for all δ ≥ N − C 1 . As λ i ( N 1 / 2 H ) = N 1 / 2 λ i \lambda_i(N^{1/2}H)=N^{1/2}\lambda_i λ i ( N 1/2 H ) = N 1/2 λ i (44) . Applying (44) with δ = N r \delta=Nr δ = N r
E N 3 ( r ) = ∑ i = 1 N − 2 P ( λ i + 2 − λ i ≤ r ) ≤ C N ( N r ) 3 , r ≥ N − 1 − C 1 . \mathbb{E}\mathcal{N}_3(r)=\sum_{i=1}^{N-2}\mathbb{P}(\lambda_{i+2}-\lambda_i\le r)\le CN(Nr)^3,\qquad r\ge N^{-1-C_1}. E N 3 ( r ) = i = 1 ∑ N − 2 P ( λ i + 2 − λ i ≤ r ) ≤ C N ( N r ) 3 , r ≥ N − 1 − C 1 . We next describe the coupling of the gaps of H t H_t H t [10 ] . Let B = ( B ( t ) ) t ≥ 0 B=(B(t))_{t\ge0} B = ( B ( t ) ) t ≥ 0 N × N N\times N N × N H H H G G G B a b B_{ab} B ab a < b a<b a < b 2 − 1 / 2 B a a 2^{-1/2}B_{aa} 2 − 1/2 B aa a ∈ [ N ] a\in[N] a ∈ [ N ] ( H t d y n ) t ≥ 0 (H_t^{\mathrm{dyn}})_{t\ge0} ( H t dyn ) t ≥ 0
H t d y n = e − t / 2 H + N − 1 / 2 ∫ 0 t e − ( t − t ′ ) / 2 d B ( t ′ ) , (45) H_t^{\mathrm{dyn}}=e^{-t/2}H+N^{-1/2}\int_0^t e^{-(t-t')/2}\,\mathrm{d}B(t'), \tag*{(45)} H t dyn = e − t /2 H + N − 1/2 ∫ 0 t e − ( t − t ′ ) /2 d B ( t ′ ) , (45) which is the solution of d H t d y n = N − 1 / 2 d B ( t ) − 2 − 1 H t d y n d t \mathrm{d}H_t^{\mathrm{dyn}}=N^{-1/2}\,\mathrm{d}B(t)-2^{-1}H_t^{\mathrm{dyn}}\,\mathrm{d}t d H t dyn = N − 1/2 d B ( t ) − 2 − 1 H t dyn d t H 0 d y n = H H_0^{\mathrm{dyn}}=H H 0 dyn = H t ≥ 0 t\ge0 t ≥ 0 (45) is a centered Gaussian real symmetric matrix, independent of H H H ( 1 − e − t ) N − 1 (1-e^{-t})N^{-1} ( 1 − e − t ) N − 1 2 ( 1 − e − t ) N − 1 2(1-e^{-t})N^{-1} 2 ( 1 − e − t ) N − 1 H t d y n H_t^{\mathrm{dyn}} H t dyn H t H_t H t t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] λ i ( t ) = λ i ( H t d y n ) \lambda_i(t)=\lambda_i(H_t^{\mathrm{dyn}}) λ i ( t ) = λ i ( H t dyn )
Let E 0 \mathcal{E}_0 E 0 H H H [10 Section 1.4] , the eigenvalues of H t d y n H_t^{\mathrm{dyn}} H t dyn E 0 \mathcal{E}_0 E 0
d λ i ( t ) = d β i ( t ) N 1 / 2 + ( 1 N ∑ j ≠ i 1 λ i ( t ) − λ j ( t ) − λ i ( t ) 2 ) d t , i ∈ [ N ] , (46) \mathrm{d}\lambda_i(t)=\frac{\mathrm{d}\beta_i(t)}{N^{1/2}}+\left(\frac{1}{N}\sum_{j\ne i}\frac{1}{\lambda_i(t)-\lambda_j(t)}-\frac{\lambda_i(t)}{2}\right)\mathrm{d}t,\qquad i\in[N], \tag*{(46)} d λ i ( t ) = N 1/2 d β i ( t ) + N 1 j = i ∑ λ i ( t ) − λ j ( t ) 1 − 2 λ i ( t ) d t , i ∈ [ N ] , (46) where β 1 , … , β N \beta_1,\ldots,\beta_N β 1 , … , β N E β i ( t ) 2 = 2 t \mathbb{E}\beta_i(t)^2=2t E β i ( t ) 2 = 2 t E 0 \mathcal{E}_0 E 0 β 1 , … , β N \beta_1,\ldots,\beta_N β 1 , … , β N μ ( t ) = ( μ 1 ( t ) , … , μ N ( t ) ) \mu(t)=(\mu_1(t),\ldots,\mu_N(t)) μ ( t ) = ( μ 1 ( t ) , … , μ N ( t )) (46) driven by the same β i \beta_i β i μ i ( 0 ) = λ i ( G ) \mu_i(0)=\lambda_i(G) μ i ( 0 ) = λ i ( G ) G G G β i \beta_i β i G G G B B B ( H , G ) (H,G) ( H , G ) G G G (46) (see the discussion of [10 (2.2)] ), so μ ( t ) \mu(t) μ ( t ) G G G t ≥ 0 t\ge0 t ≥ 0 [10 (1.7)–(1.8)] , with λ ( t ) \lambda(t) λ ( t ) μ ( t ) \mu(t) μ ( t ) 0 0 0 1 1 1
Proposition 3.5. For all c ∗ ∈ ( 0 , 1 / 2 ) c_* \in(0,1/2) c ∗ ∈ ( 0 , 1/2 ) ε > 0 \varepsilon> 0 ε > 0 D > 0 D > 0 D > 0 N 0 N_0 N 0 N ≥ N 0 N \ge N_0 N ≥ N 0 N − 1 + ε ≤ t < 1 N^{-1+\varepsilon} \le t < 1 N − 1 + ε ≤ t < 1
P ( max c ∗ N ≤ i ≤ ( 1 − c ∗ ) N ∣ ( λ i + 1 ( t ) − λ i ( t ) ) − ( μ i + 1 ( t ) − μ i ( t ) ) ∣ > N ε N 2 t ) ≤ N − D . (47) \mathbb{P}\left(\max_{c_*N\le i\le(1-c_*)N}\left|(\lambda_{i+1}(t)-\lambda_i(t))-(\mu_{i+1}(t)-\mu_i(t))\right|>\frac{N^\varepsilon}{N^2t}\right)\le N^{-D}. \tag*{(47)} P ( c ∗ N ≤ i ≤ ( 1 − c ∗ ) N max ∣ ( λ i + 1 ( t ) − λ i ( t )) − ( μ i + 1 ( t ) − μ i ( t )) ∣ > N 2 t N ε ) ≤ N − D . (47) In particular, for every such t t t H t H_t H t (47) has probability at most N − D N^{-D} N − D
Proof. This follows from [10 Corollary 3.3] and a union bound, and we now check its hypotheses. That corollary states the following. For all c ∗ , ε , D > 0 c_*,\varepsilon,D>0 c ∗ , ε , D > 0 C , N 0 > 0 C,N_0>0 C , N 0 > 0 N ≥ N 0 N\ge N_0 N ≥ N 0 t t t e C ( log log N ) 2 N − 1 < t < 1 e^{C(\log\log N)^2}N^{-1}<t<1 e C ( l o g l o g N ) 2 N − 1 < t < 1 i i i c ∗ N ≤ i ≤ ( 1 − c ∗ ) N c_*N\le i\le(1-c_*)N c ∗ N ≤ i ≤ ( 1 − c ∗ ) N
P ( ∣ ( λ i + 1 ( t ) − λ i ( t ) ) − ( μ i + 1 ( t ) − μ i ( t ) ) ∣ > N ε N 2 t ) ≤ N − D . (48) \mathbb{P}\left(\left|(\lambda_{i+1}(t)-\lambda_i(t))-(\mu_{i+1}(t)-\mu_i(t))\right|>\frac{N^\varepsilon}{N^2t}\right)\le N^{-D}. \tag*{(48)} P ( ∣ ( λ i + 1 ( t ) − λ i ( t )) − ( μ i + 1 ( t ) − μ i ( t )) ∣ > N 2 t N ε ) ≤ N − D . (48) In [10 ] , the initial condition λ ( 0 ) \lambda(0) λ ( 0 ) [10 Definition 1.1] , μ ( 0 ) \mu(0) μ ( 0 )
First, the variances of the entries of a generalized Wigner matrix in [10 Definition 1.1] have row sums exactly one, whereas those of H H H 1 + ( σ Z 2 − 1 ) / N 1+(\sigma_Z^2-1)/N 1 + ( σ Z 2 − 1 ) / N [10 Remarks 2.9 and 3.2] state that the results hold, for every value of the symmetry parameter in [10 (2.2)] that is at least one, whenever the initial conditions satisfy optimal rigidity estimates. Indeed, the proofs of [10 Theorem 3.1 and Corollary 3.3] use the initial spectra only through the rigidity event of [10 (2.7)] and the lower bound 1 − N − D 1-N^{-D} 1 − N − D [10 Lemma 2.3] . (The GOE initial condition used there also has row sums 1 + N − 1 1+N^{-1} 1 + N − 1 λ ( 0 ) \lambda(0) λ ( 0 ) μ ( 0 ) \mu(0) μ ( 0 ) x x x (46) with this initial condition satisfies
∣ x i ( t ′ ) − γ i ∣ < e C ′ ( log log N ) 2 / 2 N − 2 / 3 min ( i , N + 1 − i ) − 1 / 3 |x_i(t')-\gamma_i|<e^{C'(\log\log N)^2/2}N^{-2/3}\min(i,N+1-i)^{-1/3} ∣ x i ( t ′ ) − γ i ∣ < e C ′ ( l o g l o g N ) 2 /2 N − 2/3 min ( i , N + 1 − i ) − 1/3 for all i ∈ [ N ] i\in[N] i ∈ [ N ] t ′ ∈ [ 0 , 1 ] t'\in[0,1] t ′ ∈ [ 0 , 1 ] C ′ C' C ′
As explained before [10 Lemma 2.3] , the lower bound on the probability of this event follows from rigidity estimates at fixed times for the spectra of H t ′ d y n H_{t'}^{\mathrm{dyn}} H t ′ dyn t ′ ∈ [ 0 , 1 ] t'\in[0,1] t ′ ∈ [ 0 , 1 ] λ ( 0 ) \lambda(0) λ ( 0 ) μ ( 0 ) \mu(0) μ ( 0 ) H t ′ d y n H_{t'}^{\mathrm{dyn}} H t ′ dyn H t ′ H_{t'} H t ′ H t ′ H_{t'} H t ′ G G G
e A 0 ( log log N ) 2 ≤ 1 2 e C ′ ( log log N ) 2 / 2 (49) e^{A_0(\log\log N)^2}\le\frac{1}{2}e^{C'(\log\log N)^2/2} \tag*{(49)} e A 0 ( l o g l o g N ) 2 ≤ 2 1 e C ′ ( l o g l o g N ) 2 /2 (49) for C ′ > 2 A 0 C' > 2A_0 C ′ > 2 A 0 N N N (45) , (46) , and bounds on ∥ H t d y n ∥ \lVert H_t^{\mathrm{dyn}}\rVert ∥ H t dyn ∥ [10 Lemma 2.3] holds for our initial data as well.
Second, the system (46) is set up in [10 ] for distinct initial points, which holds on E 0 \mathcal{E}_0 E 0 [57 Corollary 2.2] , applied to N 1 / 2 H N^{1/2}H N 1/2 H C 1 > 0 C_1 > 0 C 1 > 0 C N T V > 0 C_{\mathrm{NTV}} > 0 C NTV > 0
P ( λ i + 1 − λ i ≤ N − 1 − C 1 ) ≤ C N T V N − C 1 for all i ∈ [ N − 1 ] . \mathbb{P}\left(\lambda_{i+1}-\lambda_i \leq N^{-1-C_1}\right) \leq C_{\mathrm{NTV}}N^{-C_1}\qquad\text{for all }i\in[N-1]. P ( λ i + 1 − λ i ≤ N − 1 − C 1 ) ≤ C NTV N − C 1 for all i ∈ [ N − 1 ] . A union bound over i i i
P ( E 0 c ) ≤ ∑ i = 1 N − 1 P ( λ i + 1 − λ i ≤ N − 1 − C 1 ) ≤ C N T V N 1 − C 1 , \mathbb{P}\left(\mathcal{E}_0^{c}\right) \leq\sum_{i=1}^{N-1}\mathbb{P}\left(\lambda_{i+1}-\lambda_i \leq N^{-1-C_1}\right) \leq C_{\mathrm{NTV}}N^{1-C_1}, P ( E 0 c ) ≤ i = 1 ∑ N − 1 P ( λ i + 1 − λ i ≤ N − 1 − C 1 ) ≤ C NTV N 1 − C 1 , and since C 1 C_1 C 1 E 0 \mathcal{E}_0 E 0 [71 ] .) We apply [10 Corollary 3.3] on E 0 \mathcal{E}_0 E 0 E 0 c \mathcal{E}_0^c E 0 c
Third, (48) concerns a single index i i i D + 2 D+2 D + 2 D D D N N N i i i c ∗ N ≤ i ≤ ( 1 − c ∗ ) N c_*N\leq i\leq(1-c_*)N c ∗ N ≤ i ≤ ( 1 − c ∗ ) N P ( E 0 c ) ≤ N − D − 1 \mathbb{P}(\mathcal{E}_0^c)\leq N^{-D-1} P ( E 0 c ) ≤ N − D − 1 N N N (47) by
N ⋅ N − D − 2 + P ( E 0 c ) ≤ 2 N − D − 1 ≤ N − D . N\cdot N^{-D-2}+\mathbb{P}\left(\mathcal{E}_0^c\right)\leq2N^{-D-1}\leq N^{-D}. N ⋅ N − D − 2 + P ( E 0 c ) ≤ 2 N − D − 1 ≤ N − D . Finally, e C ( log log N ) 2 N − 1 < N − 1 + ε e^{C(\log\log N)^2}N^{-1}<N^{-1+\varepsilon} e C ( l o g l o g N ) 2 N − 1 < N − 1 + ε N N N t ∈ [ N − 1 + ε , 1 ) t\in[N^{-1+\varepsilon},1) t ∈ [ N − 1 + ε , 1 ) [10 Corollary 3.3] ; the upper limit t < 1 t<1 t < 1 λ ( t ) \lambda(t) λ ( t ) H t H_t H t μ ( t ) \mu(t) μ ( t ) G G G
For α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 )
ℓ α = N − 1 − α , M α = N 1 − 2 α = N 3 ℓ α 2 . (50) \ell_\alpha=N^{-1-\alpha},\qquad M_\alpha=N^{1-2\alpha}=N^3\ell_\alpha^2. \tag*{(50)} ℓ α = N − 1 − α , M α = N 1 − 2 α = N 3 ℓ α 2 . (50) The next proposition gives the pair statistics of the GOE at the scale ℓ α \ell_\alpha ℓ α α < 1 / 4 \alpha<1/4 α < 1/4 § 12 .
Proposition 3.6. Let α ∈ ( 0 , 1 / 4 ) \alpha\in(0,1/4) α ∈ ( 0 , 1/4 ) p > 1 p>1 p > 1 f : [ 0 , ∞ ) × R → R f:[0,\infty)\times\mathbb{R}\to\mathbb{R} f : [ 0 , ∞ ) × R → R I I I C f > 0 C_f>0 C f > 0 f ( r , E ) = 0 f(r,E)=0 f ( r , E ) = 0 E ∉ I E\notin I E ∈ / I ∣ f ( r , E ) ∣ ≤ C f ( 1 + r 2 ) − p \lvert f(r,E)\rvert\leq C_f(1+r^2)^{-p} ∣ f ( r , E )∣ ≤ C f ( 1 + r 2 ) − p ( r , E ) (r,E) ( r , E ) L 1 L^1 L 1
1 M α ∑ 1 ≤ i < j ≤ N f ( λ j ( G ) − λ i ( G ) ℓ α , λ i ( G ) + λ j ( G ) 2 ) ⟶ ∫ − 2 2 ∫ 0 ∞ c ( E ) r f ( r , E ) d r d E . (51) \frac{1}{M_\alpha}\sum_{1\leq i<j\leq N}f\left(\frac{\lambda_j(G)-\lambda_i(G)}{\ell_\alpha},\frac{\lambda_i(G)+\lambda_j(G)}{2}\right)\longrightarrow\int_{-2}^{2}\int_{0}^{\infty}c(E)rf(r,E)\,\mathrm{d}r\,\mathrm{d}E. \tag*{(51)} M α 1 1 ≤ i < j ≤ N ∑ f ( ℓ α λ j ( G ) − λ i ( G ) , 2 λ i ( G ) + λ j ( G ) ) ⟶ ∫ − 2 2 ∫ 0 ∞ c ( E ) r f ( r , E ) d r d E . (51) Eigenvalues of entrywise small perturbations
We now consider a diagonal matrix perturbed by a Hermitian matrix with small entries. We group the diagonal entries into clusters of nearby points, and we compare the eigenvalues near each cluster with those of the corresponding diagonal block of the perturbed matrix. In Section 6, Section 7, and Section 9, the diagonal matrix contains the eigenvalues of H H H e − t / 2 e^{-t/2} e − t /2 Δ \Delta Δ H H H
We use the following notation in this section. For an N × N N \times N N × N A A A K , K ′ ⊂ [ N ] \mathcal{K},\mathcal{K}' \subset[N] K , K ′ ⊂ [ N ] A K K ′ = ( A i j ) i ∈ K , j ∈ K ′ A_{\mathcal{K}\mathcal{K}'} = (A_{ij})_{i\in\mathcal{K},j\in\mathcal{K}'} A K K ′ = ( A ij ) i ∈ K , j ∈ K ′ K c = [ N ] ∖ K \mathcal{K}^{c} = [N] \setminus\mathcal{K} K c = [ N ] ∖ K A A A n − ( A ) n_{-}(A) n − ( A ) A A A M M M ∥ M ∥ ∞ → ∞ = max i ∑ j ∣ M i j ∣ \lVert M\rVert_{\infty\to\infty} = \max_i \sum_j |M_{ij}| ∥ M ∥ ∞ → ∞ = max i ∑ j ∣ M ij ∣ ℓ ∞ \ell^\infty ℓ ∞
We will use the inertia additivity formula of Haynsworth [39 ] . Let M M M N × N N \times N N × N K ⊂ [ N ] \mathcal{K} \subset[N] K ⊂ [ N ] M K c K c M_{\mathcal{K}^{c}\mathcal{K}^{c}} M K c K c
M / M K c K c = M K K − M K K c M K c K c − 1 M K c K M/M_{\mathcal{K}^{c}\mathcal{K}^{c}} = M_{\mathcal{K}\mathcal{K}} - M_{\mathcal{K}\mathcal{K}^{c}}M_{\mathcal{K}^{c}\mathcal{K}^{c}}^{-1}M_{\mathcal{K}^{c}\mathcal{K}} M / M K c K c = M KK − M K K c M K c K c − 1 M K c K denote the Schur complement. After a permutation of coordinates, M M M M / M K c K c M/M_{\mathcal{K}^{c}\mathcal{K}^{c}} M / M K c K c M K c K c M_{\mathcal{K}^{c}\mathcal{K}^{c}} M K c K c [40 ] and the multiplicativity of the determinant give
n − ( M ) = n − ( M K c K c ) + n − ( M / M K c K c ) , det M = det M K c K c ⋅ det ( M / M K c K c ) . (52) n_{-}(M) = n_{-}(M_{\mathcal{K}^{c}\mathcal{K}^{c}}) + n_{-}(M/M_{\mathcal{K}^{c}\mathcal{K}^{c}}), \qquad\det M = \det M_{\mathcal{K}^{c}\mathcal{K}^{c}} \cdot\det(M/M_{\mathcal{K}^{c}\mathcal{K}^{c}}). \tag*{(52)} n − ( M ) = n − ( M K c K c ) + n − ( M / M K c K c ) , det M = det M K c K c ⋅ det ( M / M K c K c ) . (52) The following lemma compares the eigenvalues of Λ 0 + W \Lambda_0 + W Λ 0 + W Λ 0 \Lambda_0 Λ 0 ( Λ 0 + W ) K K (\Lambda_0 + W)_{\mathcal{K}\mathcal{K}} ( Λ 0 + W ) KK [50 , 55 ] , and its proof is elementary.
Lemma 3.7. Let Λ 0 \Lambda_0 Λ 0 W W W N × N N \times N N × N K ⊂ [ N ] \mathcal{K} \subset[N] K ⊂ [ N ] ( Λ 0 ) K K c = 0 (\Lambda_0)_{\mathcal{K}\mathcal{K}^{c}} = 0 ( Λ 0 ) K K c = 0 E − ≤ E + E_{-} \le E_{+} E − ≤ E + ( Λ 0 ) K K (\Lambda_0)_{\mathcal{K}\mathcal{K}} ( Λ 0 ) KK [ E − , E + ] [E_{-}, E_{+}] [ E − , E + ] ϑ > 0 \vartheta> 0 ϑ > 0 R = [ E − − ϑ , E + + ϑ ] \mathcal{R} = [E_{-} - \vartheta, E_{+} + \vartheta] R = [ E − − ϑ , E + + ϑ ] ( Λ 0 ) K c K c + W K c K c − x (\Lambda_0)_{\mathcal{K}^{c}\mathcal{K}^{c}} + W_{\mathcal{K}^{c}\mathcal{K}^{c}} - x ( Λ 0 ) K c K c + W K c K c − x x ∈ R x \in\mathcal{R} x ∈ R χ ≥ 0 \chi\ge0 χ ≥ 0
∥ W K K c ( ( Λ 0 ) K c K c + W K c K c − x ) − 1 W K c K ∥ ≤ χ for all x ∈ R , ∥ W K K ∥ + χ < ϑ . (53) \left\lVert W_{\mathcal{K}\mathcal{K}^{c}}\left((\Lambda_0)_{\mathcal{K}^{c}\mathcal{K}^{c}} + W_{\mathcal{K}^{c}\mathcal{K}^{c}} - x\right)^{-1}W_{\mathcal{K}^{c}\mathcal{K}}\right\rVert\le\chi\quad\text{for all } x \in\mathcal{R}, \qquad\lVert W_{\mathcal{K}\mathcal{K}}\rVert+ \chi< \vartheta. \tag*{(53)} W K K c ( ( Λ 0 ) K c K c + W K c K c − x ) − 1 W K c K ≤ χ for all x ∈ R , ∥ W KK ∥ + χ < ϑ . (53) Let i 0 i_0 i 0 Λ 0 + W \Lambda_0 + W Λ 0 + W E − − ϑ E_{-} - \vartheta E − − ϑ Λ 0 + W \Lambda_0 + W Λ 0 + W R \mathcal{R} R λ i 0 + 1 ( Λ 0 + W ) , … , λ i 0 + ∣ K ∣ ( Λ 0 + W ) \lambda_{i_0+1}(\Lambda_0 + W), \ldots, \lambda_{i_0+|\mathcal{K}|}(\Lambda_0 + W) λ i 0 + 1 ( Λ 0 + W ) , … , λ i 0 + ∣ K ∣ ( Λ 0 + W )
∣ λ i 0 + i ( Λ 0 + W ) − λ i ( ( Λ 0 + W ) K K ) ∣ ≤ χ for all i ∈ [ ∣ K ∣ ] , (54) \left|\lambda_{i_0+i}(\Lambda_0 + W) - \lambda_i((\Lambda_0 + W)_{\mathcal{K}\mathcal{K}})\right| \le\chi\qquad\text{for all } i \in[|\mathcal{K}|], \tag*{(54)} ∣ λ i 0 + i ( Λ 0 + W ) − λ i (( Λ 0 + W ) KK ) ∣ ≤ χ for all i ∈ [ ∣ K ∣ ] , (54) and these eigenvalues belong to [ E − − ∥ W K K ∥ − χ , E + + ∥ W K K ∥ + χ ] [E_{-} - \lVert W_{\mathcal{K}\mathcal{K}}\rVert- \chi, E_{+} + \lVert W_{\mathcal{K}\mathcal{K}}\rVert+ \chi] [ E − − ∥ W KK ∥ − χ , E + + ∥ W KK ∥ + χ ]
Proof. We abbreviate B = ( Λ 0 + W ) K K B=(\Lambda_0+W)_{K K} B = ( Λ 0 + W ) K K x ∈ R x\in\mathbb{R} x ∈ R
S ( x ) = W K K c ( ( Λ 0 ) K c K c + W K c K c − x ) − 1 W K c K , F ( x ) = B − x − S ( x ) . \mathcal{S}(x)=W_{K K^c}\left((\Lambda_0)_{K^c K^c}+W_{K^c K^c}-x\right)^{-1}W_{K^c K},\qquad\mathcal{F}(x)=B-x-\mathcal{S}(x). S ( x ) = W K K c ( ( Λ 0 ) K c K c + W K c K c − x ) − 1 W K c K , F ( x ) = B − x − S ( x ) . Observe that S ( x ) \mathcal{S}(x) S ( x ) F ( x ) \mathcal{F}(x) F ( x ) x x x W K c K = ( W K K c ) ∗ W_{K^c K}=(W_{K K^c})^* W K c K = ( W K K c ) ∗ ( Λ 0 ) K K c = 0 (\Lambda_0)_{K K^c}=0 ( Λ 0 ) K K c = 0 M = Λ 0 + W − x M=\Lambda_0+W-x M = Λ 0 + W − x M K K c = W K K c M_{K K^c}=W_{K K^c} M K K c = W K K c F ( x ) \mathcal{F}(x) F ( x ) (52) , we have
# { j ∈ [ N ] : λ j ( Λ 0 + W ) < x } = n − ( Λ 0 + W − x ) = n − ( ( Λ 0 ) K c K c + W K c K c − x ) + n − ( F ( x ) ) (55) \begin{aligned}
\#\{j\in[N]:\lambda_j(\Lambda_0+W)<x\}=n_-(\Lambda_0+W-x) \\
&=n_-\left((\Lambda_0)_{K^c K^c}+W_{K^c K^c}-x\right)+n_-(\mathcal{F}(x)) \tag*{(55)}
\end{aligned} # { j ∈ [ N ] : λ j ( Λ 0 + W ) < x } = n − ( Λ 0 + W − x ) = n − ( ( Λ 0 ) K c K c + W K c K c − x ) + n − ( F ( x )) (55) for all x ∈ R x\in\mathbb{R} x ∈ R
The matrix ( Λ 0 ) K c K c + W K c K c (\Lambda_0)_{K^c K^c}+W_{K^c K^c} ( Λ 0 ) K c K c + W K c K c R \mathcal{R} R (55) is constant on R \mathcal{R} R i 1 i_1 i 1
We next compare F ( x ) \mathcal{F}(x) F ( x ) B − x B-x B − x ∥ S ( x ) ∥ ≤ χ \|\mathcal{S}(x)\|\leq\chi ∥ S ( x ) ∥ ≤ χ
B − x − χ ≤ F ( x ) ≤ B − x + χ B-x-\chi\leq\mathcal{F}(x)\leq B-x+\chi B − x − χ ≤ F ( x ) ≤ B − x + χ in the sense of quadratic forms. By the Courant–Fischer minimax principle [40 ] , an inequality A ≤ A ′ A\leq A' A ≤ A ′ λ i ( A ) ≤ λ i ( A ′ ) \lambda_i(A)\leq\lambda_i(A') λ i ( A ) ≤ λ i ( A ′ ) i i i n − ( A ) ≥ n − ( A ′ ) n_-(A)\geq n_-(A') n − ( A ) ≥ n − ( A ′ )
# { i : λ i ( B ) < x − χ } ≤ n − ( F ( x ) ) ≤ # { i : λ i ( B ) < x + χ } for all x ∈ R . (56) \#\{i:\lambda_i(B)<x-\chi\}\leq n_-(\mathcal{F}(x))\leq\#\{i:\lambda_i(B)<x+\chi\}\quad\text{for all }x\in\mathbb{R}. \tag*{(56)} # { i : λ i ( B ) < x − χ } ≤ n − ( F ( x )) ≤ # { i : λ i ( B ) < x + χ } for all x ∈ R . (56) Using Weyl’s inequality [40 ] , the assumption on the spectrum of ( Λ 0 ) K K (\Lambda_0)_{K K} ( Λ 0 ) K K (53) , we see that every eigenvalue of B B B
E − − ϑ + χ < E − − ∥ W K K ∥ ≤ λ i ( B ) ≤ E + + ∥ W K K ∥ < E + + ϑ − χ . (57) E_- -\vartheta+\chi<E_- -\|W_{K K}\|\leq\lambda_i(B)\leq E_+ +\|W_{K K}\|<E_+ +\vartheta-\chi. \tag*{(57)} E − − ϑ + χ < E − − ∥ W K K ∥ ≤ λ i ( B ) ≤ E + + ∥ W K K ∥ < E + + ϑ − χ . (57) We now identify the eigenvalues of Λ 0 + W \Lambda_0+W Λ 0 + W R \mathcal{R} R x = E − − ϑ x=E_- -\vartheta x = E − − ϑ (57) that the right side of (56) vanishes, so (55) gives i 0 = i 1 i_0=i_1 i 0 = i 1 x = E + + ϑ x=E_+ +\vartheta x = E + + ϑ (56) is ∣ K ∣ |K| ∣ K ∣ F ( x ) \mathcal{F}(x) F ( x ) ∣ K ∣ × ∣ K ∣ |K|\times|K| ∣ K ∣ × ∣ K ∣ n − ( F ( E + + ϑ ) ) = ∣ K ∣ n_-(\mathcal{F}(E_+ +\vartheta))=|K| n − ( F ( E + + ϑ )) = ∣ K ∣ Λ 0 + W \Lambda_0+W Λ 0 + W i 0 + ∣ K ∣ i_0+|K| i 0 + ∣ K ∣ E + + ϑ E_+ +\vartheta E + + ϑ (57) gives
F ( E + + ϑ ) ≤ B − E + − ϑ + χ < 0 \mathcal{F}(E_+ +\vartheta)\leq B-E_+-\vartheta+\chi<0 F ( E + + ϑ ) ≤ B − E + − ϑ + χ < 0 in the sense of quadratic forms, so the determinant formula in (52) shows that E + + ϑ E_+ +\vartheta E + + ϑ Λ 0 + W \Lambda_0+W Λ 0 + W Λ 0 + W \Lambda_0+W Λ 0 + W R \mathcal{R} R λ i 0 + 1 ( Λ 0 + W ) , … , λ i 0 + ∣ K ∣ ( Λ 0 + W ) \lambda_{i_0+1}(\Lambda_0+W),\ldots,\lambda_{i_0+|K|}(\Lambda_0+W) λ i 0 + 1 ( Λ 0 + W ) , … , λ i 0 + ∣ K ∣ ( Λ 0 + W )
It remains to prove (54) . For brevity, we write λ i ′ = λ i 0 + i ( Λ 0 + W ) \lambda'_i=\lambda_{i_0+i}(\Lambda_0+W) λ i ′ = λ i 0 + i ( Λ 0 + W ) λ i ′ ′ = λ i ( B ) \lambda''_i=\lambda_i(B) λ i ′′ = λ i ( B ) i ∈ [ ∣ K ∣ ] i\in[|K|] i ∈ [ ∣ K ∣ ] x ∈ R x\in\mathcal{R} x ∈ R Λ 0 + W \Lambda_0+W Λ 0 + W x x x i 0 i_0 i 0 E − − ϑ E_- -\vartheta E − − ϑ λ i ′ \lambda'_i λ i ′ λ i ′ < x \lambda'_i<x λ i ′ < x (55) and using (56) , we find
# { i : λ i ′ ′ < x − χ } ≤ # { i : λ i ′ < x } ≤ # { i : λ i ′ ′ < x + χ } for all x ∈ R . (58) \#\{i:\lambda''_i<x-\chi\}\leq\#\{i:\lambda'_i<x\}\leq\#\{i:\lambda''_i<x+\chi\}\quad\text{for all }x\in\mathbb{R}. \tag*{(58)} # { i : λ i ′′ < x − χ } ≤ # { i : λ i ′ < x } ≤ # { i : λ i ′′ < x + χ } for all x ∈ R . (58) Fix i ∈ [ ∣ K ∣ ] i \in[|\mathcal{K}|] i ∈ [ ∣ K ∣ ] λ i ′ < λ i ′ ′ − χ \lambda'_i < \lambda''_i-\chi λ i ′ < λ i ′′ − χ x ∈ ( λ i ′ , λ i ′ ′ − χ ) x \in(\lambda'_i,\lambda''_i-\chi) x ∈ ( λ i ′ , λ i ′′ − χ ) R \mathcal{R} R (57) , since λ i ′ ≥ E − − ϑ \lambda'_i \ge E_- - \vartheta λ i ′ ≥ E − − ϑ x x x (58) is at least i i i λ 1 ′ ≤ ⋯ ≤ λ i ′ < x \lambda'_1 \le\cdots\le\lambda'_i < x λ 1 ′ ≤ ⋯ ≤ λ i ′ < x i i i x + χ < λ i ′ ′ ≤ ⋯ ≤ λ ∣ K ∣ ′ ′ x+\chi< \lambda''_i \le\cdots\le\lambda''_{|\mathcal{K}|} x + χ < λ i ′′ ≤ ⋯ ≤ λ ∣ K ∣ ′′ (58) . If λ i ′ > λ i ′ ′ + χ \lambda'_i > \lambda''_i+\chi λ i ′ > λ i ′′ + χ x ∈ ( λ i ′ ′ + χ , λ i ′ ) x \in(\lambda''_i+\chi,\lambda'_i) x ∈ ( λ i ′′ + χ , λ i ′ ) (58) is at least i i i i i i (54) , and the final statement follows from (54) and (57) .
We now apply Lemma 3.7 to a diagonal matrix with clustered entries. Let y 1 ≤ ⋯ ≤ y N y_1 \le\cdots\le y_N y 1 ≤ ⋯ ≤ y N D y = diag ( y 1 , … , y N ) D_y=\operatorname{diag}(y_1,\ldots,y_N) D y = diag ( y 1 , … , y N ) K \mathcal{K} K [ N ] [N] [ N ] family of clusters , and we call its elements clusters . For a cluster K \mathcal{K} K ϑ > 0 \vartheta>0 ϑ > 0
R K = { w ∈ C : min i ∈ K y i − ϑ ≤ Re w ≤ max i ∈ K y i + ϑ , ∣ Im w ∣ ≤ ϑ } (59) \mathcal{R}_{\mathcal{K}}=\left\{w\in\mathbb{C}:\min_{i\in\mathcal{K}}y_i-\vartheta\le\operatorname{Re}w\le\max_{i\in\mathcal{K}}y_i+\vartheta,\ \lvert\operatorname{Im}w\rvert\le\vartheta\right\} \tag*{(59)} R K = { w ∈ C : i ∈ K min y i − ϑ ≤ Re w ≤ i ∈ K max y i + ϑ , ∣ Im w ∣ ≤ ϑ } (59) denote the closed rectangle obtained by enlarging the interval [ min i ∈ K y i , max i ∈ K y i ] [\min_{i\in\mathcal{K}}y_i,\max_{i\in\mathcal{K}}y_i] [ min i ∈ K y i , max i ∈ K y i ] ϑ \vartheta ϑ ϑ \vartheta ϑ margin of the rectangles.
Lemma 3.8. Let y 1 ≤ ⋯ ≤ y N y_1 \le\cdots\le y_N y 1 ≤ ⋯ ≤ y N K \mathcal{K} K k 0 ≥ 1 k_0\ge1 k 0 ≥ 1 ∣ K ∣ ≤ k 0 |\mathcal{K}|\le k_0 ∣ K ∣ ≤ k 0 K ∈ K \mathcal{K}\in\mathcal{K} K ∈ K ϑ > 0 \vartheta>0 ϑ > 0 R K \mathcal{R}_{\mathcal{K}} R K K ∈ K \mathcal{K}\in\mathcal{K} K ∈ K L ∈ [ 0 , ∞ ) \mathcal{L}\in[0,\infty) L ∈ [ 0 , ∞ )
sup K ∈ K sup w ∈ R K ∑ j ∈ [ N ] ∖ K 1 ∣ y j − w ∣ ≤ L , (60) \sup_{\mathcal{K}\in\mathcal{K}}\sup_{w\in\mathcal{R}_{\mathcal{K}}}\sum_{j\in[N]\setminus\mathcal{K}}\frac{1}{|y_j-w|}\le\mathcal{L}, \tag*{(60)} K ∈ K sup w ∈ R K sup j ∈ [ N ] ∖ K ∑ ∣ y j − w ∣ 1 ≤ L , (60) with the convention 1 / 0 = ∞ 1/0=\infty 1/0 = ∞ W W W N × N N\times N N × N max i , j ∣ W i j ∣ ≤ w 0 \max_{i,j}|W_{ij}|\le w_0 max i , j ∣ W ij ∣ ≤ w 0
w 0 L ≤ 1 / 2 , k 0 w 0 + 2 k 0 w 0 2 L < ϑ . (61) w_0\mathcal{L}\le1/2,\qquad k_0w_0+2k_0w_0^2\mathcal{L}<\vartheta. \tag*{(61)} w 0 L ≤ 1/2 , k 0 w 0 + 2 k 0 w 0 2 L < ϑ . (61) Let K ∈ K \mathcal{K}\in\mathcal{K} K ∈ K K = { i 0 + 1 , … , i 0 + ∣ K ∣ } \mathcal{K}=\{i_0+1,\ldots,i_0+|\mathcal{K}|\} K = { i 0 + 1 , … , i 0 + ∣ K ∣ } D y + W D_y+W D y + W R K \mathcal{R}_{\mathcal{K}} R K λ i 0 + 1 ( D y + W ) , … , λ i 0 + ∣ K ∣ ( D y + W ) \lambda_{i_0+1}(D_y+W),\ldots,\lambda_{i_0+|\mathcal{K}|}(D_y+W) λ i 0 + 1 ( D y + W ) , … , λ i 0 + ∣ K ∣ ( D y + W )
∣ λ i 0 + i ( D y + W ) − λ i ( ( D y + W ) K K ) ∣ ≤ 2 k 0 w 0 2 L for all i ∈ [ ∣ K ∣ ] , (62) \left|\lambda_{i_0+i}(D_y+W)-\lambda_i\bigl((D_y+W)_{\mathcal{K}\mathcal{K}}\bigr)\right|\le2k_0w_0^2\mathcal{L}\qquad\text{for all }i\in[|\mathcal{K}|], \tag*{(62)} λ i 0 + i ( D y + W ) − λ i ( ( D y + W ) KK ) ≤ 2 k 0 w 0 2 L for all i ∈ [ ∣ K ∣ ] , (62) and each of these eigenvalues is within distance k 0 w 0 + 2 k 0 w 0 2 L k_0w_0+2k_0w_0^2\mathcal{L} k 0 w 0 + 2 k 0 w 0 2 L [ min i ∈ K y i , max i ∈ K y i ] [\min_{i\in\mathcal{K}}y_i,\max_{i\in\mathcal{K}}y_i] [ min i ∈ K y i , max i ∈ K y i ]
Since ∣ y j − w ∣ ≥ ∣ y j − Re w ∣ |y_j-w|\ge|y_j-\operatorname{Re}w| ∣ y j − w ∣ ≥ ∣ y j − Re w ∣ (60) may be taken over the real points of the rectangles, and since D y + W D_y+W D y + W R K \mathcal{R}_{\mathcal{K}} R K R K ∩ R \mathcal{R}_{\mathcal{K}}\cap\mathbb{R} R K ∩ R (60) , which requires that the points outside each cluster are not too dense near it.
Proof. Fix K ∈ R K \in\mathcal{R} K ∈ R E − = min i ∈ K y i E_- = \min_{i\in K} y_i E − = min i ∈ K y i E + = max i ∈ K y i E_+ = \max_{i\in K} y_i E + = max i ∈ K y i K = [ N ] K = [N] K = [ N ] N ≤ k 0 N \le k_0 N ≤ k 0
∣ λ i ( D y + W ) − y i ∣ ≤ ∥ W ∥ ≤ N w 0 ≤ k 0 w 0 < ϑ for all i ∈ [ N ] , |\lambda_i(D_y + W) - y_i| \le\|W\| \le Nw_0 \le k_0w_0 < \vartheta\qquad\text{for all } i \in[N], ∣ λ i ( D y + W ) − y i ∣ ≤ ∥ W ∥ ≤ N w 0 ≤ k 0 w 0 < ϑ for all i ∈ [ N ] , and the lemma follows, since ( D y + W ) K K = D y + W (D_y + W)_{KK} = D_y + W ( D y + W ) K K = D y + W
Now suppose that K c K^c K c Λ 0 = D y \Lambda_0 = D_y Λ 0 = D y χ = 2 k 0 w 0 2 L \chi= 2k_0w_0^2\mathcal{L} χ = 2 k 0 w 0 2 L
R = R K ∩ R = [ E − − ϑ , E + + ϑ ] . \mathcal{R} = \mathcal{R}_K \cap\mathbb{R} = [E_- - \vartheta, E_+ + \vartheta]. R = R K ∩ R = [ E − − ϑ , E + + ϑ ] . The matrix D y D_y D y ( D y ) K K c = 0 (D_y)_{K K^c} = 0 ( D y ) K K c = 0 ( D y ) K K (D_y)_{KK} ( D y ) K K y i y_i y i i ∈ K i \in K i ∈ K [ E − , E + ] [E_-, E_+] [ E − , E + ]
We now check the invertibility hypothesis of Lemma 3.7. Fix x ∈ R x \in\mathcal{R} x ∈ R ( D y − x ) K c K c (D_y - x)_{K^cK^c} ( D y − x ) K c K c y j − x y_j - x y j − x j ∈ K c j \in K^c j ∈ K c
∥ W K c K c ( D y − x ) K c K c − 1 ∥ ∞ → ∞ = max i ∈ K c ∑ j ∈ K c ∣ W i j ∣ ∣ y j − x ∣ ≤ w 0 L ≤ 1 2 . \left\|W_{K^cK^c}(D_y-x)_{K^cK^c}^{-1}\right\|_{\infty\to\infty}
= \max_{i\in K^c}\sum_{j\in K^c}\frac{|W_{ij}|}{|y_j-x|}
\le w_0\mathcal{L} \le\frac{1}{2}. W K c K c ( D y − x ) K c K c − 1 ∞ → ∞ = i ∈ K c max j ∈ K c ∑ ∣ y j − x ∣ ∣ W ij ∣ ≤ w 0 L ≤ 2 1 . Hence the Neumann series shows that Id + W K c K c ( D y − x ) K c K c − 1 \operatorname{Id}+W_{K^cK^c}(D_y-x)_{K^cK^c}^{-1} Id + W K c K c ( D y − x ) K c K c − 1 Θ ( x ) \Theta(x) Θ ( x ) ∥ Θ ( x ) ∥ ∞ → ∞ ≤ 2 \|\Theta(x)\|_{\infty\to\infty}\le2 ∥Θ ( x ) ∥ ∞ → ∞ ≤ 2
( D y + W ) K c K c − x = ( Id + W K c K c ( D y − x ) K c K c − 1 ) ( D y − x ) K c K c , (D_y+W)_{K^cK^c}-x
=
\left(\operatorname{Id}+W_{K^cK^c}(D_y-x)_{K^cK^c}^{-1}\right)(D_y-x)_{K^cK^c}, ( D y + W ) K c K c − x = ( Id + W K c K c ( D y − x ) K c K c − 1 ) ( D y − x ) K c K c , we see that this matrix is invertible, and that
( ( D y + W ) K c K c − x ) − 1 = ( D y − x ) K c K c − 1 Θ ( x ) . (63) \left((D_y+W)_{K^cK^c}-x\right)^{-1}
=
(D_y-x)_{K^cK^c}^{-1}\Theta(x). \tag*{(63)} ( ( D y + W ) K c K c − x ) − 1 = ( D y − x ) K c K c − 1 Θ ( x ) . (63) We next bound the Schur remainder
S ( x ) = W K K c ( ( D y + W ) K c K c − x ) − 1 W K c K . S(x)=W_{KK^c}\left((D_y+W)_{K^cK^c}-x\right)^{-1}W_{K^cK}. S ( x ) = W K K c ( ( D y + W ) K c K c − x ) − 1 W K c K . entrywise. Every entry of W K c K W_{K^cK} W K c K w 0 w_0 w 0
∣ ( Θ ( x ) W K c K ) j i ′ ∣ ≤ ∥ Θ ( x ) ∥ ∞ → ∞ w 0 ≤ 2 w 0 , j ∈ K c , i ′ ∈ K . \left|(\Theta(x)W_{K^cK})_{ji'}\right|
\le\|\Theta(x)\|_{\infty\to\infty}w_0
\le2w_0,\qquad j\in K^c,\ i'\in K. ∣ ( Θ ( x ) W K c K ) j i ′ ∣ ≤ ∥Θ ( x ) ∥ ∞ → ∞ w 0 ≤ 2 w 0 , j ∈ K c , i ′ ∈ K . By (3.28), for i , i ′ ∈ K i,i'\in K i , i ′ ∈ K
∣ S ( x ) i i ′ ∣ = ∣ ∑ j ∈ K c W i j y j − x ( Θ ( x ) W K c K ) j i ′ ∣ ≤ 2 w 0 2 ∑ j ∈ K c 1 ∣ y j − x ∣ ≤ 2 w 0 2 L . |S(x)_{ii'}|
=
\left|\sum_{j\in K^c}\frac{W_{ij}}{y_j-x}(\Theta(x)W_{K^cK})_{ji'}\right|
\le2w_0^2\sum_{j\in K^c}\frac{1}{|y_j-x|}
\le2w_0^2\mathcal{L}. ∣ S ( x ) i i ′ ∣ = j ∈ K c ∑ y j − x W ij ( Θ ( x ) W K c K ) j i ′ ≤ 2 w 0 2 j ∈ K c ∑ ∣ y j − x ∣ 1 ≤ 2 w 0 2 L . Bounding the operator norms by the Frobenius norms, we obtain
∥ S ( x ) ∥ ≤ ∣ K ∣ ⋅ 2 w 0 2 L ≤ χ , ∥ W K K ∥ ≤ ∣ K ∣ w 0 ≤ k 0 w 0 . \|S(x)\| \le|K|\cdot2w_0^2\mathcal{L}\le\chi,\qquad
\|W_{KK}\|\le|K|w_0\le k_0w_0. ∥ S ( x ) ∥ ≤ ∣ K ∣ ⋅ 2 w 0 2 L ≤ χ , ∥ W K K ∥ ≤ ∣ K ∣ w 0 ≤ k 0 w 0 . By (3.26), we have ∥ W K K ∥ + χ < ϑ \lVert W_{\mathcal{K}\mathcal{K}}\rVert+\chi<\vartheta ∥ W KK ∥ + χ < ϑ i 0 i_0 i 0 i 0 ′ i'_0 i 0 ′ D y + W D_y+W D y + W E − − ϑ E_- -\vartheta E − − ϑ R K ∩ R \mathcal{R}_{\mathcal{K}}\cap\mathbb{R} R K ∩ R ∣ K ∣ |\mathcal{K}| ∣ K ∣ D y + W D_y+W D y + W
It remains to show that i 0 ′ = i 0 i'_0=i_0 i 0 ′ = i 0 K ′ \mathcal{K}' K ′ ( K ′ ) c (\mathcal{K}')^c ( K ′ ) c K ′ = [ N ] \mathcal{K}'=[N] K ′ = [ N ] R K ′ ∩ R \mathcal{R}_{\mathcal{K}'}\cap\mathbb{R} R K ′ ∩ R K ′ ∈ R \mathcal{K}'\in\mathcal{R} K ′ ∈ R ∣ K ′ ∣ |\mathcal{K}'| ∣ K ′ ∣ D y + W D_y+W D y + W
Let K ′ , K ′ ′ ∈ R \mathcal{K}',\mathcal{K}''\in\mathcal{R} K ′ , K ′′ ∈ R max K ′ < min K ′ ′ \max\mathcal{K}'<\min\mathcal{K}'' max K ′ < min K ′′ max i ∈ K ′ y i ≤ min i ∈ K ′ ′ y i \max_{i\in\mathcal{K}'}y_i\le\min_{i\in\mathcal{K}''}y_i max i ∈ K ′ y i ≤ min i ∈ K ′′ y i R K ′ ∩ R \mathcal{R}_{\mathcal{K}'}\cap\mathbb{R} R K ′ ∩ R R K ′ ′ ∩ R \mathcal{R}_{\mathcal{K}''}\cap\mathbb{R} R K ′′ ∩ R N N N D y + W D_y+W D y + W E − − ϑ E_- -\vartheta E − − ϑ K ′ \mathcal{K}' K ′ max K ′ < min K \max\mathcal{K}'<\min\mathcal{K} max K ′ < min K min K − 1 = i 0 \min\mathcal{K}-1=i_0 min K − 1 = i 0
In Section 6 and Section 9, the clusters are formed by joining consecutive points at distance smaller than a threshold, and the number of points in intervals of length N − 1 N^{-1} N − 1 Figure 3.1 illustrates its proof.
Figure 3.1. The points y i y_i y i q q q R K \mathcal{R}_{\mathcal{K}} R K ϑ = q / 3 \vartheta=q/3 ϑ = q /3 q q q 2 q / 3 2q/3 2 q /3
Corollary 3.9. Let N ≥ 2 N\ge2 N ≥ 2 ε > 0 \varepsilon>0 ε > 0 q > 0 q>0 q > 0 2 N ε q < N − 1 2N^\varepsilon q<N^{-1} 2 N ε q < N − 1 y 1 ≤ ⋯ ≤ y N y_1\le\cdots\le y_N y 1 ≤ ⋯ ≤ y N [ − 4 , 4 ] [-4,4] [ − 4 , 4 ] # { i ∈ [ N ] : y i ∈ [ E , E + N − 1 ] } ≤ N ε \#\{i\in[N]:y_i\in[E,E+N^{-1}]\}\le N^\varepsilon # { i ∈ [ N ] : y i ∈ [ E , E + N − 1 ]} ≤ N ε E ∈ R E\in\mathbb{R} E ∈ R R \mathcal{R} R K \mathcal{K} K y i + 1 − y i < q y_{i+1}-y_i<q y i + 1 − y i < q i i i i , i + 1 ∈ K i,i+1\in\mathcal{K} i , i + 1 ∈ K K ∈ R \mathcal{K}\in\mathcal{R} K ∈ R ∣ K ∣ ≤ N ε |\mathcal{K}|\le N^\varepsilon ∣ K ∣ ≤ N ε y y y R \mathcal{R} R
k 0 = N ε , ϑ = q / 3 , L = 12 N ε ( q − 1 + N log N ) . (64) k_0=N^\varepsilon,\qquad\vartheta=q/3,\qquad\mathcal{L}=12N^\varepsilon(q^{-1}+N\log N). \tag*{(64)} k 0 = N ε , ϑ = q /3 , L = 12 N ε ( q − 1 + N log N ) . (64) Let W W W N × N N\times N N × N max i , j ∣ W i j ∣ ≤ w 0 \max_{i,j}|W_{ij}|\le w_0 max i , j ∣ W ij ∣ ≤ w 0
N ε w 0 ( q − 1 + N log N ) ≤ 1 / 48. (65) N^\varepsilon w_0(q^{-1}+N\log N)\le1/48. \tag*{(65)} N ε w 0 ( q − 1 + N log N ) ≤ 1/48. (65) Then for every K = { i 0 + 1 , … , i 0 + ∣ K ∣ } ∈ R \mathcal{K}=\{i_0+1,\ldots,i_0+|\mathcal{K}|\}\in\mathcal{R} K = { i 0 + 1 , … , i 0 + ∣ K ∣ } ∈ R i ∈ [ ∣ K ∣ ] i\in[|\mathcal{K}|] i ∈ [ ∣ K ∣ ]
∣ λ i 0 + i ( D y + W ) − λ i ( ( D y + W ) K K ) ∣ ≤ 24 N 2 ε w 0 2 ( q − 1 + N log N ) , (66) \left|\lambda_{i_0+i}(D_y+W)-\lambda_i\left((D_y+W)_{\mathcal{K}\mathcal{K}}\right)\right|\le24N^{2\varepsilon}w_0^2(q^{-1}+N\log N), \tag*{(66)} ∣ λ i 0 + i ( D y + W ) − λ i ( ( D y + W ) KK ) ∣ ≤ 24 N 2 ε w 0 2 ( q − 1 + N log N ) , (66) and min j ∈ K y j − q / 32 ≤ λ i 0 + i ( D y + W ) ≤ max j ∈ K y j + q / 32 \min_{j\in\mathcal{K}}y_j-q/32\le\lambda_{i_0+i}(D_y+W)\le\max_{j\in\mathcal{K}}y_j+q/32 min j ∈ K y j − q /32 ≤ λ i 0 + i ( D y + W ) ≤ max j ∈ K y j + q /32 i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] i i i i + 1 i+1 i + 1 λ i + 1 ( D y + W ) − λ i ( D y + W ) ≥ q / 2 \lambda_{i+1}(D_y+W)-\lambda_i(D_y+W)\ge q/2 λ i + 1 ( D y + W ) − λ i ( D y + W ) ≥ q /2
Proof. We first bound the size of the clusters. Let K = { i 0 + 1 , … , i 0 + ∣ K ∣ } ∈ R \mathcal{K}=\{i_0+1,\ldots,i_0+|\mathcal{K}|\}\in\mathcal{R} K = { i 0 + 1 , … , i 0 + ∣ K ∣ } ∈ R ∣ K ∣ > N ε |\mathcal{K}|>N^\varepsilon ∣ K ∣ > N ε i 1 = i 0 + ⌊ N ε ⌋ + 1 i_1=i_0+\lfloor N^\varepsilon\rfloor+1 i 1 = i 0 + ⌊ N ε ⌋ + 1 i 1 ∈ K i_1\in\mathcal{K} i 1 ∈ K K \mathcal{K} K q q q
y i 1 − y i 0 + 1 < ( i 1 − i 0 − 1 ) q ≤ N ε q < N − 1 . y_{i_1}-y_{i_0+1}<(i_1-i_0-1)q\leq N^\varepsilon q<N^{-1}. y i 1 − y i 0 + 1 < ( i 1 − i 0 − 1 ) q ≤ N ε q < N − 1 . Thus the interval [ y i 0 + 1 , y i 0 + 1 + N − 1 ] [y_{i_0+1},y_{i_0+1}+N^{-1}] [ y i 0 + 1 , y i 0 + 1 + N − 1 ] ⌊ N ε ⌋ + 1 > N ε \lfloor N^\varepsilon\rfloor+1>N^\varepsilon ⌊ N ε ⌋ + 1 > N ε y i 0 + 1 , … , y i 1 y_{i_0+1},\ldots,y_{i_1} y i 0 + 1 , … , y i 1 ∣ K ∣ ≤ N ε = k 0 |\mathcal{K}|\leq N^\varepsilon=k_0 ∣ K ∣ ≤ N ε = k 0
We next bound the distances from a rectangle to the points outside its cluster. Let K ∈ R \mathcal{K}\in\mathcal{R} K ∈ R E − = min i ∈ K y i E_-=\min_{i\in\mathcal{K}}y_i E − = min i ∈ K y i E + = max i ∈ K y i E_+=\max_{i\in\mathcal{K}}y_i E + = max i ∈ K y i j ∉ K j\notin\mathcal{K} j ∈ / K R \mathcal{R} R y i + 1 − y i ≥ q y_{i+1}-y_i\geq q y i + 1 − y i ≥ q i i i i + 1 i+1 i + 1 j < min K j<\min\mathcal{K} j < min K min K − 1 \min\mathcal{K}-1 min K − 1 min K \min\mathcal{K} min K j > max K j>\max\mathcal{K} j > max K max K \max\mathcal{K} max K max K + 1 \max\mathcal{K}+1 max K + 1 E − = y min K E_-=y_{\min\mathcal{K}} E − = y m i n K E + = y max K E_+=y_{\max\mathcal{K}} E + = y m a x K
y j ≤ y min K − 1 ≤ E − − q ( j < min K ) , y j ≥ y max K + 1 ≥ E + + q ( j > max K ) . y_j\leq y_{\min\mathcal{K}-1}\leq E_--q\quad(j<\min\mathcal{K}),\qquad y_j\geq y_{\max\mathcal{K}+1}\geq E_++q\quad(j>\max\mathcal{K}). y j ≤ y m i n K − 1 ≤ E − − q ( j < min K ) , y j ≥ y m a x K + 1 ≥ E + + q ( j > max K ) . Consequently, for every w ∈ R K w\in\mathcal{R}_{\mathcal{K}} w ∈ R K
∣ y j − w ∣ ≥ ∣ y j − Re w ∣ ≥ q − q / 3 = 2 q / 3 for all j ∉ K . (67) |y_j-w|\geq|y_j-\operatorname{Re}w|\geq q-q/3=2q/3\qquad\text{for all }j\notin\mathcal{K}. \tag*{(67)} ∣ y j − w ∣ ≥ ∣ y j − Re w ∣ ≥ q − q /3 = 2 q /3 for all j ∈ / K . (67) We next check that the rectangles are disjoint. Recall that ϑ = q / 3 \vartheta=q/3 ϑ = q /3 K ′ ∈ R \mathcal{K}'\in\mathcal{R} K ′ ∈ R min K ′ > max K \min\mathcal{K}'>\max\mathcal{K} min K ′ > max K min i ∈ K ′ y i ≥ E + + q \min_{i\in\mathcal{K}'}y_i\geq E_++q min i ∈ K ′ y i ≥ E + + q R K ′ \mathcal{R}_{\mathcal{K}'} R K ′ E + + 2 q / 3 E_++2q/3 E + + 2 q /3 R K \mathcal{R}_{\mathcal{K}} R K E + + q / 3 E_++q/3 E + + q /3
We now bound the harmonic sum. Fix K ∈ R \mathcal{K}\in\mathcal{R} K ∈ R w ∈ R K w\in\mathcal{R}_{\mathcal{K}} w ∈ R K x = Re w x=\operatorname{Re}w x = Re w ∣ x ∣ < 5 |x|<5 ∣ x ∣ < 5 ∣ y i ∣ ≤ 4 |y_i|\leq4 ∣ y i ∣ ≤ 4 i i i q < N − 1 ≤ 1 q<N^{-1}\leq1 q < N − 1 ≤ 1 ∣ y j − x ∣ < 9 |y_j-x|<9 ∣ y j − x ∣ < 9 j j j n ≥ 0 n\geq0 n ≥ 0
M n = { j ∈ [ N ] ∖ K : n / N ≤ ∣ y j − x ∣ < ( n + 1 ) / N } . \mathcal{M}_n=\{j\in[N]\setminus\mathcal{K}:n/N\leq|y_j-x|<(n+1)/N\}. M n = { j ∈ [ N ] ∖ K : n / N ≤ ∣ y j − x ∣ < ( n + 1 ) / N } . The points y j y_j y j j ∈ M n j\in\mathcal{M}_n j ∈ M n [ x − ( n + 1 ) / N , x − n / N ] [x-(n+1)/N,x-n/N] [ x − ( n + 1 ) / N , x − n / N ] [ x + n / N , x + ( n + 1 ) / N ] [x+n/N,x+(n+1)/N] [ x + n / N , x + ( n + 1 ) / N ] N − 1 N^{-1} N − 1 ∣ M n ∣ ≤ 2 N ε |\mathcal{M}_n|\leq2N^\varepsilon ∣ M n ∣ ≤ 2 N ε j ∈ M 0 j \in\mathcal{M}_0 j ∈ M 0 ∣ y j − x ∣ ≥ 2 q / 3 |y_j-x| \ge2q/3 ∣ y j − x ∣ ≥ 2 q /3 (67) , and the labels j ∈ M n j \in\mathcal{M}_n j ∈ M n n ≥ 1 n \ge1 n ≥ 1 ∣ y j − x ∣ ≥ n / N |y_j-x| \ge n/N ∣ y j − x ∣ ≥ n / N M n \mathcal{M}_n M n n ≥ 9 N n \ge9N n ≥ 9 N
∑ j ∉ K 1 ∣ y j − w ∣ ≤ ∑ n ≥ 0 ∑ j ∈ M n 1 ∣ y j − x ∣ ≤ 2 N ε ⋅ 3 2 q + ∑ n = 1 9 N 2 N ε ⋅ N n ≤ 3 N ε q + 12 N 1 + ε log N ≤ L . \sum_{j\notin K}\frac{1}{|y_j-w|} \le\sum_{n\ge0}\sum_{j\in\mathcal{M}_n}\frac{1}{|y_j-x|} \le2N^{\varepsilon}\cdot\frac{3}{2q}+\sum_{n=1}^{9N}2N^{\varepsilon}\cdot\frac{N}{n} \le\frac{3N^{\varepsilon}}{q}+12N^{1+\varepsilon}\log N \le\mathcal{L}. j ∈ / K ∑ ∣ y j − w ∣ 1 ≤ n ≥ 0 ∑ j ∈ M n ∑ ∣ y j − x ∣ 1 ≤ 2 N ε ⋅ 2 q 3 + n = 1 ∑ 9 N 2 N ε ⋅ n N ≤ q 3 N ε + 12 N 1 + ε log N ≤ L . In the third inequality, we used the bound
∑ n = 1 9 N 1 n ≤ 1 + log ( 9 N ) ≤ 6 log N ( N ≥ 2 ) . \sum_{n=1}^{9N}\frac{1}{n}\le1+\log(9N)\le6\log N \qquad(N\ge2). n = 1 ∑ 9 N n 1 ≤ 1 + log ( 9 N ) ≤ 6 log N ( N ≥ 2 ) . Hence the hypotheses of Lemma 3.8 on y y y K \mathcal{K} K (64) .
We now verify (61) . By (64) and (65) , we have
w 0 L = 12 N ε w 0 ( q − 1 + N log N ) ≤ 1 4 , k 0 w 0 = q ⋅ N ε w 0 q − 1 ≤ q 48 , w_0\mathcal{L}=12N^{\varepsilon}w_0(q^{-1}+N\log N)\le\frac{1}{4},\qquad k_0w_0=q\cdot N^{\varepsilon}w_0q^{-1}\le\frac{q}{48}, w 0 L = 12 N ε w 0 ( q − 1 + N log N ) ≤ 4 1 , k 0 w 0 = q ⋅ N ε w 0 q − 1 ≤ 48 q , 2 k 0 w 0 2 L = 2 ( k 0 w 0 ) ( w 0 L ) ≤ q 96 , 2k_0w_0^2\mathcal{L}=2(k_0w_0)(w_0\mathcal{L})\le\frac{q}{96}, 2 k 0 w 0 2 L = 2 ( k 0 w 0 ) ( w 0 L ) ≤ 96 q , and hence
k 0 w 0 + 2 k 0 w 0 2 L ≤ q 32 < ϑ . k_0w_0+2k_0w_0^2\mathcal{L}\le\frac{q}{32}<\vartheta. k 0 w 0 + 2 k 0 w 0 2 L ≤ 32 q < ϑ . Therefore Lemma 3.8 applies. By (64) , we have
2 k 0 w 0 2 L = 24 N 2 ε w 0 2 ( q − 1 + N log N ) , 2k_0w_0^2\mathcal{L}=24N^{2\varepsilon}w_0^2(q^{-1}+N\log N), 2 k 0 w 0 2 L = 24 N 2 ε w 0 2 ( q − 1 + N log N ) , so the lemma gives (66) and the location of the eigenvalues within distance q / 32 q/32 q /32 [ min j ∈ K y j , max j ∈ K y j ] [\min_{j\in\mathcal{K}}y_j,\max_{j\in\mathcal{K}}y_j] [ min j ∈ K y j , max j ∈ K y j ]
Finally, if i i i i + 1 i+1 i + 1 y i + 1 − y i ≥ q y_{i+1}-y_i\ge q y i + 1 − y i ≥ q
λ i + 1 ( D y + W ) − λ i ( D y + W ) ≥ ( y i + 1 − q / 32 ) − ( y i + q / 32 ) ≥ q − q / 16 ≥ q / 2. \lambda_{i+1}(D_y+W)-\lambda_i(D_y+W)\ge(y_{i+1}-q/32)-(y_i+q/32)\ge q-q/16\ge q/2. λ i + 1 ( D y + W ) − λ i ( D y + W ) ≥ ( y i + 1 − q /32 ) − ( y i + q /32 ) ≥ q − q /16 ≥ q /2. This completes the proof.
A Gaussian kernel estimate
We will use the following functions throughout the paper. For every integer p ≥ 1 p\ge1 p ≥ 1 f p : R → ( 0 , 1 ] f_p:\mathbb{R}\to(0,1] f p : R → ( 0 , 1 ]
f p ( y ) = ( 1 + y 2 ) − p , y ∈ R . (68) f_p(y)=(1+y^2)^{-p},\qquad y\in\mathbb{R}. \tag*{(68)} f p ( y ) = ( 1 + y 2 ) − p , y ∈ R . (68) In Section 6, the gap of a 2 × 2 2\times2 2 × 2 R 2 \mathbb{R}^2 R 2 δ \delta δ f p f_p f p n = 3 n=3 n = 3 n ≥ 1 n\ge1 n ≥ 1 v > 0 v>0 v > 0
g v ( z ) = ( 2 π v ) − n / 2 exp ( − ∣ z ∣ 2 2 v ) , z ∈ R n , (69) g_v(z)=(2\pi v)^{-n/2}\exp\left(-\frac{|z|^2}{2v}\right),\qquad z\in\mathbb{R}^n, \tag*{(69)} g v ( z ) = ( 2 π v ) − n /2 exp ( − 2 v ∣ z ∣ 2 ) , z ∈ R n , (69) denote the density of the centered Gaussian distribution on R n \mathbb{R}^n R n v Id n v\operatorname{Id}_n v Id n n n n
Lemma 3.10. Let n ∈ { 2 , 3 } n\in\{2,3\} n ∈ { 2 , 3 } p > ( n + 1 ) / 2 p>(n+1)/2 p > ( n + 1 ) /2 C > 0 C>0 C > 0 n n n p p p x ≥ 0 x\ge0 x ≥ 0 ϱ ∈ [ 1 / 2 , 2 ] \varrho\in[1/2,2] ϱ ∈ [ 1/2 , 2 ] v , v 0 ∈ [ 1 / 8 , 8 ] v,v_0\in[1/8,8] v , v 0 ∈ [ 1/8 , 8 ] δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] G x G_x G x R n \mathbb{R}^n R n ( ϱ x , 0 ) ∈ R × R n − 1 (\varrho x,0)\in\mathbb{R}\times\mathbb{R}^{n-1} ( ϱ x , 0 ) ∈ R × R n − 1 v Id n v\operatorname{Id}_n v Id n f : [ 0 , ∞ ) → R f:[0,\infty)\to\mathbb{R} f : [ 0 , ∞ ) → R ∣ f ∣ ≤ f p |f|\le f_p ∣ f ∣ ≤ f p
∣ δ − n E f ( ∣ G x ∣ / δ ) − g v 0 ( x , 0 ) ∫ R n f ( ∣ y ∣ ) d y ∣ ≤ C f p ( x ) ( δ + ∣ ϱ − 1 ∣ + ∣ v − v 0 ∣ ) . (70) \left|\delta^{-n}\mathbb{E}f(|G_x|/\delta)-g_{v_0}(x,0)\int_{\mathbb{R}^n}f(|y|)\,\mathrm{d}y\right|
\le Cf_p(x)\left(\delta+|\varrho-1|+|v-v_0|\right). \tag*{(70)} δ − n E f ( ∣ G x ∣/ δ ) − g v 0 ( x , 0 ) ∫ R n f ( ∣ y ∣ ) d y ≤ C f p ( x ) ( δ + ∣ ϱ − 1∣ + ∣ v − v 0 ∣ ) . (70) In particular, E f p ( ∣ G x ∣ / δ ) ≤ C δ n f p ( x ) \mathbb{E}f_p(|G_x|/\delta)\le C\delta^n f_p(x) E f p ( ∣ G x ∣/ δ ) ≤ C δ n f p ( x )
Proof. Throughout the proof, C C C n n n p p p x ^ = ( x , 0 ) \hat{x}=(x,0) x ^ = ( x , 0 ) 0 ∈ R n − 1 0\in\mathbb{R}^{n-1} 0 ∈ R n − 1
g v ( ϱ x ^ ) ≤ C e − x 2 / 64 , ∣ ∇ g v ( z ) ∣ = ∣ z ∣ v g v ( z ) ≤ C e − ∣ z ∣ 2 / 32 ( z ∈ R n ) , (71) g_v(\varrho\hat{x})\le Ce^{-x^2/64},\qquad|\nabla g_v(z)|=\frac{|z|}{v}g_v(z)\le Ce^{-|z|^2/32}\qquad(z\in\mathbb{R}^n), \tag*{(71)} g v ( ϱ x ^ ) ≤ C e − x 2 /64 , ∣∇ g v ( z ) ∣ = v ∣ z ∣ g v ( z ) ≤ C e − ∣ z ∣ 2 /32 ( z ∈ R n ) , (71) which hold for ϱ ∈ [ 1 / 2 , 2 ] \varrho\in[1/2,2] ϱ ∈ [ 1/2 , 2 ] v ∈ [ 1 / 8 , 8 ] v\in[1/8,8] v ∈ [ 1/8 , 8 ] e − c x 2 ≤ C f p ( x ) e^{-cx^2}\le Cf_p(x) e − c x 2 ≤ C f p ( x ) c > 0 c>0 c > 0 2 p − 1 > n 2p-1>n 2 p − 1 > n
∫ R n f p ( ∣ y ∣ ) d y + ∫ R n ∣ y ∣ f p ( ∣ y ∣ ) d y ≤ C . (72) \int_{\mathbb{R}^n}f_p(|y|)\,\mathrm{d}y+\int_{\mathbb{R}^n}|y|f_p(|y|)\,\mathrm{d}y\le C. \tag*{(72)} ∫ R n f p ( ∣ y ∣ ) d y + ∫ R n ∣ y ∣ f p ( ∣ y ∣ ) d y ≤ C . (72) (The assumption p > ( n + 1 ) / 2 p>(n+1)/2 p > ( n + 1 ) /2 2 p − n > 1 2p-n>1 2 p − n > 1
Recall that G x G_x G x ϱ x ^ \varrho\hat{x} ϱ x ^ v Id n v\operatorname{Id}_n v Id n z ↦ g v ( z − ϱ x ^ ) z\mapsto g_v(z-\varrho\hat{x}) z ↦ g v ( z − ϱ x ^ ) z = δ y z=\delta y z = δ y E f ( ∣ G x ∣ / δ ) \mathbb{E}f(|G_x|/\delta) E f ( ∣ G x ∣/ δ )
δ − n E f ( ∣ G x ∣ / δ ) = δ − n ∫ R n f ( ∣ z ∣ / δ ) g v ( z − ϱ x ^ ) d z = ∫ R n f ( ∣ y ∣ ) g v ( δ y − ϱ x ^ ) d y . \delta^{-n}\mathbb{E}f(|G_x|/\delta)=\delta^{-n}\int_{\mathbb{R}^n}f(|z|/\delta)g_v(z-\varrho\hat{x})\,\mathrm{d}z=\int_{\mathbb{R}^n}f(|y|)g_v(\delta y-\varrho\hat{x})\,\mathrm{d}y. δ − n E f ( ∣ G x ∣/ δ ) = δ − n ∫ R n f ( ∣ z ∣/ δ ) g v ( z − ϱ x ^ ) d z = ∫ R n f ( ∣ y ∣ ) g v ( δ y − ϱ x ^ ) d y . Using g v ( − ϱ x ^ ) = g v ( ϱ x ^ ) g_v(-\varrho\hat{x})=g_v(\varrho\hat{x}) g v ( − ϱ x ^ ) = g v ( ϱ x ^ ) ∣ f ∣ ≤ f p |f|\le f_p ∣ f ∣ ≤ f p
∫ ∣ y ∣ ≤ δ − 1 f p ( ∣ y ∣ ) ∣ g v ( δ y − ϱ x ^ ) − g v ( − ϱ x ^ ) ∣ d y + ∫ ∣ y ∣ > δ − 1 f p ( ∣ y ∣ ) ( g v ( δ y − ϱ x ^ ) + g v ( ϱ x ^ ) ) d y + ∣ g v ( ϱ x ^ ) − g v 0 ( x ^ ) ∣ ∫ R n f p ( ∣ y ∣ ) d y . (73) \int_{|y|\le\delta^{-1}}f_p(|y|)\left|g_v(\delta y-\varrho\hat{x})-g_v(-\varrho\hat{x})\right|\,\mathrm{d}y
+\int_{|y|>\delta^{-1}}f_p(|y|)\left(g_v(\delta y-\varrho\hat{x})+g_v(\varrho\hat{x})\right)\,\mathrm{d}y
+\left|g_v(\varrho\hat{x})-g_{v_0}(\hat{x})\right|\int_{\mathbb{R}^n}f_p(|y|)\,\mathrm{d}y. \tag*{(73)} ∫ ∣ y ∣ ≤ δ − 1 f p ( ∣ y ∣ ) ∣ g v ( δ y − ϱ x ^ ) − g v ( − ϱ x ^ ) ∣ d y + ∫ ∣ y ∣ > δ − 1 f p ( ∣ y ∣ ) ( g v ( δ y − ϱ x ^ ) + g v ( ϱ x ^ ) ) d y + ∣ g v ( ϱ x ^ ) − g v 0 ( x ^ ) ∣ ∫ R n f p ( ∣ y ∣ ) d y . (73) We bound the three terms in turn.
We begin with the first term. Let ∣ y ∣ ≤ δ − 1 |y| \le\delta^{-1} ∣ y ∣ ≤ δ − 1 [ − ϱ x ^ , δ y − ϱ x ^ ] [-\varrho\hat{x},\delta y-\varrho\hat{x}] [ − ϱ x ^ , δ y − ϱ x ^ ] z ′ z' z ′
∣ z ′ ∣ ≥ max ( ϱ x − 1 , 0 ) ≥ max ( x / 2 − 1 , 0 ) , and hence ∣ z ′ ∣ 2 ≥ x 2 8 − 1. |z'| \ge\max(\varrho x-1,0) \ge\max(x/2-1,0), \qquad\text{and hence} \qquad|z'|^2 \ge\frac{x^2}{8}-1. ∣ z ′ ∣ ≥ max ( ϱ x − 1 , 0 ) ≥ max ( x /2 − 1 , 0 ) , and hence ∣ z ′ ∣ 2 ≥ 8 x 2 − 1. where we used ( a − b ) 2 ≥ a 2 / 2 − b 2 (a-b)^2 \ge a^2/2-b^2 ( a − b ) 2 ≥ a 2 /2 − b 2 (71) ,
∣ g v ( δ y − ϱ x ^ ) − g v ( − ϱ x ^ ) ∣ ≤ δ ∣ y ∣ sup z ′ ∈ [ − ϱ x ^ , δ y − ϱ x ^ ] ∣ ∇ g v ( z ′ ) ∣ ≤ C δ ∣ y ∣ e − x 2 / 256 . |g_v(\delta y-\varrho\hat{x})-g_v(-\varrho\hat{x})| \le\delta|y| \sup_{z'\in[-\varrho\hat{x},\delta y-\varrho\hat{x}]}|\nabla g_v(z')| \le C\delta|y|e^{-x^2/256}. ∣ g v ( δ y − ϱ x ^ ) − g v ( − ϱ x ^ ) ∣ ≤ δ ∣ y ∣ z ′ ∈ [ − ϱ x ^ , δ y − ϱ x ^ ] sup ∣∇ g v ( z ′ ) ∣ ≤ C δ ∣ y ∣ e − x 2 /256 . Integrating against f p ( ∣ y ∣ ) f_p(|y|) f p ( ∣ y ∣ ) (72) , we find that the first term in (73) is bounded by
C δ e − x 2 / 256 ∫ R n ∣ y ∣ f p ( ∣ y ∣ ) d y ≤ C δ e − x 2 / 256 ≤ C δ f p ( x ) . C\delta e^{-x^2/256}\int_{\mathbb{R}^n}|y|f_p(|y|)\,dy \le C\delta e^{-x^2/256} \le C\delta f_p(x). C δ e − x 2 /256 ∫ R n ∣ y ∣ f p ( ∣ y ∣ ) d y ≤ C δ e − x 2 /256 ≤ C δ f p ( x ) . The second term requires more care, since g v ( δ y − ϱ x ^ ) g_v(\delta y-\varrho\hat{x}) g v ( δ y − ϱ x ^ ) δ y \delta y δ y ϱ x ^ \varrho\hat{x} ϱ x ^ x x x f p ( ∣ y ∣ ) f_p(|y|) f p ( ∣ y ∣ ) g v ( ϱ x ^ ) g_v(\varrho\hat{x}) g v ( ϱ x ^ ) f p ( ∣ y ∣ ) ≤ ∣ y ∣ − 2 p f_p(|y|)\le|y|^{-2p} f p ( ∣ y ∣ ) ≤ ∣ y ∣ − 2 p
g v ( ϱ x ^ ) ∫ ∣ y ∣ > δ − 1 f p ( ∣ y ∣ ) d y ≤ C e − x 2 / 64 ∫ ∣ y ∣ > δ − 1 ∣ y ∣ − 2 p d y ≤ C e − x 2 / 64 δ 2 p − n ≤ C δ f p ( x ) , g_v(\varrho\hat{x})\int_{|y|>\delta^{-1}}f_p(|y|)\,dy
\le Ce^{-x^2/64}\int_{|y|>\delta^{-1}}|y|^{-2p}\,dy
\le Ce^{-x^2/64}\delta^{2p-n}
\le C\delta f_p(x), g v ( ϱ x ^ ) ∫ ∣ y ∣ > δ − 1 f p ( ∣ y ∣ ) d y ≤ C e − x 2 /64 ∫ ∣ y ∣ > δ − 1 ∣ y ∣ − 2 p d y ≤ C e − x 2 /64 δ 2 p − n ≤ C δ f p ( x ) , where we used 2 p − n > 1 2p-n>1 2 p − n > 1 δ ≤ 1 \delta\le1 δ ≤ 1 g v ( δ y − ϱ x ^ ) g_v(\delta y-\varrho\hat{x}) g v ( δ y − ϱ x ^ ) z = δ y z=\delta y z = δ y
f p ( ∣ z ∣ / δ ) ≤ ( δ / ∣ z ∣ ) 2 p ≤ 2 p δ 2 p ( 1 + ∣ z ∣ 2 ) − p for ∣ z ∣ > 1. f_p(|z|/\delta)\le(\delta/|z|)^{2p}\le2^p\delta^{2p}(1+|z|^2)^{-p}
\qquad\text{for }|z|>1. f p ( ∣ z ∣/ δ ) ≤ ( δ /∣ z ∣ ) 2 p ≤ 2 p δ 2 p ( 1 + ∣ z ∣ 2 ) − p for ∣ z ∣ > 1. This gives
∫ ∣ y ∣ > δ − 1 f p ( ∣ y ∣ ) g v ( δ y − ϱ x ^ ) d y = δ − n ∫ ∣ z ∣ > 1 f p ( ∣ z ∣ / δ ) g v ( z − ϱ x ^ ) d z \int_{|y|>\delta^{-1}}f_p(|y|)g_v(\delta y-\varrho\hat{x})\,dy
=\delta^{-n}\int_{|z|>1}f_p(|z|/\delta)g_v(z-\varrho\hat{x})\,dz ∫ ∣ y ∣ > δ − 1 f p ( ∣ y ∣ ) g v ( δ y − ϱ x ^ ) d y = δ − n ∫ ∣ z ∣ > 1 f p ( ∣ z ∣/ δ ) g v ( z − ϱ x ^ ) d z ≤ 2 p δ 2 p − n ∫ R n g v ( z − ϱ x ^ ) ( 1 + ∣ z ∣ 2 ) p d z . \le2^p\delta^{2p-n}\int_{\mathbb{R}^n}\frac{g_v(z-\varrho\hat{x})}{(1+|z|^2)^p}\,dz. ≤ 2 p δ 2 p − n ∫ R n ( 1 + ∣ z ∣ 2 ) p g v ( z − ϱ x ^ ) d z . To bound the last integral uniformly in x x x
1 + ∣ a ∣ 2 ≤ 1 + ( ∣ a − b ∣ + ∣ b ∣ ) 2 ≤ 2 ( 1 + ∣ a − b ∣ 2 ) ( 1 + ∣ b ∣ 2 ) , a , b ∈ R n , 1+|a|^2\le1+(|a-b|+|b|)^2\le2(1+|a-b|^2)(1+|b|^2),\qquad a,b\in\mathbb{R}^n, 1 + ∣ a ∣ 2 ≤ 1 + ( ∣ a − b ∣ + ∣ b ∣ ) 2 ≤ 2 ( 1 + ∣ a − b ∣ 2 ) ( 1 + ∣ b ∣ 2 ) , a , b ∈ R n , with a = ϱ x ^ a=\varrho\hat{x} a = ϱ x ^ b = z b=z b = z 1 + ϱ 2 x 2 ≥ ( 1 + x 2 ) / 4 1+\varrho^2x^2\ge(1+x^2)/4 1 + ϱ 2 x 2 ≥ ( 1 + x 2 ) /4
( 1 + ∣ z ∣ 2 ) − p ≤ 2 p ( 1 + ∣ z − ϱ x ^ ∣ 2 ) p ( 1 + ϱ 2 x 2 ) − p ≤ 8 p ( 1 + ∣ z − ϱ x ^ ∣ 2 ) p f p ( x ) , (1+|z|^2)^{-p}\le2^p(1+|z-\varrho\hat{x}|^2)^p(1+\varrho^2x^2)^{-p}
\le8^p(1+|z-\varrho\hat{x}|^2)^pf_p(x), ( 1 + ∣ z ∣ 2 ) − p ≤ 2 p ( 1 + ∣ z − ϱ x ^ ∣ 2 ) p ( 1 + ϱ 2 x 2 ) − p ≤ 8 p ( 1 + ∣ z − ϱ x ^ ∣ 2 ) p f p ( x ) , and therefore
∫ R n g v ( z − ϱ x ^ ) ( 1 + ∣ z ∣ 2 ) p d z ≤ 8 p f p ( x ) ∫ R n ( 1 + ∣ z ′ ∣ 2 ) p g v ( z ′ ) d z ′ ≤ C f p ( x ) . \int_{\mathbb{R}^n}\frac{g_v(z-\varrho\hat{x})}{(1+|z|^2)^p}\,dz
\le8^pf_p(x)\int_{\mathbb{R}^n}(1+|z'|^2)^pg_v(z')\,dz'
\le Cf_p(x). ∫ R n ( 1 + ∣ z ∣ 2 ) p g v ( z − ϱ x ^ ) d z ≤ 8 p f p ( x ) ∫ R n ( 1 + ∣ z ′ ∣ 2 ) p g v ( z ′ ) d z ′ ≤ C f p ( x ) . Using δ 2 p − n ≤ δ \delta^{2p-n} \le\delta δ 2 p − n ≤ δ (73) is bounded by C δ f p ( x ) C\delta f_p(x) C δ f p ( x )
Next, we bound the third term. We consider
g v ′ ( ϱ ′ x ^ ) = ( 2 π v ′ ) − n / 2 exp ( − ϱ ′ 2 x 2 2 v ′ ) g_{v'}(\varrho'\hat{x}) = (2\pi v')^{-n/2}\exp\left(-\frac{\varrho'^{2}x^{2}}{2v'}\right) g v ′ ( ϱ ′ x ^ ) = ( 2 π v ′ ) − n /2 exp ( − 2 v ′ ϱ ′ 2 x 2 ) as a function of ( ϱ ′ , v ′ ) (\varrho',v') ( ϱ ′ , v ′ ) [ 1 / 2 , 2 ] × [ 1 / 8 , 8 ] [1/2,2]\times[1/8,8] [ 1/2 , 2 ] × [ 1/8 , 8 ]
∂ ϱ ′ g v ′ ( ϱ ′ x ^ ) = − ϱ ′ x 2 v ′ g v ′ ( ϱ ′ x ^ ) , ∂ v ′ g v ′ ( ϱ ′ x ^ ) = ( − n 2 v ′ + ϱ ′ 2 x 2 2 v ′ 2 ) g v ′ ( ϱ ′ x ^ ) . \partial_{\varrho'}g_{v'}(\varrho'\hat{x})=-\frac{\varrho'x^{2}}{v'}g_{v'}(\varrho'\hat{x}),\qquad\partial_{v'}g_{v'}(\varrho'\hat{x})=\left(-\frac{n}{2v'}+\frac{\varrho'^{2}x^{2}}{2v'^{2}}\right)g_{v'}(\varrho'\hat{x}). ∂ ϱ ′ g v ′ ( ϱ ′ x ^ ) = − v ′ ϱ ′ x 2 g v ′ ( ϱ ′ x ^ ) , ∂ v ′ g v ′ ( ϱ ′ x ^ ) = ( − 2 v ′ n + 2 v ′ 2 ϱ ′ 2 x 2 ) g v ′ ( ϱ ′ x ^ ) . Using (71) , we see that on this rectangle
∣ ∂ ϱ ′ g v ′ ( ϱ ′ x ^ ) ∣ + ∣ ∂ v ′ g v ′ ( ϱ ′ x ^ ) ∣ ≤ C ( 1 + x 2 ) e − x 2 / 64 ≤ C f p ( x ) . \left|\partial_{\varrho'}g_{v'}(\varrho'\hat{x})\right|+\left|\partial_{v'}g_{v'}(\varrho'\hat{x})\right|\le C(1+x^{2})e^{-x^{2}/64}\le Cf_p(x). ∣ ∂ ϱ ′ g v ′ ( ϱ ′ x ^ ) ∣ + ∣ ∂ v ′ g v ′ ( ϱ ′ x ^ ) ∣ ≤ C ( 1 + x 2 ) e − x 2 /64 ≤ C f p ( x ) . The rectangle is convex, so the mean value theorem along the segment from ( 1 , v 0 ) (1,v_0) ( 1 , v 0 ) ( ϱ , v ) (\varrho,v) ( ϱ , v )
∣ g v ( ϱ x ^ ) − g v 0 ( x ^ ) ∣ ≤ C f p ( x ) ( ∣ ϱ − 1 ∣ + ∣ v − v 0 ∣ ) . \left|g_v(\varrho\hat{x})-g_{v_0}(\hat{x})\right|\le Cf_p(x)\left(|\varrho-1|+|v-v_0|\right). ∣ g v ( ϱ x ^ ) − g v 0 ( x ^ ) ∣ ≤ C f p ( x ) ( ∣ ϱ − 1∣ + ∣ v − v 0 ∣ ) . By (72) , the third term in (73) is therefore bounded by C f p ( x ) ( ∣ ϱ − 1 ∣ + ∣ v − v 0 ∣ ) Cf_p(x)(|\varrho-1|+|v-v_0|) C f p ( x ) ( ∣ ϱ − 1∣ + ∣ v − v 0 ∣ )
Combining the bounds for the three terms in (73) , we obtain (70) . For the final claim, we apply (70) with f = f p f=f_p f = f p
δ − n E f p ( ∣ G x ∣ / δ ) ≤ g v 0 ( x ^ ) ∫ R n f p ( ∣ y ∣ ) d y + C f p ( x ) ( δ + ∣ ϱ − 1 ∣ + ∣ v − v 0 ∣ ) ≤ C f p ( x ) . \delta^{-n}\mathbb{E}f_p(|G_x|/\delta)\le g_{v_0}(\hat{x})\int_{\mathbb{R}^n}f_p(|y|)\,dy+Cf_p(x)\left(\delta+|\varrho-1|+|v-v_0|\right)\le Cf_p(x). δ − n E f p ( ∣ G x ∣/ δ ) ≤ g v 0 ( x ^ ) ∫ R n f p ( ∣ y ∣ ) d y + C f p ( x ) ( δ + ∣ ϱ − 1∣ + ∣ v − v 0 ∣ ) ≤ C f p ( x ) . In the last step, we used (72) , the bound g v 0 ( x ^ ) ≤ C f p ( x ) g_{v_0}(\hat{x})\le Cf_p(x) g v 0 ( x ^ ) ≤ C f p ( x ) v 0 ∈ [ 1 / 8 , 8 ] v_0\in[1/8,8] v 0 ∈ [ 1/8 , 8 ] δ + ∣ ϱ − 1 ∣ + ∣ v − v 0 ∣ ≤ 10 \delta+|\varrho-1|+|v-v_0|\le10 δ + ∣ ϱ − 1∣ + ∣ v − v 0 ∣ ≤ 10
Reduction to three propositions
In this section, we prove Theorem 1.2 assuming three propositions. The first, Proposition 4.1 , counts pairs of eigenvalues at a scale slightly above the critical scale N − 3 / 2 N^{-3/2} N − 3/2 Proposition 4.4 , is a local limit theorem for the linearized increments produced by resampling a small number of entries of H H H Proposition 4.5 , states that the small gaps of the resampled matrix are close to the linearized gaps; its proof is given in Section 7. We state the propositions in Section 4.1 and § 5.2 , and we deduce Theorem 1.2 from them in § 5.3 .
Throughout the rest of the paper, we use the scales
d = N − 3 / 2 , τ = N 1 / 100 , ℓ = d τ 1 / 2 = N − 299 / 200 , s = N − 7 / 5 . (74) d = N^{-3/2}, \qquad\tau= N^{1/100}, \qquad\ell= d\tau^{1/2} = N^{-299/200}, \qquad s = N^{-7/5}. \tag*{(74)} d = N − 3/2 , τ = N 1/100 , ℓ = d τ 1/2 = N − 299/200 , s = N − 7/5 . (74) The smallest gaps in Theorem 1.2 are of order d d d ℓ α = N − 1 − α \ell_\alpha= N^{-1-\alpha} ℓ α = N − 1 − α M α = N 1 − 2 α M_\alpha= N^{1-2\alpha} M α = N 1 − 2 α α ∗ = 99 / 200 \alpha_* = 99/200 α ∗ = 99/200 ℓ = ℓ α ∗ \ell= \ell_{\alpha_*} ℓ = ℓ α ∗ M α ∗ = N 3 ℓ 2 = τ M_{\alpha_*} = N^3\ell^2 = \tau M α ∗ = N 3 ℓ 2 = τ
Heuristically, a bulk interval contains of order τ \tau τ ℓ \ell ℓ s s s Proposition 4.5 . These scales satisfy
d ℓ = N − 1 / 200 , ℓ s = N − 19 / 200 , N s = N − 2 / 5 . (75) \frac{d}{\ell} = N^{-1/200}, \qquad\frac{\ell}{s} = N^{-19/200}, \qquad Ns = N^{-2/5}. \tag*{(75)} ℓ d = N − 1/200 , s ℓ = N − 19/200 , N s = N − 2/5 . (75) Recall that g i = λ i + 1 − λ i g_i = \lambda_{i+1} - \lambda_i g i = λ i + 1 − λ i E i = ( λ i + λ i + 1 ) / 2 E_i = (\lambda_i + \lambda_{i+1})/2 E i = ( λ i + λ i + 1 ) /2 i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ]
Pair counts at the scale $\ell`
Recall the functions f p ( y ) = ( 1 + y 2 ) − p f_p(y) = (1+y^2)^{-p} f p ( y ) = ( 1 + y 2 ) − p (68) . We set p ∗ = 61 p_* = 61 p ∗ = 61 α ≥ 1 / 10 \alpha\ge1/10 α ≥ 1/10 Proposition 6.7 , is chosen for the bound (230) in the proof of Proposition 6.7 , which requires 2 p ( α ′ − κ ) > 1 + 2 α ′ 2p(\alpha' - \kappa) > 1 + 2\alpha' 2 p ( α ′ − κ ) > 1 + 2 α ′ p ≥ p ∗ p \ge p_* p ≥ p ∗ κ = α − α ′ / 6 \kappa= \alpha- \alpha'/6 κ = α − α ′ /6 α ′ − κ > α / 6 ≥ 1 / 60 \alpha' - \kappa> \alpha/6 \ge1/60 α ′ − κ > α /6 ≥ 1/60 α ′ < 1 / 2 \alpha' < 1/2 α ′ < 1/2
Proposition 4.1. (i) For every continuous function f : [ 0 , ∞ ) × ( − 2 , 2 ) → R f : [0,\infty) \times(-2,2) \to\mathbb{R} f : [ 0 , ∞ ) × ( − 2 , 2 ) → R
1 τ ∑ 1 ≤ i < j ≤ N f ( λ j − λ i ℓ , λ i + λ j 2 ) ⟶ ∫ − 2 2 ∫ 0 ∞ c ( E ) r f ( r , E ) d r d E (76) \frac{1}{\tau}\sum_{1\le i<j\le N} f\left(\frac{\lambda_j-\lambda_i}{\ell},\frac{\lambda_i+\lambda_j}{2}\right) \longrightarrow\int_{-2}^{2}\int_0^\infty c(E)rf(r,E)\,\mathrm{d}r\,\mathrm{d}E \tag*{(76)} τ 1 1 ≤ i < j ≤ N ∑ f ( ℓ λ j − λ i , 2 λ i + λ j ) ⟶ ∫ − 2 2 ∫ 0 ∞ c ( E ) r f ( r , E ) d r d E (76) in probability.
(ii) For every bulk interval I I I C I > 0 C_I > 0 C I > 0
lim N → ∞ P ( ∑ 1 ≤ i < j ≤ N ( λ i + λ j ) / 2 ∈ I f p ∗ ( λ j − λ i ℓ ) ≤ C I τ ) = 1. (77) \lim_{N \to\infty} \mathbb{P}\left(\sum_{\substack{1 \le i < j \le N \\ (\lambda_i+\lambda_j)/2 \in I}} f_{p_*}\left(\frac{\lambda_j-\lambda_i}{\ell}\right) \le C_I \tau\right)=1. \tag*{(77)} N → ∞ lim P 1 ≤ i < j ≤ N ( λ i + λ j ) /2 ∈ I ∑ f p ∗ ( ℓ λ j − λ i ) ≤ C I τ = 1. (77) Compared to Theorem 1.3 with α = α ∗ \alpha=\alpha_* α = α ∗ i < j i<j i < j ( λ j − λ i ) / ℓ (\lambda_j-\lambda_i)/\ell ( λ j − λ i ) / ℓ g i ≤ R ℓ g_i \le R\ell g i ≤ R ℓ O R ( τ ) O_R(\tau) O R ( τ ) g i ≫ ℓ g_i \gg\ell g i ≫ ℓ
The resampling and the linearized increments
The resampling. Set
m = m N = 2 ⌊ τ / 8 ⌋ , so that lim N → ∞ 4 m τ = 1. (78) m=m_N=2\lfloor\tau/8\rfloor,\qquad\text{so that}\qquad\lim_{N\to\infty}\frac{4m}{\tau}=1. \tag*{(78)} m = m N = 2 ⌊ τ /8 ⌋ , so that N → ∞ lim τ 4 m = 1. (78) Let ( a 1 , b 1 ) , … , ( a m , b m ) (a_1,b_1),\ldots,(a_m,b_m) ( a 1 , b 1 ) , … , ( a m , b m ) { ( a , b ) ∈ [ N ] 2 : a < b } \{(a,b)\in[N]^2:a<b\} {( a , b ) ∈ [ N ] 2 : a < b } Y 1 , … , Y m Y_1,\ldots,Y_m Y 1 , … , Y m X X X H H H ( a l , b l ) (a_l,b_l) ( a l , b l ) Y l Y_l Y l ( a l , b l ) (a_l,b_l) ( a l , b l ) l l l selected position , and we call ( a l , b l , Y l ) l ∈ [ m ] (a_l,b_l,Y_l)_{l\in[m]} ( a l , b l , Y l ) l ∈ [ m ] resampling variables . Define real symmetric matrices H ( 0 ) , … , H ( m ) H^{(0)},\ldots,H^{(m)} H ( 0 ) , … , H ( m ) H ( 0 ) = H H^{(0)}=H H ( 0 ) = H
H ( l ) = H ( l − 1 ) + ( N − 1 / 2 Y l − H a l b l ( l − 1 ) ) V a l b l , l ∈ [ m ] , (79) H^{(l)}=H^{(l-1)}+\left(N^{-1/2}Y_l-H^{(l-1)}_{a_l b_l}\right)V^{a_l b_l},\qquad l\in[m], \tag*{(79)} H ( l ) = H ( l − 1 ) + ( N − 1/2 Y l − H a l b l ( l − 1 ) ) V a l b l , l ∈ [ m ] , (79) where V a b V^{ab} V ab H ( l ) H^{(l)} H ( l ) H ( l − 1 ) H^{(l-1)} H ( l − 1 ) ( a l , b l ) (a_l,b_l) ( a l , b l ) ( b l , a l ) (b_l,a_l) ( b l , a l ) N − 1 / 2 Y l N^{-1/2}Y_l N − 1/2 Y l
H ~ = H ( m ) , Δ = H ~ − H , (80) \widetilde H=H^{(m)},\qquad\Delta=\widetilde H-H, \tag*{(80)} H = H ( m ) , Δ = H − H , (80) and we call H ~ \widetilde H H resampled matrix . We also write λ ~ i = λ i ( H ~ ) \widetilde\lambda_i=\lambda_i(\widetilde H) λ i = λ i ( H ) i ∈ [ N ] i\in[N] i ∈ [ N ] g ~ i = λ ~ i + 1 − λ ~ i \widetilde g_i=\widetilde\lambda_{i+1}-\widetilde\lambda_i g i = λ i + 1 − λ i E ~ i = ( λ ~ i + λ ~ i + 1 ) / 2 \widetilde E_i=(\widetilde\lambda_i+\widetilde\lambda_{i+1})/2 E i = ( λ i + λ i + 1 ) /2 i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] a l < b l a_l<b_l a l < b l H H H m m m
Lemma 4.2. The resampled matrix H ~ \widetilde H H H H H
Proof. Let σ = ( ( a l ′ , b l ′ ) ) l ∈ [ m ] \sigma=((a'_l,b'_l))_{l\in[m]} σ = (( a l ′ , b l ′ ) ) l ∈ [ m ] 1 ≤ a l ′ < b l ′ ≤ N 1\le a'_l<b'_l\le N 1 ≤ a l ′ < b l ′ ≤ N H σ ( l ) H_\sigma^{(l)} H σ ( l ) ( a l ′ , b l ′ ) (a'_l,b'_l) ( a l ′ , b l ′ ) ( a l , b l ) (a_l,b_l) ( a l , b l ) H σ ( 0 ) = H H_\sigma^{(0)}=H H σ ( 0 ) = H l l l H σ ( l ) H_\sigma^{(l)} H σ ( l ) H H H l = 0 l=0 l = 0 l − 1 l-1 l − 1 H σ ( l − 1 ) H_{\sigma}^{(l-1)} H σ ( l − 1 ) H H H Y l Y_l Y l H σ ( l − 1 ) H_{\sigma}^{(l-1)} H σ ( l − 1 ) H H H Y 1 , … , Y l − 1 Y_1,\ldots,Y_{l-1} Y 1 , … , Y l − 1 H σ ( l ) H_{\sigma}^{(l)} H σ ( l ) ( a l ′ , b l ′ ) (a'_l,b'_l) ( a l ′ , b l ′ ) N − 1 / 2 Y l N^{-1/2}Y_l N − 1/2 Y l H a l ′ b l ′ H_{a'_lb'_l} H a l ′ b l ′ l l l
The selected positions are independent of H H H ( Y l ) l ∈ [ m ] (Y_l)_{l\in[m]} ( Y l ) l ∈ [ m ] H ~ = H σ ( m ) \widetilde{H}=H_{\sigma}^{(m)} H = H σ ( m ) ( a l , b l ) = ( a l ′ , b l ′ ) (a_l,b_l)=(a'_l,b'_l) ( a l , b l ) = ( a l ′ , b l ′ ) l ∈ [ m ] l\in[m] l ∈ [ m ] B \mathcal{B} B N × N N\times N N × N
P ( H ~ ∈ B ) = ∑ σ P ( ( a l , b l ) l ∈ [ m ] = σ ) P ( H σ ( m ) ∈ B ) = P ( H ∈ B ) , \mathbb{P}(\widetilde{H}\in\mathcal{B})=\sum_{\sigma}\mathbb{P}\bigl((a_l,b_l)_{l\in[m]}=\sigma\bigr)\mathbb{P}(H_{\sigma}^{(m)}\in\mathcal{B})=\mathbb{P}(H\in\mathcal{B}), P ( H ∈ B ) = σ ∑ P ( ( a l , b l ) l ∈ [ m ] = σ ) P ( H σ ( m ) ∈ B ) = P ( H ∈ B ) , where the sum runs over all sequences σ \sigma σ
Conditioning on H H H Recall that the eigenvectors u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N H H H H H H H H H H H H A \mathcal{A} A H H H { H ∈ A } \{H\in\mathcal{A}\} { H ∈ A } A \mathcal{A} A A \mathcal{A} A P ( B ∣ H ) ≥ 1 − o ( 1 ) \mathbb{P}(\mathcal{B}\mid H)\ge1-o(1) P ( B ∣ H ) ≥ 1 − o ( 1 ) A ∈ A A\in\mathcal{A} A ∈ A B \mathcal{B} B H H H A A A 1 − o ( 1 ) 1-o(1) 1 − o ( 1 ) k k k A A A
Linearized increments. For i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] N × N N\times N N × N B B B
L i ( B ) = ( u i T B u i − u i + 1 T B u i + 1 2 , u i T B u i + 1 ) ∈ R 2 . (81) L_i(B)=\left(\frac{u_i^{T}Bu_i-u_{i+1}^{T}Bu_{i+1}}{2},u_i^{T}Bu_{i+1}\right)\in\mathbb{R}^2. \tag*{(81)} L i ( B ) = ( 2 u i T B u i − u i + 1 T B u i + 1 , u i T B u i + 1 ) ∈ R 2 . (81) The map L i L_i L i L i ( H ) = ( ( λ i − λ i + 1 ) / 2 , 0 ) L_i(H)=((\lambda_i-\lambda_{i+1})/2,0) L i ( H ) = (( λ i − λ i + 1 ) /2 , 0 )
z i = d − 1 L i ( H ) = ( − g i 2 d , 0 ) , so that ∣ z i ∣ = g i 2 d . (82) z_i=d^{-1}L_i(H)=\left(-\frac{g_i}{2d},0\right),\qquad\text{so that}\qquad|z_i|=\frac{g_i}{2d}. \tag*{(82)} z i = d − 1 L i ( H ) = ( − 2 d g i , 0 ) , so that ∣ z i ∣ = 2 d g i . (82) The map L i L_i L i 2 × 2 2\times2 2 × 2 y 1 , y 2 , y 3 ∈ R y_1,y_2,y_3\in\mathbb{R} y 1 , y 2 , y 3 ∈ R
λ 2 ( y 1 y 3 y 3 y 2 ) − λ 1 ( y 1 y 3 y 3 y 2 ) = ( ( y 1 − y 2 ) 2 + 4 y 3 2 ) 1 / 2 = 2 ∣ ( y 1 − y 2 2 , y 3 ) ∣ . (83) \lambda_2\begin{pmatrix}y_1&y_3\\y_3&y_2\end{pmatrix}-\lambda_1\begin{pmatrix}y_1&y_3\\y_3&y_2\end{pmatrix}=\bigl((y_1-y_2)^2+4y_3^2\bigr)^{1/2}=2\left|\left(\frac{y_1-y_2}{2},y_3\right)\right|. \tag*{(83)} λ 2 ( y 1 y 3 y 3 y 2 ) − λ 1 ( y 1 y 3 y 3 y 2 ) = ( ( y 1 − y 2 ) 2 + 4 y 3 2 ) 1/2 = 2 ( 2 y 1 − y 2 , y 3 ) . (83) Applying (4.10) to the compression of H + B H+B H + B u i u_i u i u i + 1 u_{i+1} u i + 1 2 × 2 2\times2 2 × 2 ( u j T ( H + B ) u j ′ ) j , j ′ ∈ { i , i + 1 } (u_j^{T}(H+B)u_{j'})_{j,j'\in\{i,i+1\}} ( u j T ( H + B ) u j ′ ) j , j ′ ∈ { i , i + 1 } 2 ∣ L i ( H + B ) ∣ 2|L_i(H+B)| 2∣ L i ( H + B ) ∣
For $1 \le a < b \le N$ and i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ]
D a b ( i ) = N L i ( V a b ) = N ( u i ( a ) u i ( b ) − u i + 1 ( a ) u i + 1 ( b ) , u i ( a ) u i + 1 ( b ) + u i ( b ) u i + 1 ( a ) ) ∈ R 2 (84) D_{ab}^{(i)} = NL_i(V^{ab}) = N\bigl(u_i(a)u_i(b)-u_{i+1}(a)u_{i+1}(b), u_i(a)u_{i+1}(b)+u_i(b)u_{i+1}(a)\bigr) \in\mathbb{R}^2 \tag*{(84)} D ab ( i ) = N L i ( V ab ) = N ( u i ( a ) u i ( b ) − u i + 1 ( a ) u i + 1 ( b ) , u i ( a ) u i + 1 ( b ) + u i ( b ) u i + 1 ( a ) ) ∈ R 2 (84) where the second equality follows from (35) . For i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] l ∈ [ m ] l \in[m] l ∈ [ m ]
J i , l = ( Y l − X a l b l ) D a l b l ( i ) , S i = ∑ l = 1 m J i , l . (85) J_{i,l} = \bigl(Y_l-X_{a_lb_l}\bigr)D_{a_lb_l}^{(i)}, \qquad S_i = \sum_{l=1}^{m}J_{i,l}. \tag*{(85)} J i , l = ( Y l − X a l b l ) D a l b l ( i ) , S i = l = 1 ∑ m J i , l . (85) We call S i S_i S i linearized increment of the pair ( i , i + 1 ) (i,i+1) ( i , i + 1 ) X a l b l = N 1 / 2 H a l b l X_{a_lb_l}=N^{1/2}H_{a_lb_l} X a l b l = N 1/2 H a l b l H H H ( a l , b l ) (a_l,b_l) ( a l , b l ) l l l H H H ( a l , b l , Y l ) (a_l,b_l,Y_l) ( a l , b l , Y l ) l ∈ [ m ] l \in[m] l ∈ [ m ] S i S_i S i m m m
For several pairs, we use the following notation. For an integer k ≥ 1 k \ge1 k ≥ 1 i = ( i 1 , … , i k ) ∈ [ N − 1 ] k \mathbf{i}=(i_1,\ldots,i_k)\in[N-1]^k i = ( i 1 , … , i k ) ∈ [ N − 1 ] k
z i = ( z i 1 , … , z i k ) , D a b i = ( D a b ( i 1 ) , … , D a b ( i k ) ) , S i = ( S i 1 , … , S i k ) , (86) \mathbf{z}_{\mathbf{i}}=(z_{i_1},\ldots,z_{i_k}), \qquad D_{ab}^{\mathbf{i}}=\bigl(D_{ab}^{(i_1)},\ldots,D_{ab}^{(i_k)}\bigr), \qquad S_{\mathbf{i}}=(S_{i_1},\ldots,S_{i_k}), \tag*{(86)} z i = ( z i 1 , … , z i k ) , D ab i = ( D ab ( i 1 ) , … , D ab ( i k ) ) , S i = ( S i 1 , … , S i k ) , (86) which are vectors in R 2 k \mathbb{R}^{2k} R 2 k S i = ∑ l = 1 m ( Y l − X a l b l ) D a l b l i S_{\mathbf{i}}=\sum_{l=1}^{m}(Y_l-X_{a_lb_l})D_{a_lb_l}^{\mathbf{i}} S i = ∑ l = 1 m ( Y l − X a l b l ) D a l b l i S i 1 , … , S i k S_{i_1},\ldots,S_{i_k} S i 1 , … , S i k H H H S i S_i S i
Lemma 4.3. On the event that the selected positions ( a 1 , b 1 ) , … , ( a m , b m ) (a_1,b_1),\ldots,(a_m,b_m) ( a 1 , b 1 ) , … , ( a m , b m )
Δ = N − 1 / 2 ∑ l = 1 m ( Y l − X a l b l ) V a l b l and d − 1 L i ( Δ ) = S i for all i ∈ [ N − 1 ] . (87) \Delta=N^{-1/2}\sum_{l=1}^{m}(Y_l-X_{a_lb_l})V^{a_lb_l}
\qquad\text{and}\qquad
d^{-1}L_i(\Delta)=S_i \quad\text{for all }i\in[N-1]. \tag*{(87)} Δ = N − 1/2 l = 1 ∑ m ( Y l − X a l b l ) V a l b l and d − 1 L i ( Δ ) = S i for all i ∈ [ N − 1 ] . (87) In particular, on this event, the $2\times2$ matrix ( u j T H ~ u j ′ ) j , j ′ ∈ { i , i + 1 } (u_j^{\mathsf T}\widetilde{H}u_{j'})_{j,j'\in\{i,i+1\}} ( u j T H u j ′ ) j , j ′ ∈ { i , i + 1 } 2 d ∣ z i + S i ∣ 2d\lvert z_i+S_i\rvert 2 d ∣ z i + S i ∣
Proof. Suppose that the selected positions are distinct. Then for every l ∈ [ m ] l\in[m] l ∈ [ m ] H ( l − 1 ) H^{(l-1)} H ( l − 1 ) ( a l , b l ) (a_l,b_l) ( a l , b l ) l − 1 l-1 l − 1 (79) , we deduce that
H a l b l ( l − 1 ) = H a l b l = N − 1 / 2 X a l b l , H ( l ) − H ( l − 1 ) = N − 1 / 2 ( Y l − X a l b l ) V a l b l . H_{a_lb_l}^{(l-1)}=H_{a_lb_l}=N^{-1/2}X_{a_lb_l}, \qquad H^{(l)}-H^{(l-1)}=N^{-1/2}(Y_l-X_{a_lb_l})V^{a_lb_l}. H a l b l ( l − 1 ) = H a l b l = N − 1/2 X a l b l , H ( l ) − H ( l − 1 ) = N − 1/2 ( Y l − X a l b l ) V a l b l . Summing over l ∈ [ m ] l\in[m] l ∈ [ m ] (87) . Using the linearity of L i L_i L i (84) , and (82) , we then obtain
L i ( Δ ) = N − 1 / 2 ∑ l = 1 m ( Y l − X a l b l ) L i ( V a l b l ) = N − 3 / 2 ∑ l = 1 m J i , l = d S i , L_i(\Delta)=N^{-1/2}\sum_{l=1}^{m}(Y_l-X_{a_lb_l})L_i(V^{a_lb_l})=N^{-3/2}\sum_{l=1}^{m}J_{i,l}=dS_i, L i ( Δ ) = N − 1/2 l = 1 ∑ m ( Y l − X a l b l ) L i ( V a l b l ) = N − 3/2 l = 1 ∑ m J i , l = d S i , L i ( H ~ ) = L i ( H ) + L i ( Δ ) = d ( z i + S i ) . L_i(\widetilde{H})=L_i(H)+L_i(\Delta)=d(z_i+S_i). L i ( H ) = L i ( H ) + L i ( Δ ) = d ( z i + S i ) . The last claim now follows from the consequence of (83) stated above, applied with B = Δ B=\Delta B = Δ
We call 2 d ∣ z i + S i ∣ 2d|z_i+S_i| 2 d ∣ z i + S i ∣ linearized gap of the pair ( i , i + 1 ) (i,i+1) ( i , i + 1 ) ( i , i + 1 ) (i,i+1) ( i , i + 1 )
A local limit theorem. We now state the local limit theorem for the linearized increments. The heuristic behind its normalization is explained in § 3.2 , and the conditional covariance of the summands of S i S_i S i 4 I d 2 k 4\mathbb{I}_{d^{2k}} 4 I d 2 k Lemma 8.4 . The proposition resolves the conditional law of S i S_i S i τ 1 / 2 \tau^{1/2} τ 1/2
Proposition 4.4. Fix an integer k ≥ 1 k \ge1 k ≥ 1 A N , k \mathcal{A}_{N,k} A N , k H H H c > 0 c > 0 c > 0 lim N → ∞ P ( A N , k ) = 1 \lim_{N\to\infty}\mathbb{P}(\mathcal{A}_{N,k})=1 lim N → ∞ P ( A N , k ) = 1
(i) Let Ψ ⊂ C c ∞ ( R 2 k ) \Psi\subset C^\infty_c(\mathbb{R}^{2k}) Ψ ⊂ C c ∞ ( R 2 k ) Ψ \Psi Ψ ϵ N = ϵ N ( k , Ψ ) ≥ 0 \epsilon_N=\epsilon_N(k,\Psi)\ge0 ϵ N = ϵ N ( k , Ψ ) ≥ 0 lim N → ∞ ϵ N = 0 \lim_{N\to\infty}\epsilon_N=0 lim N → ∞ ϵ N = 0 A N , k \mathcal{A}_{N,k} A N , k
sup i , x , ψ ∣ τ k E [ ψ ( S i + x ) ∣ H ] − e − ∣ x ∣ 2 / ( 2 τ ) ( 2 π ) k ∫ R 2 k ψ ( y ) d y ] ≤ ϵ N , (88) \sup_{\mathbf{i},x,\psi}\left| \tau^k\mathbb{E}\left[\psi(S_i+x)\mid H\right]-\frac{e^{-|x|^2/(2\tau)}}{(2\pi)^k}\int_{\mathbb{R}^{2k}}\psi(y)\,\mathrm{d}y\right]\le\epsilon_N, \tag*{(88)} i , x , ψ sup τ k E [ ψ ( S i + x ) ∣ H ] − ( 2 π ) k e − ∣ x ∣ 2 / ( 2 τ ) ∫ R 2 k ψ ( y ) d y ] ≤ ϵ N , (88) where the supremum is over all ψ ∈ Ψ \psi\in\Psi ψ ∈ Ψ x ∈ R 2 k x\in\mathbb{R}^{2k} x ∈ R 2 k i = ( i 1 , … , i k ) ∈ [ N − 1 ] k \mathbf{i}=(i_1,\ldots,i_k)\in[N-1]^k i = ( i 1 , … , i k ) ∈ [ N − 1 ] k i j + 1 ≥ i j + 2 i_{j+1}\ge i_j+2 i j + 1 ≥ i j + 2 j ∈ [ k − 1 ] j\in[k-1] j ∈ [ k − 1 ]
(ii) Suppose that k = 1 k=1 k = 1 x 0 > 0 x_0>0 x 0 > 0 C x 0 > 0 C_{x_0}>0 C x 0 > 0 A N , 1 \mathcal{A}_{N,1} A N , 1
P ( 2 ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C x 0 τ − 1 e − c g i / ℓ (89) \mathbb{P}\left(2|z_i+S_i|\le x_0\mid H\right)\le C_{x_0}\tau^{-1}e^{-cg_i/\ell} \tag*{(89)} P ( 2∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C x 0 τ − 1 e − c g i / ℓ (89) for all i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] g i ≥ ℓ g_i\ge\ell g i ≥ ℓ
The uniformity in i \mathbf{i} i x x x § 5.3 we apply Proposition 4.4 with x = z i x=z_{\mathbf{i}} x = z i i \mathbf{i} i H H H g i ≫ ℓ g_i\gg\ell g i ≫ ℓ ∣ z i ∣ ≫ τ 1 / 2 |z_i|\gg\tau^{1/2} ∣ z i ∣ ≫ τ 1/2 d d d τ − 1 \tau^{-1} τ − 1
Linearization of the small gaps. Finally, we compare the gaps of H ~ \widetilde{H} H i i i λ i + 2 − λ i > 4 s \lambda_{i+2}-\lambda_i>4s λ i + 2 − λ i > 4 s i ∈ [ N − 2 ] i\in[N-2] i ∈ [ N − 2 ] 1 − C N ( N s ) 3 = 1 − C N − 1 / 5 1-CN(Ns)^3=1-CN^{-1/5} 1 − C N ( N s ) 3 = 1 − C N − 1/5 ( i , i + 1 ) (i,i+1) ( i , i + 1 ) g i ≤ s g_i\le s g i ≤ s 3 s 3s 3 s
w N = τ N − 3 / 2 + 3 / 100 , L N = 2 s − 1 log N , δ N = 8 w N 2 L N . (90) w_N=\tau N^{-3/2+3/100},\qquad\mathcal{L}_N=2s^{-1}\log N,\qquad\delta_N=8w_N^2\mathcal{L}_N. \tag*{(90)} w N = τ N − 3/2 + 3/100 , L N = 2 s − 1 log N , δ N = 8 w N 2 L N . (90) These quantities are also used in Section 7, where w N w_N w N Δ \Delta Δ H H H L N \mathcal{L}_N L N (60) , and δ N / 2 \delta_N/2 δ N /2 (62) . By (74) , we have
w N = N − 73 / 50 , δ N d = 16 N − 1 / 50 log N , (91) w_N=N^{-73/50},\qquad\frac{\delta_N}{d}=16N^{-1/50}\log N, \tag*{(91)} w N = N − 73/50 , d δ N = 16 N − 1/50 log N , (91) so w N w_N w N δ N / d \delta_N/d δ N / d
Proposition 4.5. There exist events A N \mathcal{A}_N A N H H H B N \mathcal{B}_N B N H H H lim N → ∞ P ( A N ) = 1 \lim_{N \to\infty} \mathbb{P}(\mathcal{A}_N)=1 lim N → ∞ P ( A N ) = 1 N N N
(i) On A N \mathcal{A}_N A N λ i + 2 − λ i > 4 s \lambda_{i+2}-\lambda_i>4s λ i + 2 − λ i > 4 s i ∈ [ N − 2 ] i\in[N-2] i ∈ [ N − 2 ] P ( B N ∣ H ) ≥ 1 − o ( 1 ) \mathbb{P}(\mathcal{B}_N\mid H)\geq1-o(1) P ( B N ∣ H ) ≥ 1 − o ( 1 )
(ii) On A N ∩ B N \mathcal{A}_N\cap\mathcal{B}_N A N ∩ B N i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ]
∣ d − 1 g ~ i − 2 ∣ z i + S i ∣ ∣ ≤ δ N d and ∣ E ~ i − E i ∣ ≤ w N + δ N 2 if g i ≤ s , (92) \left|d^{-1}\widetilde{g}_i-2|z_i+S_i|\right|\leq\frac{\delta_N}{d}
\quad\text{and}\quad
|\widetilde{E}_i-E_i|\leq w_N+\frac{\delta_N}{2}
\quad\text{if }g_i\leq s, \tag*{(92)} d − 1 g i − 2∣ z i + S i ∣ ≤ d δ N and ∣ E i − E i ∣ ≤ w N + 2 δ N if g i ≤ s , (92) g ~ i > s / 2 if g i > s . (93) \widetilde{g}_i>s/2\quad\text{if }g_i>s. \tag*{(93)} g i > s /2 if g i > s . (93) Thus every gap of H ~ \widetilde{H} H d d d δ N = o ( d ) \delta_N=o(d) δ N = o ( d ) g i ≤ s g_i\leq s g i ≤ s H H H P ( B N ∣ H ) \mathbb{P}(\mathcal{B}_N\mid H) P ( B N ∣ H ) o ( 1 ) o(1) o ( 1 ) A N \mathcal{A}_N A N
Proof of the main theorem
The heuristic behind the proof is explained in Section 2.2, and we proceed as follows. We introduce a point process Ξ ^ N \widehat{\Xi}_N Ξ N Ξ ^ N , R \widehat{\Xi}_{N,R} Ξ N , R g i ≤ R ℓ g_i\leq R\ell g i ≤ R ℓ Ξ ^ N , R \widehat{\Xi}_{N,R} Ξ N , R j j j g i ≤ R ℓ g_i\leq R\ell g i ≤ R ℓ O R ( τ ) O_R(\tau) O R ( τ ) H ~ \widetilde{H} H H H H
Setup. Let C c + C_c^+ C c + f : [ 0 , ∞ ) × ( − 2 , 2 ) → [ 0 , ∞ ) f:[0,\infty)\times(-2,2)\to[0,\infty) f : [ 0 , ∞ ) × ( − 2 , 2 ) → [ 0 , ∞ ) f ∈ C c + f\in C_c^+ f ∈ C c +
L ( f ) = ∫ ( 1 − e − f ) d ν = ∫ − 2 2 ∫ 0 ∞ c ( E ) r ( 1 − e − f ( r , E ) ) d r d E . (94) L(f)=\int(1-e^{-f})\,d\nu=\int_{-2}^{2}\int_{0}^{\infty}c(E)r(1-e^{-f(r,E)})\,dr\,dE. \tag*{(94)} L ( f ) = ∫ ( 1 − e − f ) d ν = ∫ − 2 2 ∫ 0 ∞ c ( E ) r ( 1 − e − f ( r , E ) ) d r d E . (94) For R > 0 R>0 R > 0 ν R = ( 1 − e − R 2 / 8 ) ν \nu_R=(1-e^{-R^2/8})\nu ν R = ( 1 − e − R 2 /8 ) ν
L R ( f ) = ∫ ( 1 − e − f ) d ν R = ( 1 − e − R 2 / 8 ) L ( f ) . (95) L_R(f)=\int(1-e^{-f})\,d\nu_R=(1-e^{-R^2/8})L(f). \tag*{(95)} L R ( f ) = ∫ ( 1 − e − f ) d ν R = ( 1 − e − R 2 /8 ) L ( f ) . (95) Let Ξ \Xi Ξ [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) ν \nu ν E e − Ξ ( f ) = e − L ( f ) \mathbb{E}e^{-\Xi(f)}=e^{-L(f)} E e − Ξ ( f ) = e − L ( f ) f ∈ C c + f\in C_c^+ f ∈ C c + [42 Chapter 3] .
Until the proof of Theorem 1.2 , I I I I ′ I' I ′ I I I
∣ I ′ ∣ = ∣ I ∣ + dist ( I , { − 2 , 2 } ) , (96) |I'|=|I|+\operatorname{dist}(I,\{-2,2\}), \tag*{(96)} ∣ I ′ ∣ = ∣ I ∣ + dist ( I , { − 2 , 2 }) , (96) where ∣ I ∣ |I| ∣ I ∣ I I I I ′ I' I ′ I ⊂ int ( I ′ ) I\subset\operatorname{int}(I') I ⊂ int ( I ′ ) int \operatorname{int} int
C = { i ∈ [ N − 1 ] : g i ≤ s , E i ∈ I ′ } , C R = { i ∈ C : g i ≤ R ℓ } ( R > 0 ) , (97) \mathcal{C}=\{i\in[N-1]:g_i\le s,\ E_i\in I'\},\qquad\mathcal{C}_R=\{i\in\mathcal{C}:g_i\le R\ell\}\quad(R>0), \tag*{(97)} C = { i ∈ [ N − 1 ] : g i ≤ s , E i ∈ I ′ } , C R = { i ∈ C : g i ≤ R ℓ } ( R > 0 ) , (97) and the point processes
Ξ ^ N = ∑ i ∈ C δ ( 2 ∣ z i + s i ∣ , E i ) , Ξ ^ N , R = ∑ i ∈ C R δ ( 2 ∣ z i + s i ∣ , E i ) (98) \widehat{\Xi}_N=\sum_{i\in\mathcal{C}}\delta_{(2|z_i+s_i|,E_i)},\qquad\widehat{\Xi}_{N,R}=\sum_{i\in\mathcal{C}_R}\delta_{(2|z_i+s_i|,E_i)} \tag*{(98)} Ξ N = i ∈ C ∑ δ ( 2∣ z i + s i ∣ , E i ) , Ξ N , R = i ∈ C R ∑ δ ( 2∣ z i + s i ∣ , E i ) (98) on [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 )
Let A N A_N A N B N B_N B N Proposition 4.5 , and for every integer k ≥ 1 k\ge1 k ≥ 1 A N , k A_{N,k} A N , k Proposition 4.4 . Let C I ′ C_{I'} C I ′ Proposition 4.1 for the interval I ′ I' I ′
A N ⋆ = A N ∩ A N , 1 ∩ { ∑ 1 ≤ i < j ≤ N ( λ i + λ j ) / 2 ∈ I ′ f p ∗ ( λ j − λ i ℓ ) ≤ C I ′ τ } . (99) A_N^\star=A_N\cap A_{N,1}\cap\left\{\sum_{\substack{1\le i<j\le N\\(\lambda_i+\lambda_j)/2\in I'}}f_{p^\ast}\left(\frac{\lambda_j-\lambda_i}{\ell}\right)\le C_{I'}\tau\right\}. \tag*{(99)} A N ⋆ = A N ∩ A N , 1 ∩ ⎩ ⎨ ⎧ 1 ≤ i < j ≤ N ( λ i + λ j ) /2 ∈ I ′ ∑ f p ∗ ( ℓ λ j − λ i ) ≤ C I ′ τ ⎭ ⎬ ⎫ . (99) All objects introduced in this paragraph, except Ξ ^ N \widehat{\Xi}_N Ξ N Ξ ^ N , R \widehat{\Xi}_{N,R} Ξ N , R B N B_N B N H H H
Lemma 4.6. We have lim N → ∞ P ( A N ⋆ ) = 1 \lim_{N\to\infty}\mathbb{P}(A_N^\star)=1 lim N → ∞ P ( A N ⋆ ) = 1 R ′ > 0 R'>0 R ′ > 0 N 0 > 0 N_0>0 N 0 > 0 R ′ R' R ′ N ≥ N 0 N\ge N_0 N ≥ N 0 A N ⋆ A_N^\star A N ⋆
(i) Distinct elements of C \mathcal{C} C
(ii) We have ∑ i ∈ C f p ∗ ( g i / ℓ ) ≤ C I ′ τ \sum_{i\in\mathcal{C}}f_{p^\ast}(g_i/\ell)\le C_{I'}\tau ∑ i ∈ C f p ∗ ( g i / ℓ ) ≤ C I ′ τ ∣ C R ∣ ≤ C I ′ ( 1 + R 2 ) p ∗ τ |\mathcal{C}_R|\le C_{I'}(1+R^2)^{p^\ast}\tau ∣ C R ∣ ≤ C I ′ ( 1 + R 2 ) p ∗ τ R > 0 R>0 R > 0
(iii) For every function Q : [ 0 , ∞ ) × ( − 2 , 2 ) → R Q:[0,\infty)\times(-2,2)\to\mathbb{R} Q : [ 0 , ∞ ) × ( − 2 , 2 ) → R Q ( y , E ) = 0 Q(y,E)=0 Q ( y , E ) = 0 y ≥ R ′ y\ge R' y ≥ R ′
∑ 1 ≤ i < j ≤ N Q ( λ j − λ i ℓ , λ i + λ j 2 ) = ∑ i = 1 N − 1 Q ( g i ℓ , E i ) . (100) \sum_{1\le i<j\le N}Q\left(\frac{\lambda_j-\lambda_i}{\ell},\frac{\lambda_i+\lambda_j}{2}\right)=\sum_{i=1}^{N-1}Q\left(\frac{g_i}{\ell},E_i\right). \tag*{(100)} 1 ≤ i < j ≤ N ∑ Q ( ℓ λ j − λ i , 2 λ i + λ j ) = i = 1 ∑ N − 1 Q ( ℓ g i , E i ) . (100) Proof. The first claim follows from Proposition 4.5 , Proposition 4.4 , the second part of Proposition 4.1 , and a union bound. Now suppose that A N ⋆ A_N^\star A N ⋆ Proposition 4.5 gives
λ i + 2 − λ i > 4 s for all i ∈ [ N − 2 ] . (101) \lambda_{i+2}-\lambda_i>4s\qquad\text{for all }i\in[N-2]. \tag*{(101)} λ i + 2 − λ i > 4 s for all i ∈ [ N − 2 ] . (101) To prove (i), suppose that i , i + 1 ∈ C i,i+1 \in C i , i + 1 ∈ C i i i
λ i + 2 − λ i = g i + g i + 1 ≤ 2 s , \lambda_{i+2}-\lambda_i=g_i+g_{i+1}\le2s, λ i + 2 − λ i = g i + g i + 1 ≤ 2 s , which contradicts (101) .
To prove (ii), note that for every i ∈ C i\in C i ∈ C ( i , i + 1 ) (i,i+1) ( i , i + 1 ) (99) , since its midpoint is E i ∈ I ′ E_i\in I' E i ∈ I ′
∑ i ∈ C f p ∗ ( g i ℓ ) ≤ C I τ . \sum_{i\in C} f_{p_*}\left(\frac{g_i}{\ell}\right)\le C_I\tau. i ∈ C ∑ f p ∗ ( ℓ g i ) ≤ C I τ . The bound on ∣ C R ∣ \lvert C_R\rvert ∣ C R ∣ f p ∗ ( g i / ℓ ) ≥ ( 1 + R 2 ) − p ∗ f_{p_*}(g_i/\ell)\ge(1+R^2)^{-p_*} f p ∗ ( g i / ℓ ) ≥ ( 1 + R 2 ) − p ∗ i ∈ C R i\in C_R i ∈ C R
Finally, to prove (iii), observe that for j ≥ i + 2 j\ge i+2 j ≥ i + 2
λ j − λ i ≥ λ i + 2 − λ i > 4 s ≥ R ′ ℓ , \lambda_j-\lambda_i\ge\lambda_{i+2}-\lambda_i>4s\ge R'\ell, λ j − λ i ≥ λ i + 2 − λ i > 4 s ≥ R ′ ℓ , where we used (101) in the second inequality, and the last inequality holds for large N N N (75) . Hence the terms with j ≥ i + 2 j\ge i+2 j ≥ i + 2 (100) vanish, and the remaining terms, with j = i + 1 j=i+1 j = i + 1
The Bonferroni inequalities. For a finite set T \mathcal{T} T y = ( y i ) i ∈ T y=(y_i)_{i\in\mathcal{T}} y = ( y i ) i ∈ T j ≥ 0 j\ge0 j ≥ 0
P j ( y ) = ∑ T ′ ⊂ T ∣ T ′ ∣ = j ∏ i ∈ T ′ y i (102) P_j(y)=\sum_{\substack{\mathcal{T}'\subset\mathcal{T}\\ |\mathcal{T}'|=j}}\prod_{i\in\mathcal{T}'}y_i \tag*{(102)} P j ( y ) = T ′ ⊂ T ∣ T ′ ∣ = j ∑ i ∈ T ′ ∏ y i (102) denote the j j j y y y P 0 ( y ) = 1 P_0(y)=1 P 0 ( y ) = 1 P j ( y ) = 0 P_j(y)=0 P j ( y ) = 0 j > ∣ T ∣ j>|\mathcal{T}| j > ∣ T ∣
Lemma 4.7. Let T \mathcal{T} T w = ( w i ) i ∈ T ∈ [ 0 , 1 ] T \mathbf{w}=(w_i)_{i\in\mathcal{T}}\in[0,1]^{\mathcal{T}} w = ( w i ) i ∈ T ∈ [ 0 , 1 ] T n ≥ 0 n\ge0 n ≥ 0
∑ j = 0 2 n + 1 ( − 1 ) j P j ( w ) ≤ ∏ i ∈ T ( 1 − w i ) ≤ ∑ j = 0 2 n ( − 1 ) j P j ( w ) . (103) \sum_{j=0}^{2n+1}(-1)^jP_j(\mathbf{w})\le\prod_{i\in\mathcal{T}}(1-w_i)\le\sum_{j=0}^{2n}(-1)^jP_j(\mathbf{w}). \tag*{(103)} j = 0 ∑ 2 n + 1 ( − 1 ) j P j ( w ) ≤ i ∈ T ∏ ( 1 − w i ) ≤ j = 0 ∑ 2 n ( − 1 ) j P j ( w ) . (103) Proof. Let ( b i ) i ∈ T (b_i)_{i\in\mathcal{T}} ( b i ) i ∈ T P ( b i = 1 ) = w i \mathbb{P}(b_i=1)=w_i P ( b i = 1 ) = w i P ( b i = 0 ) = 1 − w i \mathbb{P}(b_i=0)=1-w_i P ( b i = 0 ) = 1 − w i
M = ∑ i ∈ T b i . M=\sum_{i\in\mathcal{T}}b_i. M = i ∈ T ∑ b i . Expanding the products and using independence, we find
P j ( w ) = E ∑ T ′ ⊂ T ∣ T ′ ∣ = j ∏ i ∈ T ′ b i = E ( M j ) , ∏ i ∈ T ( 1 − w i ) = E ∏ i ∈ T ( 1 − b i ) = P ( M = 0 ) . P_j(\mathbf{w})=\mathbb{E}\sum_{\substack{\mathcal{T}'\subset\mathcal{T}\\|\mathcal{T}'|=j}}\prod_{i\in\mathcal{T}'}b_i=\mathbb{E}\binom{M}{j},\qquad
\prod_{i\in\mathcal{T}}(1-w_i)=\mathbb{E}\prod_{i\in\mathcal{T}}(1-b_i)=\mathbb{P}(M=0). P j ( w ) = E T ′ ⊂ T ∣ T ′ ∣ = j ∑ i ∈ T ′ ∏ b i = E ( j M ) , i ∈ T ∏ ( 1 − w i ) = E i ∈ T ∏ ( 1 − b i ) = P ( M = 0 ) . For integers M 0 ≥ 1 M_0 \ge1 M 0 ≥ 1 n 0 ≥ 0 n_0 \ge0 n 0 ≥ 0
∑ j = 0 n 0 ( − 1 ) j ( M 0 j ) = ( − 1 ) n 0 ( M 0 − 1 n 0 ) , \sum_{j=0}^{n_0}(-1)^j\binom{M_0}{j}=(-1)^{n_0}\binom{M_0-1}{n_0}, j = 0 ∑ n 0 ( − 1 ) j ( j M 0 ) = ( − 1 ) n 0 ( n 0 M 0 − 1 ) , whose right side is nonnegative for even n 0 n_0 n 0 n 0 n_0 n 0 M 0 = 0 M_0=0 M 0 = 0 M 0 = M M_0=M M 0 = M n 0 ∈ { 2 n , 2 n + 1 } n_0\in\{2n,2n+1\} n 0 ∈ { 2 n , 2 n + 1 } { b i = 1 } \{b_i=1\} { b i = 1 }
Counts of small linearized gaps.
Lemma 4.8. Let x 0 > 0 x_0>0 x 0 > 0 C , c > 0 C,c>0 C , c > 0 N 0 > 0 N_0>0 N 0 > 0 I I I x 0 x_0 x 0 N ≥ N 0 N\ge N_0 N ≥ N 0 A N ⋆ \mathcal{A}_N^\star A N ⋆
∑ i ∈ C g i ≥ R ℓ P ( 2 ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C sup y ≥ R ( 1 + y 2 ) p ∗ e − c y for all R ≥ 0. (104) \sum_{\substack{i\in\mathcal{C}\\g_i\ge R\ell}}\mathbb{P}(2|z_i+S_i|\le x_0\mid H)\le C\sup_{y\ge R}(1+y^2)^{p_\ast}e^{-cy}
\qquad\text{for all }R\ge0. \tag*{(104)} i ∈ C g i ≥ R ℓ ∑ P ( 2∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C y ≥ R sup ( 1 + y 2 ) p ∗ e − cy for all R ≥ 0. (104) In particular, for N ≥ N 0 N\ge N_0 N ≥ N 0 A N ⋆ \mathcal{A}_N^\star A N ⋆
E [ Ξ ^ N ( [ 0 , x 0 ] × I ′ ) ∣ H ] ≤ C ′ , (105) \mathbb{E}\left[\widehat{\Xi}_N([0,x_0]\times I')\mid H\right]\le C', \tag*{(105)} E [ Ξ N ([ 0 , x 0 ] × I ′ ) ∣ H ] ≤ C ′ , (105) E [ ( Ξ ^ N − Ξ ^ N , R ) ( [ 0 , x 0 ] × I ′ ) ∣ H ] ≤ C sup y ≥ R ( 1 + y 2 ) p ∗ e − c y for all R > 0 , (106) \mathbb{E}\left[(\widehat{\Xi}_N-\widehat{\Xi}_{N,R})([0,x_0]\times I')\mid H\right]\le C\sup_{y\ge R}(1+y^2)^{p_\ast}e^{-cy}
\qquad\text{for all }R>0, \tag*{(106)} E [ ( Ξ N − Ξ N , R ) ([ 0 , x 0 ] × I ′ ) ∣ H ] ≤ C y ≥ R sup ( 1 + y 2 ) p ∗ e − cy for all R > 0 , (106) where C ′ = C sup y ≥ 0 ( 1 + y 2 ) p ∗ e − c y C'=C\sup_{y\ge0}(1+y^2)^{p_\ast}e^{-cy} C ′ = C sup y ≥ 0 ( 1 + y 2 ) p ∗ e − cy
Proof. Let c > 0 c>0 c > 0 k = 1 k=1 k = 1 ψ 0 ∈ C c ∞ ( R 2 ) \psi_0\in C^\infty_c(\mathbb{R}^2) ψ 0 ∈ C c ∞ ( R 2 ) 0 ≤ ψ 0 ≤ 1 0\le\psi_0\le1 0 ≤ ψ 0 ≤ 1 ψ 0 ( y ) = 1 \psi_0(y)=1 ψ 0 ( y ) = 1 ∣ y ∣ ≤ x 0 / 2 |y|\le x_0/2 ∣ y ∣ ≤ x 0 /2 N N N A N ⋆ \mathcal{A}_N^\star A N ⋆ i ∈ C i\in\mathcal{C} i ∈ C k = 1 k=1 k = 1 Ψ = { ψ 0 } \Psi=\{\psi_0\} Ψ = { ψ 0 } x = z i x=z_i x = z i A N , 1 ⊃ A N ⋆ \mathcal{A}_{N,1}\supset\mathcal{A}_N^\star A N , 1 ⊃ A N ⋆
P ( 2 ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ E [ ψ 0 ( S i + z i ) ∣ H ] ≤ 1 τ ( 1 2 π ∫ R 2 ψ 0 ( y ) d y + ε N ) ≤ C τ , \mathbb{P}(2|z_i+S_i|\le x_0\mid H)\le\mathbb{E}[\psi_0(S_i+z_i)\mid H]\le\frac{1}{\tau}\left(\frac{1}{2\pi}\int_{\mathbb{R}^2}\psi_0(y)\,\mathrm{d}y+\varepsilon_N\right)\le\frac{C}{\tau}, P ( 2∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ E [ ψ 0 ( S i + z i ) ∣ H ] ≤ τ 1 ( 2 π 1 ∫ R 2 ψ 0 ( y ) d y + ε N ) ≤ τ C , where ε N = ε N ( 1 , { ψ 0 } ) \varepsilon_N=\varepsilon_N(1,\{\psi_0\}) ε N = ε N ( 1 , { ψ 0 }) g i < ℓ g_i<\ell g i < ℓ e − c g i / ℓ ≥ e − c e^{-cg_i/\ell}\ge e^{-c} e − c g i / ℓ ≥ e − c g i ≥ ℓ g_i\ge\ell g i ≥ ℓ C C C
P ( 2 ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C τ − 1 e − c g i / ℓ for all i ∈ C . \mathbb{P}(2|z_i+S_i|\le x_0\mid H)\le C\tau^{-1}e^{-cg_i/\ell}
\qquad\text{for all }i\in\mathcal{C}. P ( 2∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C τ − 1 e − c g i / ℓ for all i ∈ C . The sum over the pairs with g i ≥ R ℓ g_i\ge R\ell g i ≥ R ℓ R ≥ 0 R\ge0 R ≥ 0
∑ i ∈ C g i ≥ R ℓ P ( 2 ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C τ sup y ≥ R ( 1 + y 2 ) p ∗ e − c y ∑ i ∈ C f p ∗ ( g i ℓ ) ≤ C C I ′ sup y ≥ R ( 1 + y 2 ) p ∗ e − c y , \begin{aligned}
\sum_{\substack{i\in\mathcal{C}\\g_i\ge R\ell}}\mathbb{P}(2|z_i+S_i|\le x_0\mid H)
\le\frac{C}{\tau}\sup_{y\ge R}(1+y^2)^{p_\ast}e^{-cy}\sum_{i\in\mathcal{C}}f_{p_\ast}\left(\frac{g_i}{\ell}\right) \\
&\le CC_{I'}\sup_{y\ge R}(1+y^2)^{p_\ast}e^{-cy},
\end{aligned} i ∈ C g i ≥ R ℓ ∑ P ( 2∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ τ C y ≥ R sup ( 1 + y 2 ) p ∗ e − cy i ∈ C ∑ f p ∗ ( ℓ g i ) ≤ C C I ′ y ≥ R sup ( 1 + y 2 ) p ∗ e − cy , which is (104) .
Observe that every point of Ξ ^ N \widehat{\Xi}_N Ξ N I ′ I' I ′ C \mathcal{C} C H H H
E [ Ξ ^ N ( [ 0 , x 0 ] × I ′ ) ∣ H ] = ∑ i ∈ C P ( 2 ∣ z i + S i ∣ ≤ x 0 ∣ H ) , \mathbb{E}\left[\widehat{\Xi}_N([0,x_0]\times I')\mid H\right]=\sum_{i\in\mathcal{C}}\mathbb{P}\left(2|z_i+S_i|\leq x_0\mid H\right), E [ Ξ N ([ 0 , x 0 ] × I ′ ) ∣ H ] = i ∈ C ∑ P ( 2∣ z i + S i ∣ ≤ x 0 ∣ H ) , E [ ( Ξ ^ N − Ξ ^ N , R ) ( [ 0 , x 0 ] × I ′ ) ∣ H ] = ∑ i ∈ C g i > R ℓ P ( 2 ∣ z i + S i ∣ ≤ x 0 ∣ H ) . \mathbb{E}\left[(\widehat{\Xi}_N-\widehat{\Xi}_{N,R})([0,x_0]\times I')\mid H\right]=\sum_{\substack{i\in\mathcal{C}\\ g_i>R\ell}}\mathbb{P}\left(2|z_i+S_i|\leq x_0\mid H\right). E [ ( Ξ N − Ξ N , R ) ([ 0 , x 0 ] × I ′ ) ∣ H ] = i ∈ C g i > R ℓ ∑ P ( 2∣ z i + S i ∣ ≤ x 0 ∣ H ) . Hence (104) implies (105) and (106) .
Smooth test functions. The local limit theorem applies only to smooth test functions, but the function y ↦ f ( 2 ∣ y ∣ , E ) y\mapsto f(2|y|,E) y ↦ f ( 2∣ y ∣ , E ) R 2 \mathbb{R}^2 R 2 y = 0 y=0 y = 0 f f f f ( r , E ) = h ( r 2 , E ) f(r,E)=h(r^2,E) f ( r , E ) = h ( r 2 , E ) h h h
Lemma 4.9. Let R > 0 R>0 R > 0 h ∈ C c ∞ ( R 2 ) h\in C^\infty_c(\mathbb{R}^2) h ∈ C c ∞ ( R 2 ) supp h ⊂ R × int ( I ′ ) \operatorname{supp}h\subset\mathbb{R}\times\operatorname{int}(I') supp h ⊂ R × int ( I ′ ) f ( r , E ) = h ( r 2 , E ) f(r,E)=h(r^2,E) f ( r , E ) = h ( r 2 , E ) ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) (r,E)\in[0,\infty)\times(-2,2) ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) f ∈ C + ∞ f\in C^\infty_+ f ∈ C + ∞
E [ e − Ξ ^ N , R ( f ) ∣ H ] ⟶ e − L R ( f ) (107) \mathbb{E}\left[e^{-\widehat{\Xi}_{N,R}(f)}\mid H\right]\longrightarrow e^{-L_R(f)} \tag*{(107)} E [ e − Ξ N , R ( f ) ∣ H ] ⟶ e − L R ( f ) (107) in probability.
Proof. Fix u h > 0 u_h>0 u h > 0 I h ⊂ int ( I ′ ) I_h\subset\operatorname{int}(I') I h ⊂ int ( I ′ ) supp h ⊂ [ − u h , u h ] × I h \operatorname{supp}h\subset[-u_h,u_h]\times I_h supp h ⊂ [ − u h , u h ] × I h f f f supp f ⊂ [ 0 , u h 1 / 2 ] × I h \operatorname{supp}f\subset[0,u_h^{1/2}]\times I_h supp f ⊂ [ 0 , u h 1/2 ] × I h f ∈ C + ∞ f\in C^\infty_+ f ∈ C + ∞
We begin by introducing planar test functions. For E ∈ ( − 2 , 2 ) E\in(-2,2) E ∈ ( − 2 , 2 ) y ∈ R 2 y\in\mathbb{R}^2 y ∈ R 2
ψ E ( y ) = 1 − exp ( − h ( 4 ∣ y ∣ 2 , E ) ) , A h ( E ) = ∫ R 2 ψ E ( y ) d y . (108) \psi_E(y)=1-\exp\left(-h(4|y|^2,E)\right),\qquad A_h(E)=\int_{\mathbb{R}^2}\psi_E(y)\,\mathrm{d}y. \tag*{(108)} ψ E ( y ) = 1 − exp ( − h ( 4∣ y ∣ 2 , E ) ) , A h ( E ) = ∫ R 2 ψ E ( y ) d y . (108) Since f ( 2 ∣ y ∣ , E ) = h ( 4 ∣ y ∣ 2 , E ) f(2|y|,E)=h(4|y|^2,E) f ( 2∣ y ∣ , E ) = h ( 4∣ y ∣ 2 , E )
1 − e − f ( 2 ∣ y ∣ , E ) = ψ E ( y ) for all y ∈ R 2 and E ∈ ( − 2 , 2 ) . (109) 1-e^{-f(2|y|,E)}=\psi_E(y)\qquad\text{for all }y\in\mathbb{R}^2\text{ and }E\in(-2,2). \tag*{(109)} 1 − e − f ( 2∣ y ∣ , E ) = ψ E ( y ) for all y ∈ R 2 and E ∈ ( − 2 , 2 ) . (109) We record three properties of ψ E \psi_E ψ E A h A_h A h 0 ≤ ψ E ≤ 1 0\leq\psi_E\leq1 0 ≤ ψ E ≤ 1 ψ E ( y ) = 0 \psi_E(y)=0 ψ E ( y ) = 0 ∣ y ∣ > u h 1 / 2 / 2 |y|>u_h^{1/2}/2 ∣ y ∣ > u h 1/2 /2 E ∉ I h E\notin I_h E ∈ / I h ψ E ∈ C ∞ ( R 2 ) \psi_E\in C^\infty(\mathbb{R}^2) ψ E ∈ C ∞ ( R 2 ) ψ E \psi_E ψ E E ∈ ( − 2 , 2 ) E\in(-2,2) E ∈ ( − 2 , 2 ) A h A_h A h I h I_h I h
A h ( E ) = π 2 ∫ 0 ∞ r ( 1 − e − f ( r , E ) ) d r , (110) A_h(E)=\frac{\pi}{2}\int_0^\infty r\left(1-e^{-f(r,E)}\right)\,\mathrm{d}r, \tag*{(110)} A h ( E ) = 2 π ∫ 0 ∞ r ( 1 − e − f ( r , E ) ) d r , (110) which follows from (109) by using polar coordinates in R 2 \mathbb{R}^2 R 2 ∣ y ∣ = r / 2 |y|=r/2 ∣ y ∣ = r /2
We now apply the Bonferroni inequalities. For i ∈ C R i\in\mathcal{C}_R i ∈ C R
w i = ψ E i ( z i + S i ) ∈ [ 0 , 1 ] , (111) w_i=\psi_{E_i}(z_i+S_i)\in[0,1], \tag*{(111)} w i = ψ E i ( z i + S i ) ∈ [ 0 , 1 ] , (111) and let w = ( w i ) i ∈ C R \mathbf{w}=(w_i)_{i\in C_R} w = ( w i ) i ∈ C R (109) , we find
e − E ^ N , R ( f ) = ∏ i ∈ C R e − f ( 2 ∣ z i + S i ∣ ; E i ) = ∏ i ∈ C R ( 1 − w i ) . e^{-\widehat{\mathbb{E}}_{N,R}(f)}=\prod_{i\in C_R}e^{-f(2|z_i+S_i|;E_i)}=\prod_{i\in C_R}(1-w_i). e − E N , R ( f ) = i ∈ C R ∏ e − f ( 2∣ z i + S i ∣ ; E i ) = i ∈ C R ∏ ( 1 − w i ) . Since C R C_R C R H H H T = C R \mathcal{T}=C_R T = C R n ≥ 0 n\ge0 n ≥ 0
∑ j = 0 2 n + 1 ( − 1 ) j E [ P j ( w ) ∣ H ] ≤ E [ e − E ^ N , R ( f ) ∣ H ] ≤ ∑ j = 0 2 n ( − 1 ) j E [ P j ( w ) ∣ H ] . (112) \sum_{j=0}^{2n+1}(-1)^j\mathbb{E}\left[P_j(\mathbf{w})\mid H\right]\le\mathbb{E}\left[e^{-\widehat{\mathbb{E}}_{N,R}(f)}\mid H\right]\le\sum_{j=0}^{2n}(-1)^j\mathbb{E}\left[P_j(\mathbf{w})\mid H\right]. \tag*{(112)} j = 0 ∑ 2 n + 1 ( − 1 ) j E [ P j ( w ) ∣ H ] ≤ E [ e − E N , R ( f ) ∣ H ] ≤ j = 0 ∑ 2 n ( − 1 ) j E [ P j ( w ) ∣ H ] . (112) We claim that for every integer j ≥ 1 j\ge1 j ≥ 1
E [ P j ( w ) ∣ H ] ⟶ L R ( f ) j j ! in probability. (113) \mathbb{E}\left[P_j(\mathbf{w})\mid H\right]\longrightarrow\frac{L_R(f)^j}{j!}\qquad\text{in probability.} \tag*{(113)} E [ P j ( w ) ∣ H ] ⟶ j ! L R ( f ) j in probability. (113) We first show that (4.40) implies the lemma. Let ω > 0 \omega>0 ω > 0 ∑ j ≥ 0 ( − 1 ) j L R ( f ) j / j ! \sum_{j\ge0}(-1)^jL_R(f)^j/j! ∑ j ≥ 0 ( − 1 ) j L R ( f ) j / j ! e − L R ( f ) e^{-L_R(f)} e − L R ( f ) n ≥ 0 n\ge0 n ≥ 0
∣ ∑ j = 0 2 n ( − 1 ) j L R ( f ) j j ! − e − L R ( f ) ∣ ≤ ω , ∣ ∑ j = 0 2 n + 1 ( − 1 ) j L R ( f ) j j ! − e − L R ( f ) ∣ ≤ ω . \left|\sum_{j=0}^{2n}\frac{(-1)^jL_R(f)^j}{j!}-e^{-L_R(f)}\right|\le\omega,\qquad
\left|\sum_{j=0}^{2n+1}\frac{(-1)^jL_R(f)^j}{j!}-e^{-L_R(f)}\right|\le\omega. j = 0 ∑ 2 n j ! ( − 1 ) j L R ( f ) j − e − L R ( f ) ≤ ω , j = 0 ∑ 2 n + 1 j ! ( − 1 ) j L R ( f ) j − e − L R ( f ) ≤ ω . Then (4.40) implies that the event
∣ E [ P j ( w ) ∣ H ] − L R ( f ) j j ! ∣ ≤ ω 2 n + 2 for all j ∈ [ 2 n + 1 ] \left|\mathbb{E}\left[P_j(\mathbf{w})\mid H\right]-\frac{L_R(f)^j}{j!}\right|\le\frac{\omega}{2n+2}\qquad\text{for all }j\in[2n+1] E [ P j ( w ) ∣ H ] − j ! L R ( f ) j ≤ 2 n + 2 ω for all j ∈ [ 2 n + 1 ] has probability tending to one. On this event, (4.39) and the choice of n n n
∣ E [ e − E ^ N , R ( f ) ∣ H ] − e − L R ( f ) ∣ ≤ 2 ω . \left|\mathbb{E}\left[e^{-\widehat{\mathbb{E}}_{N,R}(f)}\mid H\right]-e^{-L_R(f)}\right|\le2\omega. E [ e − E N , R ( f ) ∣ H ] − e − L R ( f ) ≤ 2 ω . The number ω > 0 \omega>0 ω > 0
It remains to prove (4.40). Fix j ≥ 1 j\ge1 j ≥ 1 i ∈ C R i\in C_R i ∈ C R
w ˉ i : = 1 2 π τ exp ( − g i 2 8 ℓ 2 ) A h ( E i ) , (114) \bar{w}_i:=\frac{1}{2\pi\tau}\exp\left(-\frac{g_i^2}{8\ell^2}\right)A_h(E_i), \tag*{(114)} w ˉ i := 2 π τ 1 exp ( − 8 ℓ 2 g i 2 ) A h ( E i ) , (114) and let w ˉ = ( w ˉ i ) i ∈ C R \bar{\mathbf{w}}=(\bar{w}_i)_{i\in C_R} w ˉ = ( w ˉ i ) i ∈ C R E [ P j ( w ) ∣ H ] \mathbb{E}[P_j(\mathbf{w})\mid H] E [ P j ( w ) ∣ H ] P j ( w ˉ ) P_j(\bar{\mathbf{w}}) P j ( w ˉ ) P j ( w ˉ ) P_j(\bar{\mathbf{w}}) P j ( w ˉ ) ( ∑ i ∈ C R w ˉ i ) j / j ! (\sum_{i\in C_R}\bar{w}_i)^j/j! ( ∑ i ∈ C R w ˉ i ) j / j ! ∑ i ∈ C R w ˉ i \sum_{i\in C_R}\bar{w}_i ∑ i ∈ C R w ˉ i L R ( f ) L_R(f) L R ( f )
For the first comparison, let Ψ j \Psi_j Ψ j ψ E 1 ′ ⊗ ⋯ ⊗ ψ E j ′ \psi_{E'_1}\otimes\cdots\otimes\psi_{E'_j} ψ E 1 ′ ⊗ ⋯ ⊗ ψ E j ′ E 1 ′ , … , E j ′ ∈ ( − 2 , 2 ) E'_1,\ldots,E'_j\in(-2,2) E 1 ′ , … , E j ′ ∈ ( − 2 , 2 )
( ψ E 1 ′ ⊗ ⋯ ⊗ ψ E j ′ ) ( y 1 , … , y j ) = ψ E 1 ′ ( y 1 ) ⋯ ψ E j ′ ( y j ) , y 1 , … , y j ∈ R 2 . (\psi_{E'_1}\otimes\cdots\otimes\psi_{E'_j})(y_1,\ldots,y_j)=\psi_{E'_1}(y_1)\cdots\psi_{E'_j}(y_j),\qquad y_1,\ldots,y_j\in\mathbb{R}^2. ( ψ E 1 ′ ⊗ ⋯ ⊗ ψ E j ′ ) ( y 1 , … , y j ) = ψ E 1 ′ ( y 1 ) ⋯ ψ E j ′ ( y j ) , y 1 , … , y j ∈ R 2 . By the first two properties of ψ E \psi_E ψ E Ψ j \Psi_j Ψ j j j j u h 1 / 2 / 2 u_h^{1/2}/2 u h 1/2 /2 Ψ j \Psi_j Ψ j Ψ j \Psi_j Ψ j Proposition 4.4 with k = j k=j k = j N N N Lemma 4.6 applies with R ′ = R + 1 R'=R+1 R ′ = R + 1 A N ⋆ ∩ A N , j A_N^\star\cap A_{N,j} A N ⋆ ∩ A N , j
Let T ′ ⊂ C R T' \subset C_R T ′ ⊂ C R ∣ T ′ ∣ = j |T'|=j ∣ T ′ ∣ = j T ′ T' T ′ i = ( i 1 , … , i j ) \mathbf{i}=(i_1,\ldots,i_j) i = ( i 1 , … , i j ) Lemma 4.6 shows that consecutive entries of i \mathbf{i} i
ψ = ψ E i 1 ⊗ ⋯ ⊗ ψ E i j ∈ Ψ j . \psi=\psi_{E_{i_1}}\otimes\cdots\otimes\psi_{E_{i_j}}\in\Psi_j. ψ = ψ E i 1 ⊗ ⋯ ⊗ ψ E i j ∈ Ψ j . Then we have
∏ i ∈ T ′ w i = ψ ( S i + z i ) , ∫ R 2 j ψ ( y ) d y = ∏ i ∈ T ′ A h ( E i ) , ∣ z i ∣ 2 2 τ = ∑ i ∈ T ′ g i 2 8 d 2 τ = ∑ i ∈ T ′ g i 2 8 ℓ 2 , \prod_{i\in T'}w_i=\psi(S_{\mathbf{i}}+z_{\mathbf{i}}),\qquad\int_{\mathbb{R}^{2j}}\psi(y)\,\mathrm{d}y=\prod_{i\in T'}A_h(E_i),\qquad\frac{|z_{\mathbf{i}}|^2}{2\tau}=\sum_{i\in T'}\frac{g_i^2}{8d^2\tau}=\sum_{i\in T'}\frac{g_i^2}{8\ell^2}, i ∈ T ′ ∏ w i = ψ ( S i + z i ) , ∫ R 2 j ψ ( y ) d y = i ∈ T ′ ∏ A h ( E i ) , 2 τ ∣ z i ∣ 2 = i ∈ T ′ ∑ 8 d 2 τ g i 2 = i ∈ T ′ ∑ 8 ℓ 2 g i 2 , where we used (4.9) and ℓ 2 = d 2 τ \ell^2=d^2\tau ℓ 2 = d 2 τ Proposition 4.4 , applied with k = j k=j k = j x = z i x=z_{\mathbf{i}} x = z i
∣ E [ ∏ i ∈ T ′ w i ∣ H ] − ∏ i ∈ T ′ w ‾ i ∣ ≤ ε N τ − j . \left|\mathbb{E}\left[\prod_{i\in T'}w_i\mid H\right]-\prod_{i\in T'}\overline{w}_i\right|\leq\varepsilon_N\tau^{-j}. E [ i ∈ T ′ ∏ w i ∣ H ] − i ∈ T ′ ∏ w i ≤ ε N τ − j . where ε N = ε N ( j , Ψ j ) \varepsilon_N=\varepsilon_N(j,\Psi_j) ε N = ε N ( j , Ψ j ) Proposition 4.4 , which does not depend on T ′ T' T ′
We now sum over the subsets T ′ ⊂ C R T'\subset C_R T ′ ⊂ C R ∣ T ′ ∣ = j |T'|=j ∣ T ′ ∣ = j Lemma 4.6 , their number is bounded by
∣ C R ∣ j j ! ≤ ( K R τ ) j j ! , where K R = C I ′ ( 1 + R 2 ) p ∗ . \frac{|C_R|^j}{j!}\leq\frac{(K_R\tau)^j}{j!},\qquad\text{where}\qquad K_R=C_{I'}(1+R^2)^{p_*}. j ! ∣ C R ∣ j ≤ j ! ( K R τ ) j , where K R = C I ′ ( 1 + R 2 ) p ∗ . Hence, on A N ⋆ ∩ A N , j A_N^\star\cap A_{N,j} A N ⋆ ∩ A N , j
∣ E [ P j ( w ) ∣ H ] − P j ( w ‾ ) ∣ ≤ ( K R τ ) j j ! ε N τ − j = K R j ε N j ! . (115) \left|\mathbb{E}\left[P_j(\mathbf{w})\mid H\right]-P_j(\overline{\mathbf{w}})\right|\leq\frac{(K_R\tau)^j}{j!}\varepsilon_N\tau^{-j}=\frac{K_R^j\varepsilon_N}{j!}. \tag*{(115)} ∣ E [ P j ( w ) ∣ H ] − P j ( w ) ∣ ≤ j ! ( K R τ ) j ε N τ − j = j ! K R j ε N . (115) Next, we compare P j ( w ‾ ) P_j(\overline{\mathbf{w}}) P j ( w ) ( ∑ i ∈ C R w ‾ i ) j / j ! (\sum_{i\in C_R}\overline{w}_i)^j/j! ( ∑ i ∈ C R w i ) j / j ! j ≥ 2 j\geq2 j ≥ 2 j = 1 j=1 j = 1 ( ∑ i ∈ C R w ‾ i ) j (\sum_{i\in C_R}\overline{w}_i)^j ( ∑ i ∈ C R w i ) j j j j C R C_R C R j ! P j ( w ‾ ) j!P_j(\overline{\mathbf{w}}) j ! P j ( w ) ( j 2 ) \binom{j}{2} ( 2 j )
( ∑ i ∈ C R w ‾ i 2 ) ( ∑ i ∈ C R w ‾ i ) j − 2 ≤ ( max i ∈ C R w ‾ i ) ( ∑ i ∈ C R w ‾ i ) j − 1 . \left(\sum_{i\in C_R}\overline{w}_i^2\right)\left(\sum_{i\in C_R}\overline{w}_i\right)^{j-2}\leq\left(\max_{i\in C_R}\overline{w}_i\right)\left(\sum_{i\in C_R}\overline{w}_i\right)^{j-1}. ( i ∈ C R ∑ w i 2 ) ( i ∈ C R ∑ w i ) j − 2 ≤ ( i ∈ C R max w i ) ( i ∈ C R ∑ w i ) j − 1 . Let ∥ A h ∥ ∞ = sup E A h ( E ) \|A_h\|_\infty=\sup_E A_h(E) ∥ A h ∥ ∞ = sup E A h ( E ) ψ E \psi_E ψ E
max i ∈ C R w ‾ i ≤ ∥ A h ∥ ∞ 2 π τ , ∑ i ∈ C R w ‾ i ≤ K R ∥ A h ∥ ∞ 2 π τ , \max_{i\in C_R}\overline{w}_i\leq\frac{\|A_h\|_\infty}{2\pi\tau},\qquad\sum_{i\in C_R}\overline{w}_i\leq\frac{K_R\|A_h\|_\infty}{2\pi\tau}, i ∈ C R max w i ≤ 2 π τ ∥ A h ∥ ∞ , i ∈ C R ∑ w i ≤ 2 π τ K R ∥ A h ∥ ∞ , where the second bound follows from the second part of Lemma 4.6 . We conclude that
0 ≤ ( ∑ i ∈ C R w ‾ i ) j − j ! P j ( w ‾ ) ≤ ( j 2 ) ∥ A h ∥ ∞ 2 π τ ( K R ∥ A h ∥ ∞ 2 π τ ) j − 1 = O ( τ − 1 ) . (116) 0 \leq\left(\sum_{i\in C_R}\overline{w}_i\right)^j-j!P_j(\overline{w})\leq\binom{j}{2}\frac{\lVert A_h\rVert_\infty}{2\pi\tau}\left(\frac{K_R\lVert A_h\rVert_\infty}{2\pi\tau}\right)^{j-1}=O\left(\tau^{-1}\right). \tag*{(116)} 0 ≤ ( i ∈ C R ∑ w i ) j − j ! P j ( w ) ≤ ( 2 j ) 2 π τ ∥ A h ∥ ∞ ( 2 π τ K R ∥ A h ∥ ∞ ) j − 1 = O ( τ − 1 ) . (116) Finally, we show that
∑ i ∈ C R w ‾ i ⟶ L R ( f ) in probability . (117) \sum_{i\in C_R}\overline{w}_i \longrightarrow L_R(f)\qquad\text{in probability}. \tag*{(117)} i ∈ C R ∑ w i ⟶ L R ( f ) in probability . (117) For ω ∈ ( 0 , 1 ) \omega\in(0,1) ω ∈ ( 0 , 1 ) [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 )
Q ω + ( y , E ) = 1 2 π min { 1 , ( R + ω − y ) + ω } e − y 2 / 8 A h ( E ) , Q_\omega^+(y,E)=\frac{1}{2\pi}\min\left\{1,\frac{(R+\omega-y)_+}{\omega}\right\}e^{-y^2/8}A_h(E), Q ω + ( y , E ) = 2 π 1 min { 1 , ω ( R + ω − y ) + } e − y 2 /8 A h ( E ) , Q ω − ( y , E ) = 1 2 π min { 1 , ( R − y ) + ω } e − y 2 / 8 A h ( E ) . Q_\omega^-(y,E)=\frac{1}{2\pi}\min\left\{1,\frac{(R-y)_+}{\omega}\right\}e^{-y^2/8}A_h(E). Q ω − ( y , E ) = 2 π 1 min { 1 , ω ( R − y ) + } e − y 2 /8 A h ( E ) . They are continuous with support in [ 0 , R + 1 ] × I h [0,R+1]\times I_h [ 0 , R + 1 ] × I h
Q ω − ( y , E ) ≤ 1 2 π 1 { y ≤ R } e − y 2 / 8 A h ( E ) ≤ Q ω + ( y , E ) . Q_\omega^-(y,E)\leq\frac{1}{2\pi}\boldsymbol{1}\{y\leq R\}e^{-y^2/8}A_h(E)\leq Q_\omega^+(y,E). Q ω − ( y , E ) ≤ 2 π 1 1 { y ≤ R } e − y 2 /8 A h ( E ) ≤ Q ω + ( y , E ) . Note that R ℓ ≤ s R\ell\leq s R ℓ ≤ s N N N (75) . Since A h ( E i ) = 0 A_h(E_i)=0 A h ( E i ) = 0 E i ∈ I h ⊂ I ′ E_i\in I_h\subset I' E i ∈ I h ⊂ I ′
∑ i ∈ C R w ‾ i = 1 τ ∑ i = 1 N − 1 1 2 π 1 { g i ≤ R ℓ } exp ( − g i 2 8 ℓ 2 ) A h ( E i ) . \sum_{i\in C_R}\overline{w}_i=\frac{1}{\tau}\sum_{i=1}^{N-1}\frac{1}{2\pi}\boldsymbol{1}\{g_i\leq R\ell\}\exp\left(-\frac{g_i^2}{8\ell^2}\right)A_h(E_i). i ∈ C R ∑ w i = τ 1 i = 1 ∑ N − 1 2 π 1 1 { g i ≤ R ℓ } exp ( − 8 ℓ 2 g i 2 ) A h ( E i ) . Using (4.27) with $R'=R+1$, we deduce that on A N ⋆ \mathcal{A}_N^\star A N ⋆ N N N
1 τ ∑ 1 ≤ i < j ′ ≤ N Q ω − ( λ j ′ − λ i ℓ , λ i + λ j ′ 2 ) ≤ ∑ i ∈ C R w ‾ i ≤ 1 τ ∑ 1 ≤ i < j ′ ≤ N Q ω + ( λ j ′ − λ i ℓ , λ i + λ j ′ 2 ) . \frac{1}{\tau}\sum_{1\leq i<j'\leq N}Q_\omega^-\left(\frac{\lambda_{j'}-\lambda_i}{\ell},\frac{\lambda_i+\lambda_{j'}}{2}\right)\leq\sum_{i\in C_R}\overline{w}_i\leq\frac{1}{\tau}\sum_{1\leq i<j'\leq N}Q_\omega^+\left(\frac{\lambda_{j'}-\lambda_i}{\ell},\frac{\lambda_i+\lambda_{j'}}{2}\right). τ 1 1 ≤ i < j ′ ≤ N ∑ Q ω − ( ℓ λ j ′ − λ i , 2 λ i + λ j ′ ) ≤ i ∈ C R ∑ w i ≤ τ 1 1 ≤ i < j ′ ≤ N ∑ Q ω + ( ℓ λ j ′ − λ i , 2 λ i + λ j ′ ) . The first part of Proposition 4.1 implies that the left and right sides converge in probability to
∫ − 2 2 ∫ 0 ∞ c ( E ) y Q ω − ( y , E ) d y d E and ∫ − 2 2 ∫ 0 ∞ c ( E ) y Q ω + ( y , E ) d y d E , \int_{-2}^{2}\int_0^\infty c(E)yQ_\omega^-(y,E)\,\mathrm{d}y\,\mathrm{d}E
\qquad\text{and}\qquad
\int_{-2}^{2}\int_0^\infty c(E)yQ_\omega^+(y,E)\,\mathrm{d}y\,\mathrm{d}E, ∫ − 2 2 ∫ 0 ∞ c ( E ) y Q ω − ( y , E ) d y d E and ∫ − 2 2 ∫ 0 ∞ c ( E ) y Q ω + ( y , E ) d y d E , respectively. As ω → 0 \omega\to0 ω → 0
1 2 π ∫ 0 R y e − y 2 / 8 d y ∫ − 2 2 c ( E ) A h ( E ) d E = 1 2 π ⋅ 4 ( 1 − e − R 2 / 8 ) ⋅ π 2 L ( f ) = L R ( f ) , \frac{1}{2\pi}\int_0^R ye^{-y^2/8}\,\mathrm{d}y\int_{-2}^{2}c(E)A_h(E)\,\mathrm{d}E
=\frac{1}{2\pi}\cdot4\left(1-e^{-R^2/8}\right)\cdot\frac{\pi}{2}L(f)=L_R(f), 2 π 1 ∫ 0 R y e − y 2 /8 d y ∫ − 2 2 c ( E ) A h ( E ) d E = 2 π 1 ⋅ 4 ( 1 − e − R 2 /8 ) ⋅ 2 π L ( f ) = L R ( f ) , where in the first equality we used (110) and (94) . Now let ω ′ > 0 \omega'>0 ω ′ > 0 ω ∈ ( 0 , 1 ) \omega\in(0,1) ω ∈ ( 0 , 1 ) L R ( f ) L_R(f) L R ( f ) ω ′ \omega' ω ′ P ( A N ⋆ ) → 1 \mathbb{P}(\mathcal{A}_N^\star)\to1 P ( A N ⋆ ) → 1 ∑ i ∈ C R w ‾ i \sum_{i\in C_R}\overline{w}_i ∑ i ∈ C R w i
∣ ∑ i ∈ C R w ‾ i − L R ( f ) ∣ ≤ 2 ω ′ \left|\sum_{i\in C_R}\overline{w}_i-L_R(f)\right|\leq2\omega' i ∈ C R ∑ w i − L R ( f ) ≤ 2 ω ′ with probability tending to one. The number ω ′ > 0 \omega' > 0 ω ′ > 0
1 j ! ( ∑ i ∈ C R w ˉ i ) j ⟶ L R ( f ) j j ! \frac{1}{j!}\left(\sum_{i\in C_R}\bar{w}_i\right)^j \longrightarrow\frac{L_R(f)^j}{j!} j ! 1 ( i ∈ C R ∑ w ˉ i ) j ⟶ j ! L R ( f ) j in probability.
The estimates (4.42) and (4.43), whose right sides tend to zero, hold on A N ⋆ ∩ A N , j \mathcal{A}_N^\star\cap\mathcal{A}_{N,j} A N ⋆ ∩ A N , j P ( A N ⋆ ∩ A N , j ) → 1 \mathbb{P}(\mathcal{A}_N^\star\cap\mathcal{A}_{N,j}) \to1 P ( A N ⋆ ∩ A N , j ) → 1
Continuous test functions and the truncation. In the next two lemmas and in the proof of Theorem 1.2, we use the elementary inequality
∣ e − y 1 − e − y 2 ∣ ≤ min { 1 , ∣ y 1 − y 2 ∣ } , y 1 , y 2 ≥ 0. (118) \left|e^{-y_1}-e^{-y_2}\right| \le\min\{1,|y_1-y_2|\}, \qquad y_1,y_2 \ge0. \tag*{(118)} e − y 1 − e − y 2 ≤ min { 1 , ∣ y 1 − y 2 ∣ } , y 1 , y 2 ≥ 0. (118) We first extend Lemma 4.9 to continuous test functions.
Lemma 4.10. Let R > 0 R>0 R > 0 x 0 > 0 x_0>0 x 0 > 0 f ∈ C c + f\in C_c^+ f ∈ C c + [ 0 , x 0 ] × I [0,x_0]\times I [ 0 , x 0 ] × I
lim N → ∞ E e − Ξ ^ N , R ( f ) = e − L R ( f ) . (119) \lim_{N\to\infty}\mathbb{E}e^{-\widehat{\Xi}_{N,R}(f)}=e^{-L_R(f)}. \tag*{(119)} N → ∞ lim E e − Ξ N , R ( f ) = e − L R ( f ) . (119) Proof. To apply Lemma 4.9, we approximate f f f h ( r 2 , E ) h(r^2,E) h ( r 2 , E ) h ˉ : R 2 → [ 0 , ∞ ) \bar{h}:\mathbb{R}^2\to[0,\infty) h ˉ : R 2 → [ 0 , ∞ ) h ˉ ( u , E ) = 0 \bar{h}(u,E)=0 h ˉ ( u , E ) = 0 E ∉ ( − 2 , 2 ) E\notin(-2,2) E ∈ / ( − 2 , 2 )
h ˉ ( u , E ) = min { 1 , ( 1 + u ) + } f ( ( max { u , 0 } ) 1 / 2 , E ) \bar{h}(u,E)=\min\{1,(1+u)_+\}f\left((\max\{u,0\})^{1/2},E\right) h ˉ ( u , E ) = min { 1 , ( 1 + u ) + } f ( ( max { u , 0 } ) 1/2 , E ) for u ∈ R u\in\mathbb{R} u ∈ R E ∈ ( − 2 , 2 ) E\in(-2,2) E ∈ ( − 2 , 2 ) h ˉ \bar{h} h ˉ
supp h ˉ ⊂ [ − 1 , x 0 2 ] × I , h ˉ ( r 2 , E ) = f ( r , E ) for r ≥ 0 , E ∈ ( − 2 , 2 ) . \operatorname{supp}\bar{h}\subset[-1,x_0^2]\times I,\qquad\bar{h}(r^2,E)=f(r,E)\quad\text{for }r\ge0,\ E\in(-2,2). supp h ˉ ⊂ [ − 1 , x 0 2 ] × I , h ˉ ( r 2 , E ) = f ( r , E ) for r ≥ 0 , E ∈ ( − 2 , 2 ) . For ω > 0 \omega>0 ω > 0 h ω h_\omega h ω h ˉ \bar{h} h ˉ ω \omega ω h ω ∈ C ∞ ( R 2 ) h_\omega\in C^\infty(\mathbb{R}^2) h ω ∈ C ∞ ( R 2 ) ∥ h ω − h ˉ ∥ ∞ → 0 \|h_\omega-\bar{h}\|_\infty\to0 ∥ h ω − h ˉ ∥ ∞ → 0 ω → 0 \omega\to0 ω → 0 h ˉ \bar{h} h ˉ ω < 1 \omega<1 ω < 1 I I I R ∖ I ′ \mathbb{R}\setminus I' R ∖ I ′ supp h ω ⊂ [ − 2 , x 0 2 + 1 ] × int ( I ′ ) \operatorname{supp}h_\omega\subset[-2,x_0^2+1]\times\operatorname{int}(I') supp h ω ⊂ [ − 2 , x 0 2 + 1 ] × int ( I ′ )
f ω ( r , E ) = h ω ( r 2 , E ) , ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) . f_\omega(r,E)=h_\omega(r^2,E),\qquad(r,E)\in[0,\infty)\times(-2,2). f ω ( r , E ) = h ω ( r 2 , E ) , ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) . Then f f f f ω f_\omega f ω [ 0 , x 0 + 1 ] × I ′ [0,x_0+1]\times I' [ 0 , x 0 + 1 ] × I ′
sup ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) ∣ f ω ( r , E ) − f ( r , E ) ∣ ≤ ∥ h ω − h ˉ ∥ ∞ . \sup_{(r,E)\in[0,\infty)\times(-2,2)}|f_\omega(r,E)-f(r,E)|\le\|h_\omega-\bar{h}\|_\infty. ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) sup ∣ f ω ( r , E ) − f ( r , E ) ∣ ≤ ∥ h ω − h ˉ ∥ ∞ . We now compare the Laplace functionals of Ξ ^ N , R \widehat{\Xi}_{N,R} Ξ N , R f f f f ω f_\omega f ω
∣ E e − Ξ ^ N , R ( f ) − E e − Ξ ^ N , R ( f ω ) ∣ ≤ E min { 1 , ∥ h ω − h ˉ ∥ ∞ Ξ ^ N , R ( [ 0 , x 0 + 1 ] × I ′ ) } ≤ P ( ( A N ⋆ ) c ) + ∥ h ω − h ˉ ∥ ∞ E [ 1 A N ⋆ E [ Ξ ^ N ( [ 0 , x 0 + 1 ] × I ′ ) ∣ H ] ] ≤ o ( 1 ) + C ∥ h ω − h ˉ ∥ ∞ . \begin{aligned}
\left|\mathbb{E}e^{-\widehat{\Xi}_{N,R}(f)}-\mathbb{E}e^{-\widehat{\Xi}_{N,R}(f_\omega)}\right|
\le\mathbb{E}\min\left\{1,\|h_\omega-\bar{h}\|_\infty\widehat{\Xi}_{N,R}([0,x_0+1]\times I')\right\} \\
&\le\mathbb{P}\left((\mathcal{A}_N^\star)^c\right)+\|h_\omega-\bar{h}\|_\infty\mathbb{E}\left[\mathbf{1}_{\mathcal{A}_N^\star}\mathbb{E}\left[\widehat{\Xi}_N([0,x_0+1]\times I')\mid H\right]\right] \\
&\le o(1)+C\|h_\omega-\bar{h}\|_\infty.
\end{aligned} E e − Ξ N , R ( f ) − E e − Ξ N , R ( f ω ) ≤ E min { 1 , ∥ h ω − h ˉ ∥ ∞ Ξ N , R ([ 0 , x 0 + 1 ] × I ′ ) } ≤ P ( ( A N ⋆ ) c ) + ∥ h ω − h ˉ ∥ ∞ E [ 1 A N ⋆ E [ Ξ N ([ 0 , x 0 + 1 ] × I ′ ) ∣ H ] ] ≤ o ( 1 ) + C ∥ h ω − h ˉ ∥ ∞ . In the last inequality, we used (4.32) with x 0 + 1 x_0+1 x 0 + 1 x 0 x_0 x 0 Ξ ^ N , R ≤ Ξ ^ N \widehat{\Xi}_{N,R} \le\widehat{\Xi}_N Ξ N , R ≤ Ξ N
lim N → ∞ E e − Ξ ^ N , R ( f ϖ ) = e − L R ( f ϖ ) . \lim_{N\to\infty} \mathbb{E}e^{-\widehat{\Xi}_{N,R}(f_\varpi)}=e^{-L_R(f_\varpi)}. N → ∞ lim E e − Ξ N , R ( f ϖ ) = e − L R ( f ϖ ) . Moreover,
∣ L R ( f ϖ ) − L R ( f ) ∣ ≤ ∫ ∣ f ϖ − f ∣ d ν R ≤ ∥ h ϖ − h ˉ ∥ ∞ ν ( [ 0 , x 0 + 1 ] × I ′ ) . \left|L_R(f_\varpi)-L_R(f)\right|\leq\int|f_\varpi-f|\,\mathrm{d}\nu_R\leq\|h_\varpi-\bar{h}\|_\infty\nu([0,x_0+1]\times I'). ∣ L R ( f ϖ ) − L R ( f ) ∣ ≤ ∫ ∣ f ϖ − f ∣ d ν R ≤ ∥ h ϖ − h ˉ ∥ ∞ ν ([ 0 , x 0 + 1 ] × I ′ ) . Combining these estimates, we obtain
lim sup N → ∞ ∣ E e − Ξ ^ N , R ( f ) − e − L R ( f ) ∣ ≤ ( C + ν ( [ 0 , x 0 + 1 ] × I ′ ) ) ∥ h ϖ − h ˉ ∥ ∞ , \limsup_{N\to\infty}\left|\mathbb{E}e^{-\widehat{\Xi}_{N,R}(f)}-e^{-L_R(f)}\right|\leq\left(C+\nu([0,x_0+1]\times I')\right)\|h_\varpi-\bar{h}\|_\infty, N → ∞ lim sup E e − Ξ N , R ( f ) − e − L R ( f ) ≤ ( C + ν ([ 0 , x 0 + 1 ] × I ′ ) ) ∥ h ϖ − h ˉ ∥ ∞ , and letting ϖ → 0 \varpi\to0 ϖ → 0
Next, we remove the truncation, using the bound (4.33), which relies on the second part of Proposition 4.4.
Lemma 4.11. Let x 0 > 0 x_0>0 x 0 > 0 f ∈ C c + f\in C_c^+ f ∈ C c + [ 0 , x 0 ] × I [0,x_0]\times I [ 0 , x 0 ] × I
lim N → ∞ E e − Ξ ^ N ( f ) = e − L ( f ) . (120) \lim_{N\to\infty}\mathbb{E}e^{-\widehat{\Xi}_N(f)}=e^{-L(f)}. \tag*{(120)} N → ∞ lim E e − Ξ N ( f ) = e − L ( f ) . (120) Proof. Let C , c > 0 C,c>0 C , c > 0 x 0 x_0 x 0 R > 0 R>0 R > 0
∣ E e − Ξ ^ N ( f ) − E e − Ξ ^ N , R ( f ) ∣ ≤ E min { 1 , ∥ f ∥ ∞ ( Ξ ^ N − Ξ ^ N , R ) ( [ 0 , x 0 ] × I ′ ) } ≤ P ( ( A N ) c ) + C ∥ f ∥ ∞ sup y ≥ R ( ( 1 + y 2 ) p ∗ e − c y ) , \begin{aligned}
\left|\mathbb{E}e^{-\widehat{\Xi}_N(f)}-\mathbb{E}e^{-\widehat{\Xi}_{N,R}(f)}\right|\leq\mathbb{E}\min\left\{1,\|f\|_\infty(\widehat{\Xi}_N-\widehat{\Xi}_{N,R})([0,x_0]\times I')\right\} \\
&\leq\mathbb{P}\left((\mathcal{A}_N)^c\right)+C\|f\|_\infty\sup_{y\geq R}\left((1+y^2)^{p_*}e^{-cy}\right),
\end{aligned} E e − Ξ N ( f ) − E e − Ξ N , R ( f ) ≤ E min { 1 , ∥ f ∥ ∞ ( Ξ N − Ξ N , R ) ([ 0 , x 0 ] × I ′ ) } ≤ P ( ( A N ) c ) + C ∥ f ∥ ∞ y ≥ R sup ( ( 1 + y 2 ) p ∗ e − cy ) , where we used (4.45) and the fact that f f f [ 0 , x 0 ] × I [0,x_0]\times I [ 0 , x 0 ] × I
∣ e − L R ( f ) − e − L ( f ) ∣ ≤ L ( f ) − L R ( f ) = e − R 2 / 8 L ( f ) . \left|e^{-L_R(f)}-e^{-L(f)}\right|\leq L(f)-L_R(f)=e^{-R^2/8}L(f). e − L R ( f ) − e − L ( f ) ≤ L ( f ) − L R ( f ) = e − R 2 /8 L ( f ) . Together with Lemma 4.10, these two bounds yield
lim sup N → ∞ ∣ E e − Ξ ^ N ( f ) − e − L ( f ) ∣ ≤ C ∥ f ∥ ∞ sup y ≥ R ( ( 1 + y 2 ) p ∗ e − c y ) + e − R 2 / 8 L ( f ) . \limsup_{N\to\infty}\left|\mathbb{E}e^{-\widehat{\Xi}_N(f)}-e^{-L(f)}\right|\leq C\|f\|_\infty\sup_{y\geq R}\left((1+y^2)^{p_*}e^{-cy}\right)+e^{-R^2/8}L(f). N → ∞ lim sup E e − Ξ N ( f ) − e − L ( f ) ≤ C ∥ f ∥ ∞ y ≥ R sup ( ( 1 + y 2 ) p ∗ e − cy ) + e − R 2 /8 L ( f ) . Letting R → ∞ R\to\infty R → ∞
Void probabilities. We will deduce (1.6) from the convergence of Ξ N \Xi_N Ξ N
Lemma 4.12. Let S ⊂ [ 0 , ∞ ) × ( − 2 , 2 ) \mathcal{S} \subset[0,\infty) \times(-2,2) S ⊂ [ 0 , ∞ ) × ( − 2 , 2 ) [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty) \times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) ∂ S \partial\mathcal{S} ∂ S ν ( ∂ S ) = 0 \nu(\partial\mathcal{S}) = 0 ν ( ∂ S ) = 0 ( μ N ) N ≥ 2 (\mu_N)_{N\geq2} ( μ N ) N ≥ 2 [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty) \times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) Ξ \Xi Ξ
lim N → ∞ P ( μ N ( S ) = 0 ) = e − ν ( S ) . \lim_{N\to\infty} \mathbb{P}(\mu_N(\mathcal{S}) = 0) = e^{-\nu(\mathcal{S})}. N → ∞ lim P ( μ N ( S ) = 0 ) = e − ν ( S ) . Proof. As Ξ \Xi Ξ ν \nu ν ν ( ∂ S ) = 0 \nu(\partial\mathcal{S}) = 0 ν ( ∂ S ) = 0 Ξ ( ∂ S ) = 0 \Xi(\partial\mathcal{S}) = 0 Ξ ( ∂ S ) = 0 S \mathcal{S} S μ ↦ μ ( S ) \mu\mapsto\mu(\mathcal{S}) μ ↦ μ ( S ) μ \mu μ μ ( ∂ S ) = 0 \mu(\partial\mathcal{S}) = 0 μ ( ∂ S ) = 0 [42 Chapter 4] ). The continuous mapping theorem therefore shows that μ N ( S ) \mu_N(\mathcal{S}) μ N ( S ) Ξ ( S ) \Xi(\mathcal{S}) Ξ ( S ) ν ( S ) \nu(\mathcal{S}) ν ( S )
lim N → ∞ P ( μ N ( S ) = 0 ) = P ( Ξ ( S ) = 0 ) = e − ν ( S ) . \lim_{N\to\infty} \mathbb{P}(\mu_N(\mathcal{S}) = 0) = \mathbb{P}(\Xi(\mathcal{S}) = 0) = e^{-\nu(\mathcal{S})}. N → ∞ lim P ( μ N ( S ) = 0 ) = P ( Ξ ( S ) = 0 ) = e − ν ( S ) . □ \square □
Proof of Theorem 1.2. Let f ∈ C c + f \in C^+_{\mathrm{c}} f ∈ C c + f f f f f f [ 0 , x 0 ] × I [0,x_0] \times I [ 0 , x 0 ] × I x 0 > 0 x_0 > 0 x 0 > 0 I I I I ′ I' I ′ C \mathcal{C} C C R \mathcal{C}_{R} C R Ξ ^ N \widehat{\Xi}_N Ξ N Ξ ^ N , R \widehat{\Xi}_{N,R} Ξ N , R A N ⋆ \mathcal{A}^{\star}_N A N ⋆ I I I f f f [ 0 , ∞ ) × R [0,\infty) \times\mathbb{R} [ 0 , ∞ ) × R ϖ > 0 \varpi> 0 ϖ > 0
osc f ( ϖ ) = sup { ∣ f ( r , E ) − f ( r ′ , E ′ ) ∣ : ( r , E ) , ( r ′ , E ′ ) ∈ [ 0 , ∞ ) × R , ∣ ( r , E ) − ( r ′ , E ′ ) ∣ ≤ ϖ } . (121) \operatorname{osc}_{f}(\varpi) = \sup\left\{ \lvert f(r,E) - f(r',E') \rvert: (r,E),(r',E') \in[0,\infty) \times\mathbb{R},\ \lvert(r,E) - (r',E') \rvert\leq\varpi\right\}. \tag*{(121)} osc f ( ϖ ) = sup { ∣ f ( r , E ) − f ( r ′ , E ′ )∣ : ( r , E ) , ( r ′ , E ′ ) ∈ [ 0 , ∞ ) × R , ∣( r , E ) − ( r ′ , E ′ )∣ ≤ ϖ } . (121) We will show that
lim N → ∞ E e − Ξ N ( f ) = e − L ( f ) . (122) \lim_{N\to\infty} \mathbb{E}e^{-\Xi_N(f)} = e^{-L(f)}. \tag*{(122)} N → ∞ lim E e − Ξ N ( f ) = e − L ( f ) . (122) By Lemma 4.11, the limit (4.47) holds for the process Ξ ^ N \widehat{\Xi}_N Ξ N Ξ ^ N \widehat{\Xi}_N Ξ N Ξ N \Xi_N Ξ N
Passage to the resampled matrix and to H H H Define
Ξ ~ N = ∑ i = 1 N − 1 δ ( d − 1 g ~ i , λ ~ i ) , \widetilde{\Xi}_N = \sum_{i=1}^{N-1} \delta_{(d^{-1}\widetilde{g}_i,\widetilde{\lambda}_i)}, Ξ N = i = 1 ∑ N − 1 δ ( d − 1 g i , λ i ) , regarded as a measure on [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty) \times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) Ξ N \Xi_N Ξ N
Ξ ~ N ( f ) = ∑ i f ( d − 1 g ~ i , λ ~ i ) . \widetilde{\Xi}_N(f) = \sum_i f(d^{-1}\widetilde{g}_i,\widetilde{\lambda}_i). Ξ N ( f ) = i ∑ f ( d − 1 g i , λ i ) . The eigenvalues are continuous functions of the matrix entries, so Ξ N ( f ) \Xi_N(f) Ξ N ( f ) H H H Ξ ~ N ( f ) \widetilde{\Xi}_N(f) Ξ N ( f ) H ~ \widetilde{H} H
E e − Ξ N ( f ) = E e − Ξ ~ N ( f ) . (123) \mathbb{E}e^{-\Xi_N(f)}=\mathbb{E}e^{-\widetilde{\Xi}_N(f)}. \tag*{(123)} E e − Ξ N ( f ) = E e − Ξ N ( f ) . (123) We now compare Ξ ~ N ( f ) \widetilde{\Xi}_N(f) Ξ N ( f ) Ξ ^ N ( f ) \widehat{\Xi}_N(f) Ξ N ( f ) Ξ ^ N \widehat{\Xi}_N Ξ N E i E_i E i H H H Ξ ~ N \widetilde{\Xi}_N Ξ N λ ~ i \widetilde{\lambda}_i λ i H ~ \widetilde{H} H δ N ≤ d \delta_N\le d δ N ≤ d N N N
Suppose that N N N A N ⋆ ∩ B N \mathcal{A}^{\star}_N\cap\mathcal{B}_N A N ⋆ ∩ B N i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] f ( d − 1 g ~ i , λ ~ i ) ≠ 0 f(d^{-1}\widetilde{g}_i,\widetilde{\lambda}_i)\ne0 f ( d − 1 g i , λ i ) = 0 g ~ i ≤ x 0 d \widetilde{g}_i\le x_0d g i ≤ x 0 d λ ~ i ∈ I \widetilde{\lambda}_i\in I λ i ∈ I x 0 d < s / 2 x_0d<s/2 x 0 d < s /2 N N N g i ≤ s g_i\le s g i ≤ s
∣ E i − λ ~ i ∣ ≤ ∣ E i − E ~ i ∣ + g ~ i 2 ≤ w N + δ N 2 + x 0 d 2 , 2 ∣ z i + S i ∣ ≤ d − 1 g ~ i + δ N d ≤ x 0 + 1. |E_i-\widetilde{\lambda}_i|\le|E_i-\widetilde{E}_i|+\frac{\widetilde{g}_i}{2}\le w_N+\frac{\delta_N}{2}+\frac{x_0d}{2},\qquad2|z_i+S_i|\le d^{-1}\widetilde{g}_i+\frac{\delta_N}{d}\le x_0+1. ∣ E i − λ i ∣ ≤ ∣ E i − E i ∣ + 2 g i ≤ w N + 2 δ N + 2 x 0 d , 2∣ z i + S i ∣ ≤ d − 1 g i + d δ N ≤ x 0 + 1. In particular, E i ∈ I ′ E_i\in I' E i ∈ I ′ N N N i ∈ C i\in C i ∈ C
We have shown that every nonzero term of Ξ ~ N ( f ) \widetilde{\Xi}_N(f) Ξ N ( f ) i ∈ C i\in C i ∈ C 2 ∣ z i + S i ∣ ≤ x 0 + 1 2|z_i+S_i|\le x_0+1 2∣ z i + S i ∣ ≤ x 0 + 1 Ξ ^ N ( f ) \widehat{\Xi}_N(f) Ξ N ( f )
Ξ ~ N ( f ) − Ξ ^ N ( f ) = ∑ i ∈ C 2 ∣ z i + S i ∣ ≤ x 0 + 1 ( f ( d − 1 g ~ i , λ ~ i ) − f ( 2 ∣ z i + S i ∣ , E i ) ) . \widetilde{\Xi}_N(f)-\widehat{\Xi}_N(f)=\sum_{\substack{i\in C\\2|z_i+S_i|\le x_0+1}}\left(f(d^{-1}\widetilde{g}_i,\widetilde{\lambda}_i)-f(2|z_i+S_i|,E_i)\right). Ξ N ( f ) − Ξ N ( f ) = i ∈ C 2∣ z i + S i ∣ ≤ x 0 + 1 ∑ ( f ( d − 1 g i , λ i ) − f ( 2∣ z i + S i ∣ , E i ) ) . For every i i i
g ~ i ≤ d ( 2 ∣ z i + S i ∣ + δ N d ) ≤ ( x 0 + 2 ) d , \widetilde{g}_i\le d\left(2|z_i+S_i|+\frac{\delta_N}{d}\right)\le(x_0+2)d, g i ≤ d ( 2∣ z i + S i ∣ + d δ N ) ≤ ( x 0 + 2 ) d , ∣ d − 1 g ~ i − 2 ∣ z i + S i ∣ ∣ + ∣ λ ~ i − E i ∣ ≤ δ N d + g ~ i 2 + ∣ E ~ i − E i ∣ ≤ δ N d + ( x 0 + 2 ) d 2 + w N + δ N 2 . |d^{-1}\widetilde{g}_i-2|z_i+S_i||+|\widetilde{\lambda}_i-E_i|\le\frac{\delta_N}{d}+\frac{\widetilde{g}_i}{2}+|\widetilde{E}_i-E_i|\le\frac{\delta_N}{d}+\frac{(x_0+2)d}{2}+w_N+\frac{\delta_N}{2}. ∣ d − 1 g i − 2∣ z i + S i ∣∣ + ∣ λ i − E i ∣ ≤ d δ N + 2 g i + ∣ E i − E i ∣ ≤ d δ N + 2 ( x 0 + 2 ) d + w N + 2 δ N . We denote the right side of the second line by ς N \varsigma_N ς N A N ⋆ ∩ B N \mathcal{A}^{\star}_N\cap\mathcal{B}_N A N ⋆ ∩ B N
∣ Ξ ~ N ( f ) − Ξ ^ N ( f ) ∣ ≤ osc f ( ς N ) Ξ ^ N ( [ 0 , x 0 + 1 ] × I ′ ) , |\widetilde{\Xi}_N(f)-\widehat{\Xi}_N(f)|\le\operatorname{osc}_f(\varsigma_N)\widehat{\Xi}_N([0,x_0+1]\times I'), ∣ Ξ N ( f ) − Ξ N ( f ) ∣ ≤ osc f ( ς N ) Ξ N ([ 0 , x 0 + 1 ] × I ′ ) , and osc f ( ς N ) = o ( 1 ) \operatorname{osc}_f(\varsigma_N)=o(1) osc f ( ς N ) = o ( 1 ) f f f
We can now bound the difference of the Laplace functionals. Using the first part of Proposition 4.5 and (4.32) with x 0 + 1 x_0+1 x 0 + 1 x 0 x_0 x 0
∣ E e − Ξ ~ N ( f ) − E e − Ξ ^ N ( f ) ∣ ≤ P ( ( A N ⋆ ) c ) + E [ 1 A N ⋆ P ( B N c ∣ H ) ] |\mathbb{E}e^{-\widetilde{\Xi}_N(f)}-\mathbb{E}e^{-\widehat{\Xi}_N(f)}|
\le\mathbb{P}((\mathcal{A}^{\star}_N)^c)+\mathbb{E}\left[1_{\mathcal{A}^{\star}_N}\mathbb{P}(\mathcal{B}_N^c\mid H)\right] ∣ E e − Ξ N ( f ) − E e − Ξ N ( f ) ∣ ≤ P (( A N ⋆ ) c ) + E [ 1 A N ⋆ P ( B N c ∣ H ) ] + osc f ( ς N ) E [ 1 A N ⋆ E [ Ξ ^ N ( [ 0 , x 0 + 1 ] × I ′ ) ∣ H ] ] ≤ o ( 1 ) + o ( 1 ) + C osc f ( ς N ) = o ( 1 ) . \begin{aligned}
\qquad+\operatorname{osc}_f(\varsigma_N)\mathbb{E}\left[1_{\mathcal{A}^{\star}_N}\mathbb{E}\left[\widehat{\Xi}_N([0,x_0+1]\times I')\mid H\right]\right] \\
&\le o(1)+o(1)+C\operatorname{osc}_f(\varsigma_N)=o(1).
\end{aligned} + osc f ( ς N ) E [ 1 A N ⋆ E [ Ξ N ([ 0 , x 0 + 1 ] × I ′ ) ∣ H ] ] ≤ o ( 1 ) + o ( 1 ) + C osc f ( ς N ) = o ( 1 ) . Here the conditional probability of B N C \mathcal{B}_N^C B N C ∣ e − Ξ ~ N ( f ) − e − Ξ ‾ N ( f ) ∣ ≤ 1 \lvert e^{-\widetilde{\Xi}_N(f)}-e^{-\overline{\Xi}_N(f)}\rvert\leq1 ∣ e − Ξ N ( f ) − e − Ξ N ( f ) ∣ ≤ 1
Vague convergence. The test function f ∈ C c + f\in C_c^+ f ∈ C c + f ∈ C c + f\in C_c^+ f ∈ C c + e − L ( f ) = E e − Ξ ( f ) e^{-L(f)}=\mathbb{E}e^{-\Xi(f)} e − L ( f ) = E e − Ξ ( f ) [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) Ξ N \Xi_N Ξ N ν \nu ν [42 Chapter 4] ), the convergence of E e − Ξ N ( f ) \mathbb{E}e^{-\Xi_N(f)} E e − Ξ N ( f ) E e − Ξ ( f ) \mathbb{E}e^{-\Xi(f)} E e − Ξ ( f ) f ∈ C c + f\in C_c^+ f ∈ C c + Ξ N \Xi_N Ξ N Ξ \Xi Ξ
The smallest gap. In this step, I I I I ‾ ⊂ ( − 2 , 2 ) \overline{I}\subset(-2,2) I ⊂ ( − 2 , 2 ) x ≥ 0 x\geq0 x ≥ 0 S = [ 0 , x ] × I \mathcal{S}=[0,x]\times I S = [ 0 , x ] × I Ξ N \Xi_N Ξ N
{ N 3 / 2 min λ i ∈ I g i > x } = { Ξ N ( S ) = 0 } . \left\{N^{3/2}\min_{\lambda_i\in I}g_i>x\right\}=\{\Xi_N(\mathcal{S})=0\}. { N 3/2 λ i ∈ I min g i > x } = { Ξ N ( S ) = 0 } . where the minimum is over i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] λ i ∈ I \lambda_i\in I λ i ∈ I + ∞ +\infty + ∞ S \mathcal{S} S [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) [ 0 , x ] × I ‾ [0,x]\times\overline{I} [ 0 , x ] × I S \mathcal{S} S [ 0 , x ) × int ( I ) [0,x)\times\operatorname{int}(I) [ 0 , x ) × int ( I ) ( { x } × I ‾ ) ∪ ( [ 0 , x ] × ( I ‾ ∖ int ( I ) ) ) \bigl(\{x\}\times\overline{I}\bigr)\cup\bigl([0,x]\times(\overline{I}\setminus\operatorname{int}(I))\bigr) ( { x } × I ) ∪ ( [ 0 , x ] × ( I ∖ int ( I )) ) I ‾ ∖ int ( I ) \overline{I}\setminus\operatorname{int}(I) I ∖ int ( I ) ν \nu ν ν ( ∂ S ) = 0 \nu(\partial\mathcal{S})=0 ν ( ∂ S ) = 0
ν ( S ) = ∫ I ∫ 0 x c ( E ) y d y d E = π 2 x 2 12 ∫ I ρ s c ( E ) 3 d E (124) \nu(\mathcal{S})=\int_I\int_0^x c(E)y\,\mathrm{d}y\,\mathrm{d}E=\frac{\pi^2x^2}{12}\int_I\rho_{\mathrm{sc}}(E)^3\,\mathrm{d}E \tag*{(124)} ν ( S ) = ∫ I ∫ 0 x c ( E ) y d y d E = 12 π 2 x 2 ∫ I ρ sc ( E ) 3 d E (124) where we used the definition of c ( E ) c(E) c ( E ) μ N = Ξ N \mu_N=\Xi_N μ N = Ξ N
lim N → ∞ P ( Ξ N ( S ) = 0 ) = e − ν ( S ) . \lim_{N\to\infty}\mathbb{P}\bigl(\Xi_N(\mathcal{S})=0\bigr)=e^{-\nu(\mathcal{S})}. N → ∞ lim P ( Ξ N ( S ) = 0 ) = e − ν ( S ) . Together with (4.51), this proves the second claim of Theorem 1.2.
A comparison estimate for pair statistics
In this section, we compare Laplace transforms of pair statistics of H H H H t H_t H t (36) . In Section 6, pair statistics of H t H_t H t t t t H t H_t H t H H H ℓ α = N − 1 − α \ell_\alpha= N^{-1-\alpha} ℓ α = N − 1 − α
Our argument follows the continuity estimate of Bourgade and Yau [11 Lemma A.1] (see also [23 Chapter 15] ), which is a continuous version of the Lindeberg replacement strategy [15 , 70 ] along the Ornstein–Uhlenbeck interpolation. It requires bounds on the derivatives of the observable in one matrix entry, up to order three, or four when E X 3 = 0 \mathbb{E}X^3 = 0 E X 3 = 0 ( λ i − λ j ) − 1 (\lambda_i - \lambda_j)^{-1} ( λ i − λ j ) − 1 ℓ α \ell_\alpha ℓ α [ − 4 , 4 ] [-4,4] [ − 4 , 4 ] N − α N^{-\alpha} N − α N − 1 / 2 N^{-1/2} N − 1/2
§ 7 introduces the smoothed energy weights. In § 7.1 , we show that a complex perturbation of one entry of a regular matrix keeps its spectrum in the thin rectangle; this is the only place in this section where we use the conditions of Definition 3.1. § 7.2 constructs the holomorphic extension, and § 7.3 proves the comparison estimate. The proof uses the observable only through the two properties listed in Remark 5.10.
Smoothed energy weights
Recall the scale ℓ α = N − 1 − α \ell_\alpha= N^{-1-\alpha} ℓ α = N − 1 − α α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) N × N N \times N N × N A A A φ : R → R \varphi:\mathbb{R}\to\mathbb{R} φ : R → R f : [ 0 , ∞ ) → R f:[0,\infty)\to\mathbb{R} f : [ 0 , ∞ ) → R
X α ( A ; φ , f ) = ∑ 1 ≤ i < j ≤ N φ ( λ i ( A ) + λ j ( A ) 2 ) f ( λ j ( A ) − λ i ( A ) ℓ α ) . (125) \mathcal{X}_\alpha(A;\varphi,f)=\sum_{1\leq i<j\leq N}\varphi\left(\frac{\lambda_i(A)+\lambda_j(A)}{2}\right)f\left(\frac{\lambda_j(A)-\lambda_i(A)}{\ell_\alpha}\right). \tag*{(125)} X α ( A ; φ , f ) = 1 ≤ i < j ≤ N ∑ φ ( 2 λ i ( A ) + λ j ( A ) ) f ( ℓ α λ j ( A ) − λ i ( A ) ) . (125) Throughout the rest of this section, we fix α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) f p f_p f p (68) . For an integer p ≥ 1 p \geq1 p ≥ 1 f p f_p f p C ∖ { i , − i } \mathbb{C}\setminus\{i,-i\} C ∖ { i , − i } f p ( z ) = ( 1 + z 2 ) − p f_p(z)=(1+z^2)^{-p} f p ( z ) = ( 1 + z 2 ) − p
We set h N = N − 1 / 10 h_N=N^{-1/10} h N = N − 1/10 φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 ))
φ N ( w ) = 1 2 π h N ∫ R φ ( y ) exp ( − ( w − y ) 2 2 h N 2 ) d y , w ∈ C . (126) \varphi_N(w)=\frac{1}{\sqrt{2\pi}h_N}\int_{\mathbb{R}}\varphi(y)\exp\left(-\frac{(w-y)^2}{2h_N^2}\right)\,\mathrm{d}y,\qquad w\in\mathbb{C}. \tag*{(126)} φ N ( w ) = 2 π h N 1 ∫ R φ ( y ) exp ( − 2 h N 2 ( w − y ) 2 ) d y , w ∈ C . (126) We use this notation for every nonnegative function in C c ∞ ( ( − 2 , 2 ) ) C^\infty_c((-2,2)) C c ∞ (( − 2 , 2 )) φ j , N \varphi_{j,N} φ j , N Definition 5.4 below denotes the function defined by (126) with φ j \varphi_j φ j φ \varphi φ φ N \varphi_N φ N x x x φ N ( x ) \varphi_N(x) φ N ( x ) φ \varphi φ x x x h N 2 h_N^2 h N 2 z ∈ C ∖ { 0 } z\in\mathbb{C}\setminus\{0\} z ∈ C ∖ { 0 } arg z ∈ ( − π , π ] \arg z\in(-\pi,\pi] arg z ∈ ( − π , π ]
Part (3) of the following lemma controls the argument of φ N \varphi_N φ N § 7.2 , and we will use parts (1), (2), (4), and (5) in Section 6 and Section 9.
Lemma 5.1. Let φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) N ≥ 1 N\geq1 N ≥ 1
(1) For all x ∈ R x\in\mathbb{R} x ∈ R 0 ≤ φ N ( x ) ≤ ∥ φ ∥ ∞ 0\leq\varphi_N(x)\leq\|\varphi\|_\infty 0 ≤ φ N ( x ) ≤ ∥ φ ∥ ∞ φ \varphi φ φ N ( x ) > 0 \varphi_N(x)>0 φ N ( x ) > 0 x ∈ R x\in\mathbb{R} x ∈ R
(2) For all x ∈ R x\in\mathbb{R} x ∈ R ∣ x ∣ ≤ 5 |x|\leq5 ∣ x ∣ ≤ 5 w ∈ C w\in\mathbb{C} w ∈ C ∣ w − x ∣ ≤ 1 |w-x|\leq1 ∣ w − x ∣ ≤ 1
∣ φ N ( w ) − φ N ( x ) ∣ ≤ ( exp ( 8 ∣ w − x ∣ h N − 2 ) − 1 ) φ N ( x ) . (127) |\varphi_N(w)-\varphi_N(x)|\leq\left(\exp(8|w-x|h_N^{-2})-1\right)\varphi_N(x). \tag*{(127)} ∣ φ N ( w ) − φ N ( x ) ∣ ≤ ( exp ( 8∣ w − x ∣ h N − 2 ) − 1 ) φ N ( x ) . (127) If moreover w ∈ R w\in\mathbb{R} w ∈ R
exp ( − 8 ∣ w − x ∣ h N − 2 ) φ N ( x ) ≤ φ N ( w ) ≤ exp ( 8 ∣ w − x ∣ h N − 2 ) φ N ( x ) . (128) \exp(-8|w-x|h_N^{-2})\varphi_N(x)\leq\varphi_N(w)\leq\exp(8|w-x|h_N^{-2})\varphi_N(x). \tag*{(128)} exp ( − 8∣ w − x ∣ h N − 2 ) φ N ( x ) ≤ φ N ( w ) ≤ exp ( 8∣ w − x ∣ h N − 2 ) φ N ( x ) . (128) (3) For all w ∈ C w\in\mathbb{C} w ∈ C ( ∣ Re w ∣ + 2 ) ∣ Im w ∣ ≤ h N 2 (|\operatorname{Re}w|+2)|\operatorname{Im}w|\leq h_N^2 ( ∣ Re w ∣ + 2 ) ∣ Im w ∣ ≤ h N 2 φ N ( w ) = 0 \varphi_N(w)=0 φ N ( w ) = 0
∣ arg φ N ( w ) ∣ ≤ ( ∣ Re w ∣ + 2 ) ∣ Im w ∣ h N − 2 . (129) |\arg\varphi_N(w)|\leq(|\operatorname{Re}w|+2)|\operatorname{Im}w|h_N^{-2}. \tag*{(129)} ∣ arg φ N ( w ) ∣ ≤ ( ∣ Re w ∣ + 2 ) ∣ Im w ∣ h N − 2 . (129) (4) For all x ∈ R x\in\mathbb{R} x ∈ R
φ N ( x ) ≤ 2 ∥ φ ∥ ∞ exp ( − dist ( x , supp φ ) 2 2 h N 2 ) . (130) \varphi_N(x)\leq2\|\varphi\|_\infty\exp\left(-\frac{\operatorname{dist}(x,\operatorname{supp}\varphi)^2}{2h_N^2}\right). \tag*{(130)} φ N ( x ) ≤ 2∥ φ ∥ ∞ exp ( − 2 h N 2 dist ( x , supp φ ) 2 ) . (130) sup x ∈ R ∣ φ N ( x ) − φ ( x ) ∣ ≤ ∥ φ ′ ∥ ∞ h N . (131) \sup_{x\in\mathbb{R}} \lvert\varphi_N(x)-\varphi(x)\rvert\le\lVert\varphi' \rVert_\infty h_N. \tag*{(131)} x ∈ R sup ∣ φ N ( x ) − φ ( x )∣ ≤ ∥ φ ′ ∥ ∞ h N . (131) Proof. The first claim in part (1) holds since φ N ( x ) \varphi_N(x) φ N ( x ) φ \varphi φ φ \varphi φ
We now prove part (2). Fix x ∈ R x\in\mathbb{R} x ∈ R ∣ x ∣ ≤ 5 \lvert x\rvert\le5 ∣ x ∣ ≤ 5 w ∈ C w\in\mathbb{C} w ∈ C ∣ w − x ∣ ≤ 1 \lvert w-x\rvert\le1 ∣ w − x ∣ ≤ 1 y ∈ supp φ y\in\operatorname{supp}\varphi y ∈ supp φ ∣ x − y ∣ ≤ 7 \lvert x-y\rvert\le7 ∣ x − y ∣ ≤ 7 ( w − y ) 2 (w-y)^2 ( w − y ) 2 x − y x-y x − y
exp ( − ( w − y ) 2 2 h N 2 ) = exp ( − ( x − y ) 2 2 h N 2 ) exp ( − 2 ( x − y ) ( w − x ) + ( w − x ) 2 2 h N 2 ) . (132) \exp\left(-\frac{(w-y)^2}{2h_N^2}\right)
=
\exp\left(-\frac{(x-y)^2}{2h_N^2}\right)
\exp\left(-\frac{2(x-y)(w-x)+(w-x)^2}{2h_N^2}\right). \tag*{(132)} exp ( − 2 h N 2 ( w − y ) 2 ) = exp ( − 2 h N 2 ( x − y ) 2 ) exp ( − 2 h N 2 2 ( x − y ) ( w − x ) + ( w − x ) 2 ) . (132) The exponent in the second factor satisfies
∣ 2 ( x − y ) ( w − x ) + ( w − x ) 2 ∣ 2 h N 2 ≤ 14 ∣ w − x ∣ + ∣ w − x ∣ 2 2 h N 2 ≤ 8 ∣ w − x ∣ h N − 2 . (133) \frac{\left\lvert2(x-y)(w-x)+(w-x)^2\right\rvert}{2h_N^2}
\le
\frac{14\lvert w-x\rvert+\lvert w-x\rvert^2}{2h_N^2}
\le
8\lvert w-x\rvert h_N^{-2}. \tag*{(133)} 2 h N 2 2 ( x − y ) ( w − x ) + ( w − x ) 2 ≤ 2 h N 2 14 ∣ w − x ∣ + ∣ w − x ∣ 2 ≤ 8 ∣ w − x ∣ h N − 2 . (133) Using ∣ e z − 1 ∣ ≤ e ∣ z ∣ − 1 \lvert e^z-1\rvert\le e^{\lvert z\rvert}-1 ∣ e z − 1 ∣ ≤ e ∣ z ∣ − 1 z ∈ C z\in\mathbb{C} z ∈ C (132) against φ ( y ) d y \varphi(y)\,\mathrm{d}y φ ( y ) d y
∣ φ N ( w ) − φ N ( x ) ∣ ≤ 1 2 π h N ∫ R φ ( y ) exp ( − ( x − y ) 2 2 h N 2 ) ( e 8 ∣ w − x ∣ h N − 2 − 1 ) d y = ( e 8 ∣ w − x ∣ h N − 2 − 1 ) φ N ( x ) , \begin{aligned}
\lvert\varphi_N(w)-\varphi_N(x)\rvert
&\le
\frac{1}{\sqrt{2\pi}h_N}
\int_{\mathbb{R}}\varphi(y)\exp\left(-\frac{(x-y)^2}{2h_N^2}\right)
\left(e^{8\lvert w-x\rvert h_N^{-2}}-1\right)\,\mathrm{d}y \\
&=
\left(e^{8\lvert w-x\rvert h_N^{-2}}-1\right)\varphi_N(x),
\end{aligned} ∣ φ N ( w ) − φ N ( x )∣ ≤ 2 π h N 1 ∫ R φ ( y ) exp ( − 2 h N 2 ( x − y ) 2 ) ( e 8 ∣ w − x ∣ h N − 2 − 1 ) d y = ( e 8 ∣ w − x ∣ h N − 2 − 1 ) φ N ( x ) , which is (127) .
When w w w (132) is real, so by (133) , we have
e − 8 ∣ w − x ∣ h N − 2 ≤ exp ( − 2 ( x − y ) ( w − x ) + ( w − x ) 2 2 h N 2 ) ≤ e 8 ∣ w − x ∣ h N − 2 , y ∈ supp φ . e^{-8\lvert w-x\rvert h_N^{-2}}
\le
\exp\left(-\frac{2(x-y)(w-x)+(w-x)^2}{2h_N^2}\right)
\le
e^{8\lvert w-x\rvert h_N^{-2}},
\qquad y\in\operatorname{supp}\varphi. e − 8 ∣ w − x ∣ h N − 2 ≤ exp ( − 2 h N 2 2 ( x − y ) ( w − x ) + ( w − x ) 2 ) ≤ e 8 ∣ w − x ∣ h N − 2 , y ∈ supp φ . Integrating (132) against φ ( y ) d y \varphi(y)\,\mathrm{d}y φ ( y ) d y (128) .
To prove part (3), we expand
( w − y ) 2 = ( Re w − y ) 2 − ( Im w ) 2 + 2 i Im w ( Re w − y ) , (w-y)^2=(\operatorname{Re}w-y)^2-(\operatorname{Im}w)^2+2i\operatorname{Im}w(\operatorname{Re}w-y), ( w − y ) 2 = ( Re w − y ) 2 − ( Im w ) 2 + 2 i Im w ( Re w − y ) , which gives
φ N ( w ) = 1 2 π h N ∫ R φ ( y ) exp ( − ( Re w − y ) 2 − ( Im w ) 2 2 h N 2 ) × exp ( − i Im w ( Re w − y ) h N 2 ) d y . \varphi_N(w)
=
\frac{1}{\sqrt{2\pi}h_N}
\int_{\mathbb{R}}\varphi(y)
\exp\left(-\frac{(\operatorname{Re}w-y)^2-(\operatorname{Im}w)^2}{2h_N^2}\right)
\times
\exp\left(-\frac{i\operatorname{Im}w(\operatorname{Re}w-y)}{h_N^2}\right)\,\mathrm{d}y. φ N ( w ) = 2 π h N 1 ∫ R φ ( y ) exp ( − 2 h N 2 ( Re w − y ) 2 − ( Im w ) 2 ) × exp ( − h N 2 i Im w ( Re w − y ) ) d y . The first two factors in the integrand are nonnegative. For y ∈ supp φ y\in\operatorname{supp}\varphi y ∈ supp φ
β = ( ∣ Re w ∣ + 2 ) ∣ Im w ∣ h N − 2 ≤ 1. \beta=(\lvert\operatorname{Re}w\rvert+2)\lvert\operatorname{Im}w\rvert h_N^{-2}\le1. β = (∣ Re w ∣ + 2 ) ∣ Im w ∣ h N − 2 ≤ 1. Since β < π / 2 \beta< \pi/2 β < π /2 β \beta β
{ 0 } ∪ { z ≠ 0 : ∣ arg z ∣ ≤ β } = { z : Re ( e i ( π / 2 − β ) z ) ≥ 0 } ∩ { z : Re ( e − i ( π / 2 − β ) z ) ≥ 0 } . \{0\} \cup\{z \ne0 : |\arg z| \le\beta\} = \{z : \operatorname{Re}(e^{\mathrm{i}(\pi/2-\beta)}z) \ge0\} \cap\{z : \operatorname{Re}(e^{-\mathrm{i}(\pi/2-\beta)}z) \ge0\}. { 0 } ∪ { z = 0 : ∣ arg z ∣ ≤ β } = { z : Re ( e i ( π /2 − β ) z ) ≥ 0 } ∩ { z : Re ( e − i ( π /2 − β ) z ) ≥ 0 } . Each of these half-planes is preserved by integration against a nonnegative weight, so φ N ( w ) \varphi_N(w) φ N ( w ) (129) .
For parts (4) and (5), we change variables in (126) and write, for x ∈ R x \in\mathbb{R} x ∈ R
φ N ( x ) = 1 2 π ∫ R φ ( x + h N y ) e − y 2 / 2 d y . \varphi_N(x) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \varphi(x+h_Ny)e^{-y^2/2}\,dy. φ N ( x ) = 2 π 1 ∫ R φ ( x + h N y ) e − y 2 /2 d y . The integrand vanishes unless h N ∣ y ∣ ≥ dist ( x , supp φ ) h_N|y| \ge\operatorname{dist}(x,\operatorname{supp}\varphi) h N ∣ y ∣ ≥ dist ( x , supp φ ) (130) follows from the Gaussian tail bound
∫ ∣ y ∣ ≥ y 0 e − y 2 / 2 d y ≤ 2 2 π e − y 0 2 / 2 , y 0 ≥ 0. \int_{|y|\ge y_0} e^{-y^2/2}\,dy \le2\sqrt{2\pi}e^{-y_0^2/2}, \qquad y_0 \ge0. ∫ ∣ y ∣ ≥ y 0 e − y 2 /2 d y ≤ 2 2 π e − y 0 2 /2 , y 0 ≥ 0. Further, the mean value theorem gives
∣ φ N ( x ) − φ ( x ) ∣ ≤ 1 2 π ∫ R ∣ φ ( x + h N y ) − φ ( x ) ∣ e − y 2 / 2 d y ≤ ( 2 π ) 1 / 2 ∥ φ ′ ∥ ∞ h N ≤ ∥ φ ′ ∥ ∞ h N , |\varphi_N(x)-\varphi(x)| \le\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}|\varphi(x+h_Ny)-\varphi(x)|e^{-y^2/2}\,dy \le\left(\frac{2}{\pi}\right)^{1/2}\|\varphi'\|_\infty h_N \le\|\varphi'\|_\infty h_N, ∣ φ N ( x ) − φ ( x ) ∣ ≤ 2 π 1 ∫ R ∣ φ ( x + h N y ) − φ ( x ) ∣ e − y 2 /2 d y ≤ ( π 2 ) 1/2 ∥ φ ′ ∥ ∞ h N ≤ ∥ φ ′ ∥ ∞ h N , which is part (5).
Spectra under a perturbation of one entry
For an integer p ≥ 1 p \ge1 p ≥ 1
ω p = 1 1000 ( p + 1 ) , c p = ω p 100 , (134) \omega_p = \frac{1}{1000(p+1)}, \qquad c_p = \frac{\omega_p}{100}, \tag*{(134)} ω p = 1000 ( p + 1 ) 1 , c p = 100 ω p , (134) and we define the open rectangle
R α , p = { w ∈ C : ∣ Re w ∣ < 4 , ∣ Im w ∣ < ω p ℓ α } . (135) \mathcal{R}_{\alpha,p} = \{w \in\mathbb{C} : |\operatorname{Re} w| < 4,\ |\operatorname{Im} w| < \omega_p\ell_\alpha\}. \tag*{(135)} R α , p = { w ∈ C : ∣ Re w ∣ < 4 , ∣ Im w ∣ < ω p ℓ α } . (135) Recall the set G ε \mathcal{G}_\varepsilon G ε Definition 3.1 and the matrices V a b V^{ab} V ab (35) . In the next lemma, the size 8 c p N − α − 4 ε 8c_pN^{-\alpha-4\varepsilon} 8 c p N − α − 4 ε N − 1 / 2 N^{-1/2} N − 1/2 H t H_t H t α + 4 ε < 1 / 2 \alpha+4\varepsilon< 1/2 α + 4 ε < 1/2
Lemma 5.2. Let ε > 0 \varepsilon> 0 ε > 0 p ≥ 1 p \ge1 p ≥ 1 N 0 = N 0 ( α , p ) N_0 = N_0(\alpha,p) N 0 = N 0 ( α , p ) N ≥ N 0 N \ge N_0 N ≥ N 0 A ∈ G ε A \in\mathcal{G}_\varepsilon A ∈ G ε a , b ∈ [ N ] a,b \in[N] a , b ∈ [ N ] a ≤ b a \le b a ≤ b ζ ∈ C \zeta\in\mathbb{C} ζ ∈ C ∣ ζ ∣ ≤ 8 c p N − α − 4 ε |\zeta| \le8c_pN^{-\alpha-4\varepsilon} ∣ ζ ∣ ≤ 8 c p N − α − 4 ε Spec ( A + ζ V a b ) ⊂ R α , p \operatorname{Spec}(A+\zeta V^{ab}) \subset\mathcal{R}_{\alpha,p} Spec ( A + ζ V ab ) ⊂ R α , p
Proof. We fix a spectral decomposition
A = ∑ i = 1 N λ i ( A ) v i v i T , A = \sum_{i=1}^{N}\lambda_i(A)v_i v_i^{T}, A = i = 1 ∑ N λ i ( A ) v i v i T , where ( v i ) i ∈ [ N ] (v_i)_{i\in[N]} ( v i ) i ∈ [ N ] R N \mathbb{R}^N R N G ε \mathcal{G}_{\varepsilon} G ε ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3 ∥ v i ∥ ∞ ≤ N − 1 / 2 + ε \lVert v_i\rVert_{\infty} \le N^{-1/2+\varepsilon} ∥ v i ∥ ∞ ≤ N − 1/2 + ε i ∈ [ N ] i\in[N] i ∈ [ N ]
We write ∂ R α , p \partial\mathcal{R}_{\alpha,p} ∂ R α , p R α , p \mathcal{R}_{\alpha,p} R α , p Im w = ± ω p ℓ α \operatorname{Im} w=\pm\omega_p\ell_{\alpha} Im w = ± ω p ℓ α Re w = ± 4 \operatorname{Re} w=\pm4 Re w = ± 4
We begin by bounding the entries of the resolvent ( A − w ) − 1 (A-w)^{-1} ( A − w ) − 1 w ∈ ∂ R α , p w\in\partial\mathcal{R}_{\alpha,p} w ∈ ∂ R α , p w w w ∣ Re w ∣ ≤ 4 \lvert\operatorname{Re} w\rvert\le4 ∣ Re w ∣ ≤ 4 ∣ Im w ∣ = ω p ℓ α \lvert\operatorname{Im} w\rvert=\omega_p\ell_{\alpha} ∣ Im w ∣ = ω p ℓ α a ′ , b ′ ∈ { a , b } a',b'\in\{a,b\} a ′ , b ′ ∈ { a , b }
∣ ( ( A − w ) − 1 ) a ′ b ′ ∣ ≤ ∑ i = 1 N ∣ v i ( a ′ ) ∣ ∣ v i ( b ′ ) ∣ ∣ λ i ( A ) − w ∣ ≤ N − 1 + 2 ε ∑ i = 1 N 1 ∣ λ i ( A ) − w ∣ . (136) \left\lvert\left((A-w)^{-1}\right)_{a'b'}\right\rvert
\le\sum_{i=1}^{N}\frac{\lvert v_i(a')\rvert\lvert v_i(b')\rvert}{\lvert\lambda_i(A)-w\rvert}
\le N^{-1+2\varepsilon}\sum_{i=1}^{N}\frac{1}{\lvert\lambda_i(A)-w\rvert}. \tag*{(136)} ( ( A − w ) − 1 ) a ′ b ′ ≤ i = 1 ∑ N ∣ λ i ( A ) − w ∣ ∣ v i ( a ′ )∣ ∣ v i ( b ′ )∣ ≤ N − 1 + 2 ε i = 1 ∑ N ∣ λ i ( A ) − w ∣ 1 . (136) To bound the last sum, we partition the real line into the intervals
I n = [ Re w + n N , Re w + n + 1 N ) , n ∈ Z . \mathcal{I}_n=\left[\operatorname{Re} w+\frac{n}{N},\operatorname{Re} w+\frac{n+1}{N}\right),\qquad n\in\mathbb{Z}. I n = [ Re w + N n , Re w + N n + 1 ) , n ∈ Z . Every eigenvalue of A A A I n \mathcal{I}_n I n ∣ n ∣ ≤ 7 N \lvert n\rvert\le7N ∣ n ∣ ≤ 7 N λ i ( A ) ∈ [ − 3 , 3 ] \lambda_i(A)\in[-3,3] λ i ( A ) ∈ [ − 3 , 3 ] ∣ Re w ∣ ≤ 4 \lvert\operatorname{Re} w\rvert\le4 ∣ Re w ∣ ≤ 4 G ε \mathcal{G}_{\varepsilon} G ε I n \mathcal{I}_n I n N ε N^{\varepsilon} N ε A A A λ i ( A ) ∈ I n \lambda_i(A)\in\mathcal{I}_n λ i ( A ) ∈ I n
∣ λ i ( A ) − w ∣ ≥ { ω p ℓ α if n ∈ { − 1 , 0 } , n / N if n ≥ 1 , ( ∣ n ∣ − 1 ) / N if n ≤ − 2 , (137) \lvert\lambda_i(A)-w\rvert\ge
\begin{cases}
\omega_p\ell_{\alpha} & \text{if } n\in\{-1,0\},\\
n/N & \text{if } n\ge1,\\
(\lvert n\rvert-1)/N & \text{if } n\le-2,
\end{cases} \tag*{(137)} ∣ λ i ( A ) − w ∣ ≥ ⎩ ⎨ ⎧ ω p ℓ α n / N (∣ n ∣ − 1 ) / N if n ∈ { − 1 , 0 } , if n ≥ 1 , if n ≤ − 2 , (137) where the first bound uses ∣ Im w ∣ = ω p ℓ α \lvert\operatorname{Im} w\rvert=\omega_p\ell_{\alpha} ∣ Im w ∣ = ω p ℓ α ∣ λ i ( A ) − w ∣ ≥ ∣ λ i ( A ) − Re w ∣ \lvert\lambda_i(A)-w\rvert\ge\lvert\lambda_i(A)-\operatorname{Re} w\rvert ∣ λ i ( A ) − w ∣ ≥ ∣ λ i ( A ) − Re w ∣ n n n
∑ i = 1 N 1 ∣ λ i ( A ) − w ∣ ≤ 2 N ε ω p ℓ α + 2 N ε ∑ n = 1 7 N N n ≤ 2 N 1 + α + ε ω p + 2 N 1 + ε ( 1 + log ( 7 N ) ) . \sum_{i=1}^{N}\frac{1}{\lvert\lambda_i(A)-w\rvert}
\le\frac{2N^{\varepsilon}}{\omega_p\ell_{\alpha}}+2N^{\varepsilon}\sum_{n=1}^{7N}\frac{N}{n}
\le\frac{2N^{1+\alpha+\varepsilon}}{\omega_p}+2N^{1+\varepsilon}(1+\log(7N)). i = 1 ∑ N ∣ λ i ( A ) − w ∣ 1 ≤ ω p ℓ α 2 N ε + 2 N ε n = 1 ∑ 7 N n N ≤ ω p 2 N 1 + α + ε + 2 N 1 + ε ( 1 + log ( 7 N )) . We now choose N 0 = N 0 ( α , p ) N_0=N_0(\alpha,p) N 0 = N 0 ( α , p )
2 ω p ( 1 + log ( 7 N ) ) ≤ N α for all N ≥ N 0 . 2\omega_p(1+\log(7N))\le N^{\alpha}\qquad\text{for all }N\ge N_0. 2 ω p ( 1 + log ( 7 N )) ≤ N α for all N ≥ N 0 . For such N N N 3 N 1 + α + ε / ω p 3N^{1+\alpha+\varepsilon}/\omega_p 3 N 1 + α + ε / ω p (136) , we obtain
∣ ( ( A − w ) − 1 ) a ′ b ′ ∣ ≤ 3 N α + 3 ε ω p , a ′ , b ′ ∈ { a , b } . (138) \left\lvert\left((A-w)^{-1}\right)_{a'b'}\right\rvert\le\frac{3N^{\alpha+3\varepsilon}}{\omega_p},\qquad a',b'\in\{a,b\}. \tag*{(138)} ( ( A − w ) − 1 ) a ′ b ′ ≤ ω p 3 N α + 3 ε , a ′ , b ′ ∈ { a , b } . (138) It remains to treat the vertical sides. There we have ∣ Re w ∣ = 4 \lvert\operatorname{Re} w\rvert=4 ∣ Re w ∣ = 4 dist ( w , Spec A ) ≥ 1 \operatorname{dist}(w,\operatorname{Spec} A)\ge1 dist ( w , Spec A ) ≥ 1 ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3 ∥ ( A − w ) − 1 ∥ ≤ 1 \lVert(A-w)^{-1}\rVert\le1 ∥( A − w ) − 1 ∥ ≤ 1 (138) holds on all of ∂ R α , p \partial\mathcal{R}_{\alpha,p} ∂ R α , p
Next, we reduce the characteristic polynomial of A + ζ V a b A+\zeta V^{ab} A + ζ V ab
det ( Id N + Q 1 Q 2 ) = det ( Id 2 + Q 2 Q 1 ) , Q 1 ∈ C N × 2 , Q 2 ∈ C 2 × N . \det(\operatorname{Id}_N+Q_1Q_2)=\det(\operatorname{Id}_2+Q_2Q_1),\qquad Q_1\in\mathbb{C}^{N\times2},\quad Q_2\in\mathbb{C}^{2\times N}. det ( Id N + Q 1 Q 2 ) = det ( Id 2 + Q 2 Q 1 ) , Q 1 ∈ C N × 2 , Q 2 ∈ C 2 × N . For a < b a < b a < b
ζ V a b = Q 1 Q 2 , Q 1 = ( e a , e b ) , Q 2 = ζ ( e b , e a ) T . \zeta V^{ab} = Q_1Q_2, \qquad Q_1 = (e_a,e_b), \qquad Q_2 = \zeta(e_b,e_a)^{T}. ζ V ab = Q 1 Q 2 , Q 1 = ( e a , e b ) , Q 2 = ζ ( e b , e a ) T . Factoring out A − w A-w A − w w ∉ Spec A w \notin\operatorname{Spec} A w ∈ / Spec A x ∈ [ 0 , 1 ] x \in[0,1] x ∈ [ 0 , 1 ]
det ( A + x ζ V a b − w ) = det ( A − w ) det ( Id 2 + x ζ ( ( ( A − w ) − 1 ) b a ( ( A − w ) − 1 ) b b ( ( A − w ) − 1 ) a a ( ( A − w ) − 1 ) a b ) ) . (139) \det(A+x\zeta V^{ab}-w)=\det(A-w)\det\left(\operatorname{Id}_2+x\zeta\begin{pmatrix} ((A-w)^{-1})_{ba} & ((A-w)^{-1})_{bb} \\ ((A-w)^{-1})_{aa} & ((A-w)^{-1})_{ab} \end{pmatrix}\right). \tag*{(139)} det ( A + x ζ V ab − w ) = det ( A − w ) det ( Id 2 + x ζ ( (( A − w ) − 1 ) ba (( A − w ) − 1 ) aa (( A − w ) − 1 ) bb (( A − w ) − 1 ) ab ) ) . (139) For a = b a=b a = b Q 1 = e a Q_1=e_a Q 1 = e a Q 2 = ζ e a T Q_2=\zeta e_a^{T} Q 2 = ζ e a T
det ( A + x ζ V a a − w ) = det ( A − w ) ( 1 + x ζ ( ( A − w ) − 1 ) a a ) . (140) \det(A+x\zeta V^{aa}-w)=\det(A-w)\left(1+x\zeta((A-w)^{-1})_{aa}\right). \tag*{(140)} det ( A + x ζ V aa − w ) = det ( A − w ) ( 1 + x ζ (( A − w ) − 1 ) aa ) . (140) Now let w ∈ ∂ R α , p w \in\partial\mathcal{R}_{\alpha,p} w ∈ ∂ R α , p x ∈ [ 0 , 1 ] x \in[0,1] x ∈ [ 0 , 1 ] ∂ R α , p \partial\mathcal{R}_{\alpha,p} ∂ R α , p [ − 3 , 3 ] [-3,3] [ − 3 , 3 ] w ∉ Spec A w \notin\operatorname{Spec} A w ∈ / Spec A ζ \zeta ζ 2 × 2 2 \times2 2 × 2 x ζ x\zeta x ζ
8 c p N − α − 4 ε ⋅ 3 N α + 3 ε ω p = 24 c p ω p N − ε ≤ 1 4 , 8c_pN^{-\alpha-4\varepsilon}\cdot\frac{3N^{\alpha+3\varepsilon}}{\omega_p}=\frac{24c_p}{\omega_p}N^{-\varepsilon}\leq\frac{1}{4}, 8 c p N − α − 4 ε ⋅ ω p 3 N α + 3 ε = ω p 24 c p N − ε ≤ 4 1 , where we used c p = ω p / 100 c_p=\omega_p/100 c p = ω p /100 2 × 2 2 \times2 2 × 2 Q ′ Q' Q ′ 1 / 4 1/4 1/4
∣ det ( Id 2 + Q ′ ) ∣ = ∣ 1 + Tr Q ′ + det Q ′ ∣ ≥ 1 − 1 2 − 1 8 > 0. \left|\det(\operatorname{Id}_2+Q')\right|=\left|1+\operatorname{Tr}Q'+\det Q'\right|\geq1-\frac{1}{2}-\frac{1}{8}>0. ∣ det ( Id 2 + Q ′ ) ∣ = ∣ 1 + Tr Q ′ + det Q ′ ∣ ≥ 1 − 2 1 − 8 1 > 0. In the case a = b a=b a = b 3 / 4 3/4 3/4
det ( A + x ζ V a b − w ) ≠ 0 for all w ∈ ∂ R α , p and x ∈ [ 0 , 1 ] . (141) \det(A+x\zeta V^{ab}-w)\ne0 \qquad\text{for all } w\in\partial\mathcal{R}_{\alpha,p}\text{ and }x\in[0,1]. \tag*{(141)} det ( A + x ζ V ab − w ) = 0 for all w ∈ ∂ R α , p and x ∈ [ 0 , 1 ] . (141) Finally, we use the argument principle. For x ∈ [ 0 , 1 ] x \in[0,1] x ∈ [ 0 , 1 ]
χ x ( w ) = det ( w Id N − A − x ζ V a b ) , w ∈ C . \chi_x(w)=\det(w\operatorname{Id}_N-A-x\zeta V^{ab}), \qquad w\in\mathbb{C}. χ x ( w ) = det ( w Id N − A − x ζ V ab ) , w ∈ C . This is a monic polynomial of degree N N N w w w x x x [ 0 , 1 ] × ∂ R α , p [0,1]\times\partial\mathcal{R}_{\alpha,p} [ 0 , 1 ] × ∂ R α , p ∣ χ x ( w ) ∣ |\chi_x(w)| ∣ χ x ( w ) ∣ ( x , w ) (x,w) ( x , w ) ∂ R α , p \partial\mathcal{R}_{\alpha,p} ∂ R α , p
1 2 π i ∮ ∂ R α , p ∂ w χ x ( w ) χ x ( w ) d w \frac{1}{2\pi i}\oint_{\partial\mathcal{R}_{\alpha,p}}\frac{\partial_w\chi_x(w)}{\chi_x(w)}\,dw 2 π i 1 ∮ ∂ R α , p χ x ( w ) ∂ w χ x ( w ) d w which counts the zeros of χ x \chi_x χ x R α , p \mathcal{R}_{\alpha,p} R α , p x ∈ [ 0 , 1 ] x \in[0,1] x ∈ [ 0 , 1 ]
For x = 0 x=0 x = 0 χ 0 \chi_0 χ 0 A A A [ − 3 , 3 ] ⊂ R α , p [-3,3]\subset\mathcal{R}_{\alpha,p} [ − 3 , 3 ] ⊂ R α , p N N N x x x N N N χ 1 \chi_1 χ 1 A + ζ V a b A+\zeta V^{ab} A + ζ V ab R α , p \mathcal{R}_{\alpha,p} R α , p
A holomorphic extension of the pair statistic
We will need the following elementary fact.
Lemma 5.3. Let O ⊂ C \mathcal{O} \subset\mathbb{C} O ⊂ C A ∈ C N × N A \in\mathbb{C}^{N \times N} A ∈ C N × N Spec A ⊂ O \operatorname{Spec} A \subset\mathcal{O} Spec A ⊂ O C N × N \mathbb{C}^{N \times N} C N × N
Proof. Let ( A n ) n ≥ 1 (A_n)_{n \ge1} ( A n ) n ≥ 1 A A A A n A_n A n w n ∉ O w_n \notin\mathcal{O} w n ∈ / O ( w n ) (w_n) ( w n ) ∣ w n ∣ ≤ ∥ A n ∥ |w_n| \le\lVert A_n\rVert ∣ w n ∣ ≤ ∥ A n ∥ w n → w w_n \to w w n → w w ∉ O w \notin\mathcal{O} w ∈ / O C ∖ O \mathbb{C} \setminus\mathcal{O} C ∖ O
det ( A − w ) = lim n → ∞ det ( A n − w n ) = 0. \det(A-w)=\lim_{n\to\infty}\det(A_n-w_n)=0. det ( A − w ) = n → ∞ lim det ( A n − w n ) = 0. Hence the set of matrices with an eigenvalue outside O \mathcal{O} O
We now introduce the tensor products used to define the extension. We identify C N ⊗ C N \mathbb{C}^{N} \otimes\mathbb{C}^{N} C N ⊗ C N C N 2 \mathbb{C}^{N^2} C N 2 ( e i ⊗ e j ) i , j ∈ [ N ] (e_i \otimes e_j)_{i,j \in[N]} ( e i ⊗ e j ) i , j ∈ [ N ] ( i , j ) (i,j) ( i , j ) A ∈ C N × N A \in\mathbb{C}^{N \times N} A ∈ C N × N
D ( A ) = A ⊗ Id − Id ⊗ A , M ( A ) = 1 2 ( A ⊗ Id + Id ⊗ A ) . (142) \mathcal{D}(A)=A\otimes\operatorname{Id}-\operatorname{Id}\otimes A,\qquad\mathcal{M}(A)=\frac{1}{2}(A\otimes\operatorname{Id}+\operatorname{Id}\otimes A). \tag*{(142)} D ( A ) = A ⊗ Id − Id ⊗ A , M ( A ) = 2 1 ( A ⊗ Id + Id ⊗ A ) . (142) By Schur’s theorem, there exist a unitary matrix O O O T T T A = O T O ∗ A=OTO^* A = O T O ∗ T i i = w i T_{ii}=w_i T ii = w i A A A
( O ⊗ O ) ∗ D ( A ) ( O ⊗ O ) = T ⊗ Id − Id ⊗ T , (O\otimes O)^*\mathcal{D}(A)(O\otimes O)=T\otimes\operatorname{Id}-\operatorname{Id}\otimes T, ( O ⊗ O ) ∗ D ( A ) ( O ⊗ O ) = T ⊗ Id − Id ⊗ T , ( O ⊗ O ) ∗ M ( A ) ( O ⊗ O ) = 1 2 ( T ⊗ Id + Id ⊗ T ) . (143) (O\otimes O)^*\mathcal{M}(A)(O\otimes O)=\frac{1}{2}(T\otimes\operatorname{Id}+\operatorname{Id}\otimes T). \tag*{(143)} ( O ⊗ O ) ∗ M ( A ) ( O ⊗ O ) = 2 1 ( T ⊗ Id + Id ⊗ T ) . (143) The matrices T ⊗ Id T\otimes\operatorname{Id} T ⊗ Id Id ⊗ T \operatorname{Id}\otimes T Id ⊗ T
( T ⊗ Id ) ( e i ⊗ e j ) = ∑ i ′ ≤ i T i ′ i e i ′ ⊗ e j , ( Id ⊗ T ) ( e i ⊗ e j ) = ∑ j ′ ≤ j T j ′ j e i ⊗ e j ′ . (T\otimes\operatorname{Id})(e_i\otimes e_j)=\sum_{i'\le i}T_{i'i}e_{i'}\otimes e_j,\qquad
(\operatorname{Id}\otimes T)(e_i\otimes e_j)=\sum_{j'\le j}T_{j'j}e_i\otimes e_{j'}. ( T ⊗ Id ) ( e i ⊗ e j ) = i ′ ≤ i ∑ T i ′ i e i ′ ⊗ e j , ( Id ⊗ T ) ( e i ⊗ e j ) = j ′ ≤ j ∑ T j ′ j e i ⊗ e j ′ . and their diagonal entries at e i ⊗ e j e_i\otimes e_j e i ⊗ e j w i w_i w i w j w_j w j (143) are upper triangular, with diagonal entries w i − w j w_i-w_j w i − w j ( w i + w j ) / 2 (w_i+w_j)/2 ( w i + w j ) /2 e i ⊗ e j e_i\otimes e_j e i ⊗ e j
For a square matrix M M M
φ N ( M ) = ∑ n = 0 ∞ φ N ( n ) ( 0 ) n ! M n , (144) \varphi_N(M)=\sum_{n=0}^{\infty}\frac{\varphi_N^{(n)}(0)}{n!}M^n, \tag*{(144)} φ N ( M ) = n = 0 ∑ ∞ n ! φ N ( n ) ( 0 ) M n , (144) where the series converges absolutely since φ N \varphi_N φ N M M M φ N ( M ) \varphi_N(M) φ N ( M ) φ N ( M i i ) \varphi_N(M_{ii}) φ N ( M ii ) M M M Q Q Q
φ N ( Q ∗ M Q ) = Q ∗ φ N ( M ) Q . (145) \varphi_N(Q^*MQ)=Q^*\varphi_N(M)Q. \tag*{(145)} φ N ( Q ∗ M Q ) = Q ∗ φ N ( M ) Q . (145) Suppose now that every eigenvalue w j w_j w j A A A ∣ Im w j ∣ < ω p ℓ α |\operatorname{Im} w_j|<\omega_p\ell_\alpha ∣ Im w j ∣ < ω p ℓ α (143) , the matrix Id + ℓ α − 2 D ( A ) 2 \operatorname{Id}+\ell_\alpha^{-2}\mathcal{D}(A)^2 Id + ℓ α − 2 D ( A ) 2 1 + ( ( w i − w j ) / ℓ α ) 2 1+((w_i-w_j)/\ell_\alpha)^2 1 + (( w i − w j ) / ℓ α ) 2 z = ( w i − w j ) / ℓ α z=(w_i-w_j)/\ell_\alpha z = ( w i − w j ) / ℓ α ∣ Im z ∣ < 2 ω p |\operatorname{Im}z|<2\omega_p ∣ Im z ∣ < 2 ω p
Re ( 1 + z 2 ) = 1 + ( Re z ) 2 − ( Im z ) 2 ≥ 1 − 4 ω p 2 > 0. \operatorname{Re}(1+z^2)=1+(\operatorname{Re}z)^2-(\operatorname{Im}z)^2\ge1-4\omega_p^2>0. Re ( 1 + z 2 ) = 1 + ( Re z ) 2 − ( Im z ) 2 ≥ 1 − 4 ω p 2 > 0. Therefore Id + ℓ α − 2 D ( A ) 2 \operatorname{Id}+\ell_\alpha^{-2}\mathcal{D}(A)^2 Id + ℓ α − 2 D ( A ) 2
Definition 5.4. For an integer p ≥ 1 p\ge1 p ≥ 1
U α , p = { A ∈ C N × N : ∣ Im w ∣ < ω p ℓ α for every eigenvalue w of A } . (146) \mathcal{U}_{\alpha,p}=\{A\in\mathbb{C}^{N\times N}:|\operatorname{Im}w|<\omega_p\ell_\alpha\text{ for every eigenvalue }w\text{ of }A\}. \tag*{(146)} U α , p = { A ∈ C N × N : ∣ Im w ∣ < ω p ℓ α for every eigenvalue w of A } . (146) For a nonnegative φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) p ≥ 1 p\ge1 p ≥ 1 A ∈ U α , p A\in\mathcal{U}_{\alpha,p} A ∈ U α , p
X α ( A ; φ N , f p ) = 1 2 Tr [ φ N ( M ( A ) ) ( Id + ℓ α − 2 D ( A ) 2 ) − p ] − 1 2 Tr φ N ( A ) . (147) \mathcal{X}_\alpha(A;\varphi_N,f_p)=\frac{1}{2}\operatorname{Tr}\left[\varphi_N(\mathcal{M}(A))\left(\operatorname{Id}+\ell_\alpha^{-2}\mathcal{D}(A)^2\right)^{-p}\right]-\frac{1}{2}\operatorname{Tr}\varphi_N(A). \tag*{(147)} X α ( A ; φ N , f p ) = 2 1 Tr [ φ N ( M ( A )) ( Id + ℓ α − 2 D ( A ) 2 ) − p ] − 2 1 Tr φ N ( A ) . (147) Let J \mathcal{J} J j ∈ J j\in\mathcal{J} j ∈ J φ j ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi_j\in C^\infty_c((-2,2)) φ j ∈ C c ∞ (( − 2 , 2 )) p j ≥ 1 p_j\ge1 p j ≥ 1 θ j ≥ 0 \theta_j\ge0 θ j ≥ 0 p = max j ∈ J p j p=\max_{j\in\mathcal{J}}p_j p = max j ∈ J p j
F ( A ) = exp ( − ∑ j ∈ J θ j X α ( A ; φ j , N , f p j ) ) , A ∈ U α , p . (148) F(A)=\exp\left(-\sum_{j\in\mathcal{J}}\theta_j\mathcal{X}_\alpha(A;\varphi_{j,N},f_{p_j})\right),\qquad A\in\mathcal{U}_{\alpha,p}. \tag*{(148)} F ( A ) = exp − j ∈ J ∑ θ j X α ( A ; φ j , N , f p j ) , A ∈ U α , p . (148) Since ω p ≤ ω p j \omega_p\le\omega_{p_j} ω p ≤ ω p j U α , p ⊂ U α , p j \mathcal{U}_{\alpha,p}\subset\mathcal{U}_{\alpha,p_j} U α , p ⊂ U α , p j j ∈ J j\in\mathcal{J} j ∈ J (148) is well defined. By Lemma 5.3 , the set U α , p \mathcal{U}_{\alpha,p} U α , p R α , p \mathbb{R}_{\alpha,p} R α , p F F F N N N ( φ j , p j , θ j ) j ∈ J (\varphi_j,p_j,\theta_j)_{j\in\mathcal{J}} ( φ j , p j , θ j ) j ∈ J θ j \theta_j θ j N N N
Lemma 5.5. Let p ≥ 1 p\ge1 p ≥ 1 φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 ))
(1) The function A ↦ X α ( A ; φ N , f p ) A\mapsto\mathcal{X}_\alpha(A;\varphi_N,f_p) A ↦ X α ( A ; φ N , f p ) A A A U α , p \mathcal{U}_{\alpha,p} U α , p A ∈ U α , p A\in\mathcal{U}_{\alpha,p} A ∈ U α , p w 1 , … , w N w_1,\ldots,w_N w 1 , … , w N
X α ( A ; φ N , f p ) = ∑ 1 ≤ i < j ≤ N φ N ( w i + w j 2 ) f p ( w j − w i ℓ α ) . (149) \mathcal{X}_\alpha(A;\varphi_N,f_p)=\sum_{1\le i<j\le N}\varphi_N\left(\frac{w_i+w_j}{2}\right)f_p\left(\frac{w_j-w_i}{\ell_\alpha}\right). \tag*{(149)} X α ( A ; φ N , f p ) = 1 ≤ i < j ≤ N ∑ φ N ( 2 w i + w j ) f p ( ℓ α w j − w i ) . (149) In particular, (5.22) agrees with (5.1) for every real symmetric matrix A A A
(2) Let A ∈ U α , p A \in\mathcal{U}_{\alpha,p} A ∈ U α , p ( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ ≤ h N 2 / 200 (|\operatorname{Re} w|+2)|\operatorname{Im} w'| \le h_N^2/200 ( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ ≤ h N 2 /200 w w w w ′ w' w ′ A A A (149) is either zero or has argument of absolute value at most 1 / 100 1/100 1/100 Re X α ( A ; φ N , f p ) ≥ 0 \operatorname{Re}\mathcal{X}_{\alpha}(A;\varphi_N,f_p) \ge0 Re X α ( A ; φ N , f p ) ≥ 0
(3) If Spec A ⊂ R α , p \operatorname{Spec} A \subset\mathcal{R}_{\alpha,p} Spec A ⊂ R α , p A ∈ U α , p A \in\mathcal{U}_{\alpha,p} A ∈ U α , p A A A
(4) Let F F F p = max j ∈ J p j p=\max_{j\in J}p_j p = max j ∈ J p j F F F A A A U α , p \mathcal{U}_{\alpha,p} U α , p ∣ F ( A ) ∣ ≤ 1 |F(A)|\le1 ∣ F ( A ) ∣ ≤ 1 A ∈ U α , p A\in\mathcal{U}_{\alpha,p} A ∈ U α , p ∣ F ( A ) ∣ ≤ 1 |F(A)|\le1 ∣ F ( A ) ∣ ≤ 1 Spec A ⊂ R α , p \operatorname{Spec} A\subset\mathcal{R}_{\alpha,p} Spec A ⊂ R α , p A A A
Proof. We begin with holomorphy. The entries of M ( A ) n \mathcal{M}(A)^n M ( A ) n A A A ∥ M ( A ) n ∥ ≤ ∥ A ∥ n \|\mathcal{M}(A)^n\|\le\|A\|^n ∥ M ( A ) n ∥ ≤ ∥ A ∥ n φ N \varphi_N φ N φ N ( M ( A ) ) \varphi_N(\mathcal{M}(A)) φ N ( M ( A )) C N × N \mathbb{C}^{N\times N} C N × N φ N ( M ( A ) ) \varphi_N(\mathcal{M}(A)) φ N ( M ( A )) A A A φ N ( A ) \varphi_N(A) φ N ( A ) ( Id + ℓ α − 2 Q ( A ) 2 ) − 1 (\operatorname{Id}+\ell_\alpha^{-2}\mathcal{Q}(A)^2)^{-1} ( Id + ℓ α − 2 Q ( A ) 2 ) − 1 A A A det ( Id + ℓ α − 2 Q ( A ) 2 ) \det(\operatorname{Id}+\ell_\alpha^{-2}\mathcal{Q}(A)^2) det ( Id + ℓ α − 2 Q ( A ) 2 ) U α , p \mathcal{U}_{\alpha,p} U α , p (147) is holomorphic on U α , p \mathcal{U}_{\alpha,p} U α , p
Next, we prove (149) . Let A = O T O ∗ A=OTO^* A = O T O ∗ T i i = w i T_{ii}=w_i T ii = w i (143) and (145) with Q = O ⊗ O Q=O\otimes O Q = O ⊗ O
( O ⊗ O ) ∗ φ N ( M ( A ) ) ( Id + ℓ α − 2 Q ( A ) 2 ) − p ( O ⊗ O ) = φ N ( 1 2 ( T ⊗ Id + Id ⊗ T ) ) ( Id + ℓ α − 2 ( T ⊗ Id − Id ⊗ T ) 2 ) − p . (O\otimes O)^*\varphi_N(\mathcal{M}(A))(\operatorname{Id}+\ell_\alpha^{-2}\mathcal{Q}(A)^2)^{-p}(O\otimes O)
=
\varphi_N\left(\frac{1}{2}(T\otimes\operatorname{Id}+\operatorname{Id}\otimes T)\right)
\left(\operatorname{Id}+\ell_\alpha^{-2}(T\otimes\operatorname{Id}-\operatorname{Id}\otimes T)^2\right)^{-p}. ( O ⊗ O ) ∗ φ N ( M ( A )) ( Id + ℓ α − 2 Q ( A ) 2 ) − p ( O ⊗ O ) = φ N ( 2 1 ( T ⊗ Id + Id ⊗ T ) ) ( Id + ℓ α − 2 ( T ⊗ Id − Id ⊗ T ) 2 ) − p . By the facts on triangular matrices recalled after (144) , the right side is a product of upper triangular matrices. The diagonal entry of this product at e i ⊗ e j e_i\otimes e_j e i ⊗ e j
φ N ( w i + w j 2 ) f p ( w i − w j ℓ α ) . \varphi_N\left(\frac{w_i+w_j}{2}\right)f_p\left(\frac{w_i-w_j}{\ell_\alpha}\right). φ N ( 2 w i + w j ) f p ( ℓ α w i − w j ) . Similarly, O ∗ φ N ( A ) O = φ N ( T ) O^*\varphi_N(A)O=\varphi_N(T) O ∗ φ N ( A ) O = φ N ( T ) φ N ( w i ) \varphi_N(w_i) φ N ( w i )
X α ( A ; φ N , f p ) = 1 2 ∑ i , j = 1 N φ N ( w i + w j 2 ) f p ( w i − w j ℓ α ) − 1 2 ∑ i = 1 N φ N ( w i ) = ∑ 1 ≤ i < j ≤ N φ N ( w i + w j 2 ) f p ( w j − w i ℓ α ) . \begin{aligned}
\mathcal{X}_{\alpha}(A;\varphi_N,f_p)
=
\frac{1}{2}\sum_{i,j=1}^{N}\varphi_N\left(\frac{w_i+w_j}{2}\right)f_p\left(\frac{w_i-w_j}{\ell_\alpha}\right)
-\frac{1}{2}\sum_{i=1}^{N}\varphi_N(w_i) \\
&=
\sum_{1\le i<j\le N}\varphi_N\left(\frac{w_i+w_j}{2}\right)f_p\left(\frac{w_j-w_i}{\ell_\alpha}\right).
\end{aligned} X α ( A ; φ N , f p ) = 2 1 i , j = 1 ∑ N φ N ( 2 w i + w j ) f p ( ℓ α w i − w j ) − 2 1 i = 1 ∑ N φ N ( w i ) = 1 ≤ i < j ≤ N ∑ φ N ( 2 w i + w j ) f p ( ℓ α w j − w i ) . where we used that f p ( 0 ) = 1 f_p(0)=1 f p ( 0 ) = 1 i = j i=j i = j ( i , j ) (i,j) ( i , j ) f p ( − z ) = f p ( z ) f_p(-z)=f_p(z) f p ( − z ) = f p ( z ) A A A w i = λ i ( A ) w_i=\lambda_i(A) w i = λ i ( A ) (149) becomes (125) .
To prove part (2), we first bound the argument of f p f_p f p z ∈ C z\in\mathbb{C} z ∈ C ∣ Im z ∣ ≤ 1 / 2 |\operatorname{Im}z|\le1/2 ∣ Im z ∣ ≤ 1/2
Re ( 1 + z 2 ) = 1 + ( Re z ) 2 − ( Im z ) 2 ≥ 3 4 ( 1 + ( Re z ) 2 ) > 0 , (150) \operatorname{Re}(1+z^2)=1+(\operatorname{Re}z)^2-(\operatorname{Im}z)^2\ge\frac{3}{4}(1+(\operatorname{Re}z)^2)>0, \tag*{(150)} Re ( 1 + z 2 ) = 1 + ( Re z ) 2 − ( Im z ) 2 ≥ 4 3 ( 1 + ( Re z ) 2 ) > 0 , (150) ∣ Im ( 1 + z 2 ) ∣ = 2 ∣ Re z ∣ ∣ Im z ∣ ≤ ( 1 + ( Re z ) 2 ) ∣ Im z ∣ . |\operatorname{Im}(1+z^2)|=2|\operatorname{Re}z||\operatorname{Im}z|\le(1+(\operatorname{Re}z)^2)|\operatorname{Im}z|. ∣ Im ( 1 + z 2 ) ∣ = 2∣ Re z ∣∣ Im z ∣ ≤ ( 1 + ( Re z ) 2 ) ∣ Im z ∣. The real part of 1 + z 2 1+z^2 1 + z 2 ∣ arctan y ∣ ≤ ∣ y ∣ |\arctan y|\leq|y| ∣ arctan y ∣ ≤ ∣ y ∣
∣ arg ( 1 + z 2 ) ∣ ≤ ∣ Im ( 1 + z 2 ) ∣ Re ( 1 + z 2 ) ≤ 4 3 ∣ Im z ∣ . (151) |\arg(1+z^2)|\leq\frac{|\operatorname{Im}(1+z^2)|}{\operatorname{Re}(1+z^2)}\leq\frac{4}{3}|\operatorname{Im}z|. \tag*{(151)} ∣ arg ( 1 + z 2 ) ∣ ≤ Re ( 1 + z 2 ) ∣ Im ( 1 + z 2 ) ∣ ≤ 3 4 ∣ Im z ∣. (151) For such z z z ( 4 / 3 ) p ∣ Im z ∣ < π (4/3)p|\operatorname{Im}z|<\pi ( 4/3 ) p ∣ Im z ∣ < π arg f p ( z ) = − p arg ( 1 + z 2 ) \arg f_p(z)=-p\arg(1+z^2) arg f p ( z ) = − p arg ( 1 + z 2 )
∣ arg f p ( z ) ∣ ≤ 4 3 p ∣ Im z ∣ . (152) |\arg f_p(z)|\leq\frac{4}{3}p|\operatorname{Im}z|. \tag*{(152)} ∣ arg f p ( z ) ∣ ≤ 3 4 p ∣ Im z ∣. (152) Now let A A A i < j i<j i < j z = ( w j − w i ) / ℓ α z=(w_j-w_i)/\ell_\alpha z = ( w j − w i ) / ℓ α A ∈ U α , p A\in\mathcal{U}_{\alpha,p} A ∈ U α , p
∣ Im z ∣ < 2 ω p ≤ 1 1000 , |\operatorname{Im}z|<2\omega_p\leq\frac{1}{1000}, ∣ Im z ∣ < 2 ω p ≤ 1000 1 , and in particular ( 4 / 3 ) p ∣ Im z ∣ < π (4/3)p|\operatorname{Im}z|<\pi ( 4/3 ) p ∣ Im z ∣ < π
∣ arg f p ( z ) ∣ ≤ 8 p ω p 3 = 8 p 3000 ( p + 1 ) < 1 375 . |\arg f_p(z)|\leq\frac{8p\omega_p}{3}=\frac{8p}{3000(p+1)}<\frac{1}{375}. ∣ arg f p ( z ) ∣ ≤ 3 8 p ω p = 3000 ( p + 1 ) 8 p < 375 1 . For the midpoint ( w i + w j ) / 2 (w_i+w_j)/2 ( w i + w j ) /2
∣ Re w i + w j 2 ∣ ≤ max ( ∣ Re w i ∣ , ∣ Re w j ∣ ) , ∣ Im w i + w j 2 ∣ ≤ max ( ∣ Im w i ∣ , ∣ Im w j ∣ ) , \left|\operatorname{Re}\frac{w_i+w_j}{2}\right|\leq\max(|\operatorname{Re}w_i|,|\operatorname{Re}w_j|),\qquad\left|\operatorname{Im}\frac{w_i+w_j}{2}\right|\leq\max(|\operatorname{Im}w_i|,|\operatorname{Im}w_j|), Re 2 w i + w j ≤ max ( ∣ Re w i ∣ , ∣ Re w j ∣ ) , Im 2 w i + w j ≤ max ( ∣ Im w i ∣ , ∣ Im w j ∣ ) , so that our assumption gives
( ∣ Re w i + w j 2 ∣ + 2 ) ∣ Im w i + w j 2 ∣ ≤ max w , w ′ ∈ { w i , w j } ( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ ≤ h N 2 200 . \left(\left|\operatorname{Re}\frac{w_i+w_j}{2}\right|+2\right)\left|\operatorname{Im}\frac{w_i+w_j}{2}\right|\leq\max_{w,w'\in\{w_i,w_j\}} (|\operatorname{Re}w|+2)|\operatorname{Im}w'|\leq\frac{h_N^2}{200}. ( Re 2 w i + w j + 2 ) Im 2 w i + w j ≤ w , w ′ ∈ { w i , w j } max ( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ ≤ 200 h N 2 . Applying part (3) of Lemma 5.1 , we find that either φ N ( ( w i + w j ) / 2 ) = 0 \varphi_N((w_i+w_j)/2)=0 φ N (( w i + w j ) /2 ) = 0 ∣ arg φ N ( ( w i + w j ) / 2 ) ∣ ≤ 1 / 200 |\arg\varphi_N((w_i+w_j)/2)|\leq1/200 ∣ arg φ N (( w i + w j ) /2 ) ∣ ≤ 1/200
∣ arg ( φ N ( w i + w j 2 ) f p ( z ) ) ∣ ≤ ∣ arg φ N ( w i + w j 2 ) ∣ + ∣ arg f p ( z ) ∣ ≤ 1 200 + 1 375 < 1 100 , \left|\arg\left(\varphi_N\left(\frac{w_i+w_j}{2}\right)f_p(z)\right)\right|\leq\left|\arg\varphi_N\left(\frac{w_i+w_j}{2}\right)\right|+|\arg f_p(z)|\leq\frac{1}{200}+\frac{1}{375}<\frac{1}{100}, arg ( φ N ( 2 w i + w j ) f p ( z ) ) ≤ arg φ N ( 2 w i + w j ) + ∣ arg f p ( z ) ∣ ≤ 200 1 + 375 1 < 100 1 , where the first inequality holds because the middle expression is smaller than π \pi π 1 / 100 1/100 1/100
For part (3), let Spec A ⊂ R α , p \operatorname{Spec} A\subset\mathcal{R}_{\alpha,p} Spec A ⊂ R α , p A ∈ U α , p A\in\mathcal{U}_{\alpha,p} A ∈ U α , p w w w w ′ w' w ′ A A A
( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ < 6 ω p ℓ α ≤ 6 2000 N − 1 ≤ N − 1 / 5 200 = h N 2 200 . (|\operatorname{Re}w|+2)|\operatorname{Im}w'|<6\omega_p\ell_\alpha\leq\frac{6}{2000}N^{-1}\leq\frac{N^{-1/5}}{200}=\frac{h_N^2}{200}. ( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ < 6 ω p ℓ α ≤ 2000 6 N − 1 ≤ 200 N − 1/5 = 200 h N 2 . It remains to prove part (4). The function F F F U α , p \mathcal{U}_{\alpha,p} U α , p U α , p ⊂ U α , p j \mathcal{U}_{\alpha,p}\subset\mathcal{U}_{\alpha,p_j} U α , p ⊂ U α , p j j ∈ J j\in\mathcal{J} j ∈ J A ∈ U α , p A\in\mathcal{U}_{\alpha,p} A ∈ U α , p p j p_j p j φ j \varphi_j φ j p p p φ \varphi φ Re X α ( A ; φ j , N , f p j ) ≥ 0 \operatorname{Re}\mathcal{X}_{\alpha}(A;\varphi_{j,N},f_{p_j}) \ge0 Re X α ( A ; φ j , N , f p j ) ≥ 0 j ∈ J j \in\mathcal{J} j ∈ J θ j ≥ 0 \theta_j \ge0 θ j ≥ 0
∣ F ( A ) ∣ = exp ( − ∑ j ∈ J θ j Re X α ( A ; φ j , N , f p j ) ) ≤ 1. |F(A)|=\exp\left(-\sum_{j\in\mathcal{J}}\theta_j\operatorname{Re}\mathcal{X}_{\alpha}(A;\varphi_{j,N},f_{p_j})\right)\le1. ∣ F ( A ) ∣ = exp − j ∈ J ∑ θ j Re X α ( A ; φ j , N , f p j ) ≤ 1. The last claim follows from part (3) and, for real symmetric A A A X α ( A ; φ j , N , f p j ) ≥ 0 \mathcal{X}_{\alpha}(A;\varphi_{j,N},f_{p_j})\ge0 X α ( A ; φ j , N , f p j ) ≥ 0 f p j f_{p_j} f p j R \mathbb{R} R
For a function Q Q Q N × N N\times N N × N B B B a ≤ b a\le b a ≤ b n ≥ 1 n\ge1 n ≥ 1
∂ a b n Q ( B ) = d n d x n Q ( B + x V a b ) ∣ x = 0 (153) \partial_{ab}^{n}Q(B)=\left.\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}Q(B+xV^{ab})\right|_{x=0} \tag*{(153)} ∂ ab n Q ( B ) = d x n d n Q ( B + x V ab ) x = 0 (153) whenever this derivative exists, and ∂ a b = ∂ a b 1 \partial_{ab}=\partial_{ab}^{1} ∂ ab = ∂ ab 1 x a b x_{ab} x ab ∑ a ≤ b x a b V a b \sum_{a\le b}x_{ab}V^{ab} ∑ a ≤ b x ab V ab ∂ a b \partial_{ab} ∂ ab x a b x_{ab} x ab F F F
Lemma 5.6. Let F F F p = max j ∈ J p j p=\max_{j\in\mathcal{J}}p_j p = max j ∈ J p j B B B a ≤ b a\le b a ≤ b x ↦ F ( B + x V a b ) x\mapsto F(B+xV^{ab}) x ↦ F ( B + x V ab ) R \mathbb{R} R n ≥ 1 n\ge1 n ≥ 1
∣ ∂ a b n F ( B ) ∣ ≤ n ! ( 2 ω p ) n ( 1 + ∥ B ∥ ℓ α h N 2 ) n . (154) |\partial_{ab}^{n}F(B)|\le n!\left(\frac{2}{\omega_p}\right)^{n}\left(\frac{1+\|B\|}{\ell_{\alpha}h_N^{2}}\right)^{n}. \tag*{(154)} ∣ ∂ ab n F ( B ) ∣ ≤ n ! ( ω p 2 ) n ( ℓ α h N 2 1 + ∥ B ∥ ) n . (154) Proof. Fix B B B a ≤ b a\le b a ≤ b
r B = ω p ℓ α h N 2 2 ( 1 + ∥ B ∥ ) . r_B=\frac{\omega_p\ell_{\alpha}h_N^{2}}{2(1+\|B\|)}. r B = 2 ( 1 + ∥ B ∥ ) ω p ℓ α h N 2 . We first locate the eigenvalues of B + ζ V a b B+\zeta V^{ab} B + ζ V ab ζ ∈ C \zeta\in\mathbb{C} ζ ∈ C w ∈ C w\in\mathbb{C} w ∈ C dist ( w , Spec B ) > ∣ ζ ∣ \operatorname{dist}(w,\operatorname{Spec}B)>|\zeta| dist ( w , Spec B ) > ∣ ζ ∣ B B B ∥ V a b ∥ = 1 \|V^{ab}\|=1 ∥ V ab ∥ = 1
∥ ζ ( B − w ) − 1 V a b ∥ ≤ ∣ ζ ∣ dist ( w , Spec B ) − 1 < 1 , \|\zeta(B-w)^{-1}V^{ab}\|\le|\zeta|\operatorname{dist}(w,\operatorname{Spec}B)^{-1}<1, ∥ ζ ( B − w ) − 1 V ab ∥ ≤ ∣ ζ ∣ dist ( w , Spec B ) − 1 < 1 , so the matrix
B + ζ V a b − w = ( B − w ) ( Id + ζ ( B − w ) − 1 V a b ) B+\zeta V^{ab}-w=(B-w)\left(\operatorname{Id}+\zeta(B-w)^{-1}V^{ab}\right) B + ζ V ab − w = ( B − w ) ( Id + ζ ( B − w ) − 1 V ab ) is invertible. Hence every eigenvalue w w w B + ζ V a b B+\zeta V^{ab} B + ζ V ab dist ( w , Spec B ) ≤ ∣ ζ ∣ \operatorname{dist}(w,\operatorname{Spec}B)\le|\zeta| dist ( w , Spec B ) ≤ ∣ ζ ∣
∣ Im w ∣ ≤ ∣ ζ ∣ , ∣ Re w ∣ ≤ ∥ B ∥ + ∣ ζ ∣ . (155) |\operatorname{Im}w|\le|\zeta|,\qquad|\operatorname{Re}w|\le\|B\|+|\zeta|. \tag*{(155)} ∣ Im w ∣ ≤ ∣ ζ ∣ , ∣ Re w ∣ ≤ ∥ B ∥ + ∣ ζ ∣. (155) The bound (155) shows that B + ζ V a b ∈ U α , p B+\zeta V^{ab}\in U_{\alpha,p} B + ζ V ab ∈ U α , p ∣ ζ ∣ < ω p ℓ α |\zeta|<\omega_p\ell_\alpha ∣ ζ ∣ < ω p ℓ α ζ ↦ F ( B + ζ V a b ) \zeta\mapsto F(B+\zeta V^{ab}) ζ ↦ F ( B + ζ V ab ) { ∣ ζ ∣ < ω p ℓ α } \{|\zeta|<\omega_p\ell_\alpha\} { ∣ ζ ∣ < ω p ℓ α } B + x 0 V a b B+x_0V^{ab} B + x 0 V ab B B B x 0 ∈ R x_0\in\mathbb{R} x 0 ∈ R x ↦ F ( B + x V a b ) x\mapsto F(B+xV^{ab}) x ↦ F ( B + x V ab ) R \mathbb{R} R
Now let ∣ ζ ∣ ≤ r B |\zeta|\le r_B ∣ ζ ∣ ≤ r B w w w w ′ w' w ′ B + ζ V a b B+\zeta V^{ab} B + ζ V ab (155) , r B ≤ 1 r_B\le1 r B ≤ 1 r B < ω p ℓ α r_B<\omega_p\ell_\alpha r B < ω p ℓ α B + ζ V a b ∈ U α , p B+\zeta V^{ab}\in U_{\alpha,p} B + ζ V ab ∈ U α , p
( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ ≤ ( ∥ B ∥ + 3 ) r B ≤ 3 ( 1 + ∥ B ∥ ) r B = 3 2 ω p ℓ α h N 2 ≤ h N 2 200 . (|\operatorname{Re}w|+2)|\operatorname{Im}w'|\leq(\lVert B\rVert+3)r_B\leq3(1+\lVert B\rVert)r_B=\frac{3}{2}\omega_p\ell_\alpha h_N^2\leq\frac{h_N^2}{200}. ( ∣ Re w ∣ + 2 ) ∣ Im w ′ ∣ ≤ (∥ B ∥ + 3 ) r B ≤ 3 ( 1 + ∥ B ∥) r B = 2 3 ω p ℓ α h N 2 ≤ 200 h N 2 . Hence B + ζ V a b B+\zeta V^{ab} B + ζ V ab ∣ F ( B + ζ V a b ) ∣ ≤ 1 |F(B+\zeta V^{ab})|\leq1 ∣ F ( B + ζ V ab ) ∣ ≤ 1 ∣ ζ ∣ ≤ r B |\zeta|\leq r_B ∣ ζ ∣ ≤ r B r B r_B r B
∣ ∂ a b n F ( B ) ∣ ≤ n ! r B − n , |\partial_{ab}^nF(B)|\leq n!r_B^{-n}, ∣ ∂ ab n F ( B ) ∣ ≤ n ! r B − n , which is (154) .
Comparison along the Ornstein–Uhlenbeck interpolation
For a ≤ b a\leq b a ≤ b t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] x a b ( t ) = ( H t ) a b x_{ab}(t)=(H_t)_{ab} x ab ( t ) = ( H t ) ab ( a , b ) (a,b) ( a , b ) H t H_t H t
H t = ∑ a ≤ b x a b ( t ) V a b , x a b ( t ) = e − t / 2 H a b + ( 1 − e − t ) 1 / 2 G a b , (156) H_t=\sum_{a\leq b}x_{ab}(t)V^{ab},\qquad x_{ab}(t)=e^{-t/2}H_{ab}+(1-e^{-t})^{1/2}G_{ab}, \tag*{(156)} H t = a ≤ b ∑ x ab ( t ) V ab , x ab ( t ) = e − t /2 H ab + ( 1 − e − t ) 1/2 G ab , (156) and we let H t [ a b ] = H t − x a b ( t ) V a b H_t^{[ab]}=H_t-x_{ab}(t)V^{ab} H t [ ab ] = H t − x ab ( t ) V ab H t H_t H t ( a , b ) (a,b) ( a , b ) ( b , a ) (b,a) ( b , a ) v a b G = E G a b 2 v_{ab}^G=\mathbb{E}G_{ab}^2 v ab G = E G ab 2 v a b G = 1 / N v_{ab}^G=1/N v ab G = 1/ N a < b a<b a < b v a a G = 2 / N v_{aa}^G=2/N v aa G = 2/ N ( H a b ) a ≤ b (H_{ab})_{a\leq b} ( H ab ) a ≤ b ( G a b ) a ≤ b (G_{ab})_{a\leq b} ( G ab ) a ≤ b t t t ( x a b ( t ) ) a ≤ b (x_{ab}(t))_{a\leq b} ( x ab ( t ) ) a ≤ b x a b ( t ) x_{ab}(t) x ab ( t ) H t [ a b ] H_t^{[ab]} H t [ ab ]
We record some properties of the coordinates. Recall from Definition 1.1 that H a b = N − 1 / 2 X a b H_{ab}=N^{-1/2}X_{ab} H ab = N − 1/2 X ab a < b a<b a < b H a a = N − 1 / 2 Z a H_{aa}=N^{-1/2}Z_a H aa = N − 1/2 Z a P ( ∣ X ∣ > y ) + P ( ∣ Z ∣ > y ) ≤ C 0 e − c 0 y 2 \mathbb{P}(|X|>y)+\mathbb{P}(|Z|>y)\leq C_0e^{-c_0y^2} P ( ∣ X ∣ > y ) + P ( ∣ Z ∣ > y ) ≤ C 0 e − c 0 y 2 y ≥ 0 y\geq0 y ≥ 0 G a b G_{ab} G ab 2 / N 2/N 2/ N t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] a ≤ b a\leq b a ≤ b y ≥ 0 y\geq0 y ≥ 0
P ( ∣ x a b ( t ) ∣ > y ) ≤ P ( ∣ H a b ∣ > y / 2 ) + P ( ∣ G a b ∣ > y / 2 ) ≤ C 0 e − c 0 N y 2 / 4 + 2 e − N y 2 / 16 . (157) \mathbb{P}(|x_{ab}(t)|>y)\leq\mathbb{P}(|H_{ab}|>y/2)+\mathbb{P}(|G_{ab}|>y/2)\leq C_0e^{-c_0Ny^2/4}+2e^{-Ny^2/16}. \tag*{(157)} P ( ∣ x ab ( t ) ∣ > y ) ≤ P ( ∣ H ab ∣ > y /2 ) + P ( ∣ G ab ∣ > y /2 ) ≤ C 0 e − c 0 N y 2 /4 + 2 e − N y 2 /16 . (157) In particular, for every integer n ≥ 1 n\geq1 n ≥ 1 C n > 0 C_n>0 C n > 0
E ∣ x a b ( t ) ∣ n ≤ C n N − n / 2 (158) \mathbb{E}|x_{ab}(t)|^n\leq C_nN^{-n/2} \tag*{(158)} E ∣ x ab ( t ) ∣ n ≤ C n N − n /2 (158) for all t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] a ≤ b a\leq b a ≤ b H a b H_{ab} H ab G a b G_{ab} G ab E H a b = E G a b = E G a b 3 = 0 \mathbb{E}H_{ab}=\mathbb{E}G_{ab}=\mathbb{E}G_{ab}^3=0 E H ab = E G ab = E G ab 3 = 0 E H a b 2 = 1 / N \mathbb{E}H_{ab}^2=1/N E H ab 2 = 1/ N a < b a<b a < b E H a a 2 = σ Z 2 / N \mathbb{E}H_{aa}^2=\sigma_Z^2/N E H aa 2 = σ Z 2 / N
E x a b ( t ) 2 = v a b G , E x a b ( t ) 3 = e − 3 t / 2 E X 3 N 3 / 2 ( a < b ) , E x a a ( t ) 2 = v a a G + e − t ( σ Z 2 − 2 ) N . (159) \mathbb{E}x_{ab}(t)^2=v_{ab}^G,\qquad\mathbb{E}x_{ab}(t)^3=\frac{e^{-3t/2}\mathbb{E}X^3}{N^{3/2}}\quad(a<b),\qquad\mathbb{E}x_{aa}(t)^2=v_{aa}^G+\frac{e^{-t}(\sigma_Z^2-2)}{N}. \tag*{(159)} E x ab ( t ) 2 = v ab G , E x ab ( t ) 3 = N 3/2 e − 3 t /2 E X 3 ( a < b ) , E x aa ( t ) 2 = v aa G + N e − t ( σ Z 2 − 2 ) . (159) The following lemma expresses E Q ( H t ) − E Q ( H ) \mathbb{E}Q(H_t)-\mathbb{E}Q(H) E Q ( H t ) − E Q ( H ) Q Q Q
Lemma 5.7. Let Q Q Q N × N N \times N N × N K 0 > 0 K_0>0 K 0 > 0
∣ ∂ a b Q ( A ) ∣ + ∣ ∂ a b 2 Q ( A ) ∣ ≤ K 0 ( 1 + ∥ A ∥ ) 2 (160) \lvert\partial_{ab}Q(A)\rvert+\lvert\partial_{ab}^{2}Q(A)\rvert\le K_0(1+\lVert A\rVert)^2 \tag*{(160)} ∣ ∂ ab Q ( A )∣ + ∣ ∂ ab 2 Q ( A )∣ ≤ K 0 ( 1 + ∥ A ∥ ) 2 (160) for every real symmetric matrix A A A a ≤ b a\le b a ≤ b t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ]
E Q ( H t ) − E Q ( H ) = 1 2 ∫ 0 t ∑ a ≤ b E [ v a b G ∂ a b 2 Q ( H t ′ ) − x a b ( t ′ ) ∂ a b Q ( H t ′ ) ] d t ′ . (161) \mathbb{E}Q(H_t)-\mathbb{E}Q(H)=\frac{1}{2}\int_0^t\sum_{a\le b}\mathbb{E}\left[v_{ab}^{G}\partial_{ab}^{2}Q(H_{t'})-x_{ab}(t')\partial_{ab}Q(H_{t'})\right]\,\mathrm{d}t'. \tag*{(161)} E Q ( H t ) − E Q ( H ) = 2 1 ∫ 0 t a ≤ b ∑ E [ v ab G ∂ ab 2 Q ( H t ′ ) − x ab ( t ′ ) ∂ ab Q ( H t ′ ) ] d t ′ . (161) Proof. Fix realizations of H H H G G G t ′ ↦ H t ′ t'\mapsto H_{t'} t ′ ↦ H t ′ [ 0 , 1 ] [0,1] [ 0 , 1 ] ( 0 , 1 ] (0,1] ( 0 , 1 ]
e − t ′ ( 1 − e − t ′ ) − 1 / 2 = ( 1 − e − t ′ ) − 1 / 2 − ( 1 − e − t ′ ) 1 / 2 , e^{-t'}(1-e^{-t'})^{-1/2}=(1-e^{-t'})^{-1/2}-(1-e^{-t'})^{1/2}, e − t ′ ( 1 − e − t ′ ) − 1/2 = ( 1 − e − t ′ ) − 1/2 − ( 1 − e − t ′ ) 1/2 , we find that
d d t ′ x a b ( t ′ ) = − 1 2 x a b ( t ′ ) + 1 2 ( 1 − e − t ′ ) − 1 / 2 G a b , t ′ ∈ ( 0 , 1 ] . \frac{\mathrm{d}}{\mathrm{d}t'}x_{ab}(t')=-\frac{1}{2}x_{ab}(t')+\frac{1}{2}(1-e^{-t'})^{-1/2}G_{ab},\qquad t'\in(0,1]. d t ′ d x ab ( t ′ ) = − 2 1 x ab ( t ′ ) + 2 1 ( 1 − e − t ′ ) − 1/2 G ab , t ′ ∈ ( 0 , 1 ] . The chain rule gives, for t ′ ∈ ( 0 , 1 ] t'\in(0,1] t ′ ∈ ( 0 , 1 ]
d d t ′ Q ( H t ′ ) = ∑ a ≤ b ∂ a b Q ( H t ′ ) ( − 1 2 x a b ( t ′ ) + 1 2 ( 1 − e − t ′ ) − 1 / 2 G a b ) . (162) \frac{\mathrm{d}}{\mathrm{d}t'}Q(H_{t'})=\sum_{a\le b}\partial_{ab}Q(H_{t'})\left(-\frac{1}{2}x_{ab}(t')+\frac{1}{2}(1-e^{-t'})^{-1/2}G_{ab}\right). \tag*{(162)} d t ′ d Q ( H t ′ ) = a ≤ b ∑ ∂ ab Q ( H t ′ ) ( − 2 1 x ab ( t ′ ) + 2 1 ( 1 − e − t ′ ) − 1/2 G ab ) . (162) We now justify integrating (162) . For t ′ ∈ ( 0 , 1 ] t'\in(0,1] t ′ ∈ ( 0 , 1 ]
∥ H t ′ ∥ ≤ ∥ H ∥ + ∥ G ∥ , ∣ x a b ( t ′ ) ∣ ≤ ∣ H a b ∣ + ∣ G a b ∣ , 1 − e − t ′ ≥ t ′ / e . \lVert H_{t'}\rVert\le\lVert H\rVert+\lVert G\rVert,\qquad\lvert x_{ab}(t')\rvert\le\lvert H_{ab}\rvert+\lvert G_{ab}\rvert,\qquad1-e^{-t'}\ge t'/e. ∥ H t ′ ∥ ≤ ∥ H ∥ + ∥ G ∥ , ∣ x ab ( t ′ )∣ ≤ ∣ H ab ∣ + ∣ G ab ∣ , 1 − e − t ′ ≥ t ′ / e . Together with (5.35), these bounds give
∣ d d t ′ Q ( H t ′ ) ∣ ≤ K 0 ( 1 + ∥ H ∥ + ∥ G ∥ ) 2 ( 1 + ( e / t ′ ) 1 / 2 ) ∑ a ≤ b ( ∣ H a b ∣ + ∣ G a b ∣ ) . (163) \left\lvert\frac{\mathrm{d}}{\mathrm{d}t'}Q(H_{t'})\right\rvert\le K_0(1+\lVert H\rVert+\lVert G\rVert)^2(1+(e/t')^{1/2})\sum_{a\le b}(\lvert H_{ab}\rvert+\lvert G_{ab}\rvert). \tag*{(163)} d t ′ d Q ( H t ′ ) ≤ K 0 ( 1 + ∥ H ∥ + ∥ G ∥ ) 2 ( 1 + ( e / t ′ ) 1/2 ) a ≤ b ∑ (∣ H ab ∣ + ∣ G ab ∣) . (163) The entries of H H H G G G ∥ H ∥ 2 ≤ ∑ a , b H a b 2 \lVert H\rVert^2\le\sum_{a,b}H_{ab}^2 ∥ H ∥ 2 ≤ ∑ a , b H ab 2 t ′ ∈ ( 0 , 1 ] t'\in(0,1] t ′ ∈ ( 0 , 1 ] t ′ t' t ′ ( 0 , 1 ] (0,1] ( 0 , 1 ] ( t ′ ) − 1 / 2 (t')^{-1/2} ( t ′ ) − 1/2 t ′ = 0 t'=0 t ′ = 0 [ t 0 , t ] [t_0,t] [ t 0 , t ] 0 < t 0 < t 0<t_0<t 0 < t 0 < t t 0 → 0 t_0\to0 t 0 → 0 t ′ ↦ Q ( H t ′ ) t'\mapsto Q(H_{t'}) t ′ ↦ Q ( H t ′ ) t ′ = 0 t'=0 t ′ = 0
Q ( H t ) − Q ( H ) = ∫ 0 t d d t ′ Q ( H t ′ ) d t ′ , t ∈ [ 0 , 1 ] . Q(H_t)-Q(H)=\int_0^t\frac{\mathrm{d}}{\mathrm{d}t'}Q(H_{t'})\,\mathrm{d}t',\qquad t\in[0,1]. Q ( H t ) − Q ( H ) = ∫ 0 t d t ′ d Q ( H t ′ ) d t ′ , t ∈ [ 0 , 1 ] . Taking expectations and using Fubini’s theorem, we find
E Q ( H t ) − E Q ( H ) = ∫ 0 t E [ d d t ′ Q ( H t ′ ) ] d t ′ . (164) \mathbb{E}Q(H_t)-\mathbb{E}Q(H)=\int_0^t \mathbb{E}\left[\frac{\mathrm{d}}{\mathrm{d}t'}Q(H_{t'})\right]\mathrm{d}t'. \tag*{(164)} E Q ( H t ) − E Q ( H ) = ∫ 0 t E [ d t ′ d Q ( H t ′ ) ] d t ′ . (164) Next, we compute the expectation of (5.37) with Gaussian integration by parts. Fix t ′ ∈ ( 0 , 1 ] t' \in(0,1] t ′ ∈ ( 0 , 1 ] a ≤ b a \le b a ≤ b H H H ( G a ′ b ′ ) ( a ′ , b ′ ) ≠ ( a , b ) (G_{a'b'})_{(a',b')\ne(a,b)} ( G a ′ b ′ ) ( a ′ , b ′ ) = ( a , b ) H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] H a b H_{ab} H ab
H t ′ = H t ′ [ a b ] + ( e − t ′ / 2 H a b + ( 1 − e − t ′ ) 1 / 2 G a b ) V a b . H_{t'}=H_{t'}^{[ab]}+\left(e^{-t'/2}H_{ab}+\left(1-e^{-t'}\right)^{1/2}G_{ab}\right)V^{ab}. H t ′ = H t ′ [ ab ] + ( e − t ′ /2 H ab + ( 1 − e − t ′ ) 1/2 G ab ) V ab . Hence ∂ a b Q ( H t ′ ) \partial_{ab}Q(H_{t'}) ∂ ab Q ( H t ′ ) G a b G_{ab} G ab
∂ ∂ G a b ∂ a b Q ( H t ′ ) = ( 1 − e − t ′ ) 1 / 2 ∂ a b 2 Q ( H t ′ ) . \frac{\partial}{\partial G_{ab}}\partial_{ab}Q(H_{t'})=\left(1-e^{-t'}\right)^{1/2}\partial_{ab}^{2}Q(H_{t'}). ∂ G ab ∂ ∂ ab Q ( H t ′ ) = ( 1 − e − t ′ ) 1/2 ∂ ab 2 Q ( H t ′ ) . Moreover, (5.35) implies that this function and its derivative grow at most polynomially in G a b G_{ab} G ab
The variable G a b G_{ab} G ab v a b G v_{ab}^{G} v ab G
E [ G a b ψ ( G a b ) ] = v a b G E [ ψ ′ ( G a b ) ] , \mathbb{E}[G_{ab}\psi(G_{ab})]=v_{ab}^{G}\mathbb{E}[\psi'(G_{ab})], E [ G ab ψ ( G ab )] = v ab G E [ ψ ′ ( G ab )] , valid for every continuously differentiable ψ : R → C \psi:\mathbb{R}\to\mathbb{C} ψ : R → C ψ \psi ψ ψ ′ \psi' ψ ′
E [ G a b ∂ a b Q ( H t ′ ) ] = ( 1 − e − t ′ ) 1 / 2 v a b G E [ ∂ a b 2 Q ( H t ′ ) ] . (165) \mathbb{E}\left[G_{ab}\partial_{ab}Q(H_{t'})\right]=\left(1-e^{-t'}\right)^{1/2}v_{ab}^{G}\mathbb{E}\left[\partial_{ab}^{2}Q(H_{t'})\right]. \tag*{(165)} E [ G ab ∂ ab Q ( H t ′ ) ] = ( 1 − e − t ′ ) 1/2 v ab G E [ ∂ ab 2 Q ( H t ′ ) ] . (165) Taking expectations in (5.37) and inserting (5.40), we see that the integrands on the right sides of (5.39) and (5.36) coincide, which proves (5.36).
We now explain how the integrand in (5.36) will be estimated for Q = F Q=F Q = F
E [ v a b G Υ ′ ′ ( x a b ( t ′ ) ) − x a b ( t ′ ) Υ ′ ( x a b ( t ′ ) ) ] . \mathbb{E}\left[v_{ab}^{G}\Upsilon''(x_{ab}(t'))-x_{ab}(t')\Upsilon'(x_{ab}(t'))\right]. E [ v ab G Υ ′′ ( x ab ( t ′ )) − x ab ( t ′ ) Υ ′ ( x ab ( t ′ )) ] . for a random function Υ \Upsilon Υ x a b ( t ′ ) x_{ab}(t') x ab ( t ′ ) F F F H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] V a b V^{ab} V ab v a b G = E x a b ( t ′ ) 2 v_{ab}^{G}=\mathbb{E}x_{ab}(t')^{2} v ab G = E x ab ( t ′ ) 2 Υ ′ \Upsilon' Υ ′ Υ ′ ′ \Upsilon'' Υ ′′ 0 0 0 E ∣ x a b ( t ′ ) ∣ 3 \mathbb{E}|x_{ab}(t')|^{3} E ∣ x ab ( t ′ ) ∣ 3 Υ \Upsilon Υ [44 ] (see also [54 Proposition 3.1] ), makes this precise. We state it with a cutoff, since good bounds on the derivatives of Υ \Upsilon Υ
Lemma 5.8. Let κ \kappa κ E κ = 0 \mathbb{E}\kappa= 0 E κ = 0 E κ 2 ≤ 1 \mathbb{E}\kappa^2 \le1 E κ 2 ≤ 1 E κ 4 < ∞ \mathbb{E}\kappa^4 < \infty E κ 4 < ∞ r ∈ ( 0 , 1 ] r \in(0,1] r ∈ ( 0 , 1 ] K 0 ≥ 0 K_0 \ge0 K 0 ≥ 0 Υ : R → C \Upsilon:\mathbb{R} \to\mathbb{C} Υ : R → C
∣ Υ ( n ) ( x ) ∣ ≤ n ! r − n for ∣ x ∣ ≤ r and n ∈ { 1 , 2 , 3 , 4 } , (166) \lvert\Upsilon^{(n)}(x) \rvert\le n!r^{-n} \qquad\text{for } \lvert x\rvert\le r \text{ and } n \in\{1,2,3,4\}, \tag*{(166)} ∣ Υ ( n ) ( x )∣ ≤ n ! r − n for ∣ x ∣ ≤ r and n ∈ { 1 , 2 , 3 , 4 } , (166) ∣ Υ ′ ( x ) ∣ + ∣ Υ ′ ′ ( x ) ∣ ≤ K 0 ( 1 + ∣ x ∣ ) 2 for x ∈ R . (167) \lvert\Upsilon'(x) \rvert+ \lvert\Upsilon''(x) \rvert\le K_0(1+\lvert x\rvert)^2 \qquad\text{for } x \in\mathbb{R}. \tag*{(167)} ∣ Υ ′ ( x )∣ + ∣ Υ ′′ ( x )∣ ≤ K 0 ( 1 + ∣ x ∣ ) 2 for x ∈ R . (167) Set t = ( 9 r − 3 + 2 K 0 ) E [ ( 1 + ∣ κ ∣ ) 4 1 ∣ κ ∣ > r ] t=(9r^{-3}+2K_0)\mathbb{E}[(1+\lvert\kappa\rvert)^4\mathbf{1}_{\lvert\kappa\rvert>r}] t = ( 9 r − 3 + 2 K 0 ) E [( 1 + ∣ κ ∣ ) 4 1 ∣ κ ∣ > r ]
∣ E [ κ Υ ′ ( κ ) ] − E [ κ 2 ] E [ Υ ′ ′ ( κ ) ] ∣ ≤ 9 r − 3 E ∣ κ ∣ 3 + t . (168) \left\lvert\mathbb{E}[\kappa\Upsilon'(\kappa)]-\mathbb{E}[\kappa^2]\mathbb{E}[\Upsilon''(\kappa)]\right\rvert\le9r^{-3}\mathbb{E}\lvert\kappa\rvert^3+t. \tag*{(168)} E [ κ Υ ′ ( κ )] − E [ κ 2 ] E [ Υ ′′ ( κ )] ≤ 9 r − 3 E ∣ κ ∣ 3 + t . (168) If moreover E κ 3 = 0 \mathbb{E}\kappa^3=0 E κ 3 = 0
∣ E [ κ Υ ′ ( κ ) ] − E [ κ 2 ] E [ Υ ′ ′ ( κ ) ] ∣ ≤ 16 r − 4 E κ 4 + t . (169) \left\lvert\mathbb{E}[\kappa\Upsilon'(\kappa)]-\mathbb{E}[\kappa^2]\mathbb{E}[\Upsilon''(\kappa)]\right\rvert\le16r^{-4}\mathbb{E}\kappa^4+t. \tag*{(169)} E [ κ Υ ′ ( κ )] − E [ κ 2 ] E [ Υ ′′ ( κ )] ≤ 16 r − 4 E κ 4 + t . (169) Finally, we have
∣ E Υ ′ ′ ( κ ) ∣ ≤ 2 r − 2 + t . (170) \lvert\mathbb{E}\Upsilon''(\kappa)\rvert\le2r^{-2}+t. \tag*{(170)} ∣ E Υ ′′ ( κ )∣ ≤ 2 r − 2 + t . (170) Proof. For notational convenience, we abbreviate 1 − = 1 ∣ κ ∣ ≤ r \mathbf{1}_{-}=\mathbf{1}_{\lvert\kappa\rvert\le r} 1 − = 1 ∣ κ ∣ ≤ r 1 + = 1 ∣ κ ∣ > r \mathbf{1}_{+}=\mathbf{1}_{\lvert\kappa\rvert>r} 1 + = 1 ∣ κ ∣ > r v = E κ 2 v=\mathbb{E}\kappa^2 v = E κ 2 n = 3 n=3 n = 3 { ∣ κ ∣ ≤ r } \{\lvert\kappa\rvert\le r\} {∣ κ ∣ ≤ r }
Υ ′ ( κ ) = Υ ′ ( 0 ) + κ Υ ′ ′ ( 0 ) + ρ 1 , Υ ′ ′ ( κ ) = Υ ′ ′ ( 0 ) + ρ 2 , \Upsilon'(\kappa)=\Upsilon'(0)+\kappa\Upsilon''(0)+\rho_1,\qquad\Upsilon''(\kappa)=\Upsilon''(0)+\rho_2, Υ ′ ( κ ) = Υ ′ ( 0 ) + κ Υ ′′ ( 0 ) + ρ 1 , Υ ′′ ( κ ) = Υ ′′ ( 0 ) + ρ 2 , ∣ ρ 1 ∣ ≤ 3 r − 3 κ 2 , ∣ ρ 2 ∣ ≤ 6 r − 3 ∣ κ ∣ . \lvert\rho_1\rvert\le3r^{-3}\kappa^2,\qquad\lvert\rho_2\rvert\le6r^{-3}\lvert\kappa\rvert. ∣ ρ 1 ∣ ≤ 3 r − 3 κ 2 , ∣ ρ 2 ∣ ≤ 6 r − 3 ∣ κ ∣ . Therefore
E [ ( κ Υ ′ ( κ ) − v Υ ′ ′ ( κ ) ) 1 − ] = Υ ′ ( 0 ) E [ κ 1 − ] + Υ ′ ′ ( 0 ) E [ ( κ 2 − v ) 1 − ] + E [ ( κ ρ 1 − v ρ 2 ) 1 − ] . (171) \mathbb{E}\left[(\kappa\Upsilon'(\kappa)-v\Upsilon''(\kappa))\mathbf{1}_{-}\right]
=\Upsilon'(0)\mathbb{E}[\kappa\mathbf{1}_{-}]
+\Upsilon''(0)\mathbb{E}[(\kappa^2-v)\mathbf{1}_{-}]
+\mathbb{E}[(\kappa\rho_1-v\rho_2)\mathbf{1}_{-}]. \tag*{(171)} E [ ( κ Υ ′ ( κ ) − v Υ ′′ ( κ )) 1 − ] = Υ ′ ( 0 ) E [ κ 1 − ] + Υ ′′ ( 0 ) E [( κ 2 − v ) 1 − ] + E [( κ ρ 1 − v ρ 2 ) 1 − ] . (171) The identities E κ = 0 \mathbb{E}\kappa=0 E κ = 0 E [ κ 2 − v ] = 0 \mathbb{E}[\kappa^2-v]=0 E [ κ 2 − v ] = 0
E [ κ 1 − ] = − E [ κ 1 + ] , E [ ( κ 2 − v ) 1 − ] = − E [ ( κ 2 − v ) 1 + ] . \mathbb{E}[\kappa\mathbf{1}_{-}]=-\mathbb{E}[\kappa\mathbf{1}_{+}],\qquad
\mathbb{E}[(\kappa^2-v)\mathbf{1}_{-}]=-\mathbb{E}[(\kappa^2-v)\mathbf{1}_{+}]. E [ κ 1 − ] = − E [ κ 1 + ] , E [( κ 2 − v ) 1 − ] = − E [( κ 2 − v ) 1 + ] . Using ∣ Υ ′ ( 0 ) ∣ ≤ r − 1 \lvert\Upsilon'(0)\rvert\le r^{-1} ∣ Υ ′ ( 0 )∣ ≤ r − 1 ∣ Υ ′ ′ ( 0 ) ∣ ≤ 2 r − 2 \lvert\Upsilon''(0)\rvert\le2r^{-2} ∣ Υ ′′ ( 0 )∣ ≤ 2 r − 2 v ≤ 1 v\le1 v ≤ 1 r ≤ 1 r\le1 r ≤ 1
r − 1 E [ ∣ κ ∣ 1 + ] + 2 r − 2 E [ ( κ 2 + 1 ) 1 + ] ≤ 3 r − 3 E [ ( 1 + ∣ κ ∣ ) 4 1 + ] . r^{-1}\mathbb{E}[\lvert\kappa\rvert\mathbf{1}_{+}]
+2r^{-2}\mathbb{E}[(\kappa^2+1)\mathbf{1}_{+}]
\le3r^{-3}\mathbb{E}[(1+\lvert\kappa\rvert)^4\mathbf{1}_{+}]. r − 1 E [∣ κ ∣ 1 + ] + 2 r − 2 E [( κ 2 + 1 ) 1 + ] ≤ 3 r − 3 E [( 1 + ∣ κ ∣ ) 4 1 + ] . Using Hölder’s inequality in the form v E ∣ κ ∣ ≤ E ∣ κ ∣ 3 v\mathbb{E}\lvert\kappa\rvert\le\mathbb{E}\lvert\kappa\rvert^3 v E ∣ κ ∣ ≤ E ∣ κ ∣ 3
∣ E [ ( κ ρ 1 − v ρ 2 ) 1 − ] ∣ ≤ 3 r − 3 E ∣ κ ∣ 3 + 6 r − 3 v E ∣ κ ∣ ≤ 9 r − 3 E ∣ κ ∣ 3 . \left\lvert\mathbb{E}[(\kappa\rho_1-v\rho_2)\mathbf{1}_{-}]\right\rvert
\le3r^{-3}\mathbb{E}\lvert\kappa\rvert^3+6r^{-3}v\mathbb{E}\lvert\kappa\rvert
\le9r^{-3}\mathbb{E}\lvert\kappa\rvert^3. ∣ E [( κ ρ 1 − v ρ 2 ) 1 − ] ∣ ≤ 3 r − 3 E ∣ κ ∣ 3 + 6 r − 3 v E ∣ κ ∣ ≤ 9 r − 3 E ∣ κ ∣ 3 . To bound the contribution of the event { ∣ κ ∣ > r } \{\lvert\kappa\rvert>r\} {∣ κ ∣ > r }
∣ E [ ( κ Υ ′ ( κ ) − v Υ ′ ′ ( κ ) ) 1 + ] ∣ ≤ K 0 E [ ( ∣ κ ∣ + 1 ) ( 1 + ∣ κ ∣ ) 2 1 + ] ≤ K 0 E [ ( 1 + ∣ κ ∣ ) 4 1 + ] . \left\lvert\mathbb{E}[(\kappa\Upsilon'(\kappa)-v\Upsilon''(\kappa))\mathbf{1}_{+}]\right\rvert
\le K_0\mathbb{E}[(\lvert\kappa\rvert+1)(1+\lvert\kappa\rvert)^2\mathbf{1}_{+}]
\le K_0\mathbb{E}[(1+\lvert\kappa\rvert)^4\mathbf{1}_{+}]. ∣ E [( κ Υ ′ ( κ ) − v Υ ′′ ( κ )) 1 + ] ∣ ≤ K 0 E [(∣ κ ∣ + 1 ) ( 1 + ∣ κ ∣ ) 2 1 + ] ≤ K 0 E [( 1 + ∣ κ ∣ ) 4 1 + ] . Adding the contributions of the two events, we obtain (168) .
When E ϰ 3 = 0 \mathbb{E}\varkappa^3=0 E ϰ 3 = 0 { ∣ ϰ ∣ ≤ r } \{|\varkappa|\le r\} { ∣ ϰ ∣ ≤ r } (166) with n = 4 n=4 n = 4
Υ ′ ( ϰ ) = Υ ′ ( 0 ) + ϰ Υ ′ ′ ( 0 ) + ϰ 2 2 Υ ′ ′ ′ ( 0 ) + ρ 3 , Υ ′ ′ ( ϰ ) = Υ ′ ′ ( 0 ) + ϰ Υ ′ ′ ′ ( 0 ) + ρ 4 , \Upsilon'(\varkappa)=\Upsilon'(0)+\varkappa\Upsilon''(0)+\frac{\varkappa^2}{2}\Upsilon'''(0)+\rho_3,\qquad\Upsilon''(\varkappa)=\Upsilon''(0)+\varkappa\Upsilon'''(0)+\rho_4, Υ ′ ( ϰ ) = Υ ′ ( 0 ) + ϰ Υ ′′ ( 0 ) + 2 ϰ 2 Υ ′′′ ( 0 ) + ρ 3 , Υ ′′ ( ϰ ) = Υ ′′ ( 0 ) + ϰ Υ ′′′ ( 0 ) + ρ 4 , ∣ ρ 3 ∣ ≤ 4 r − 4 ∣ ϰ ∣ 3 , ∣ ρ 4 ∣ ≤ 12 r − 4 ϰ 2 . |\rho_3|\le4r^{-4}|\varkappa|^3,\qquad|\rho_4|\le12r^{-4}\varkappa^2. ∣ ρ 3 ∣ ≤ 4 r − 4 ∣ ϰ ∣ 3 , ∣ ρ 4 ∣ ≤ 12 r − 4 ϰ 2 . Compared with (171) , there is an additional term involving Υ ′ ′ ′ ( 0 ) \Upsilon'''(0) Υ ′′′ ( 0 ) E [ ϰ 3 / 2 − v ϰ ] = 0 \mathbb{E}[\varkappa^3/2-v\varkappa]=0 E [ ϰ 3 /2 − v ϰ ] = 0 ∣ Υ ′ ′ ′ ( 0 ) ∣ ≤ 6 r − 3 |\Upsilon'''(0)|\le6r^{-3} ∣ Υ ′′′ ( 0 ) ∣ ≤ 6 r − 3
∣ Υ ′ ′ ′ ( 0 ) E [ ( ϰ 3 / 2 − v ϰ ) 1 − ] ∣ = ∣ Υ ′ ′ ′ ( 0 ) E [ ( ϰ 3 / 2 − v ϰ ) 1 + ] ∣ ≤ 6 r − 3 E [ ( 1 + ∣ ϰ ∣ ) 4 1 + ] . |\Upsilon'''(0)\mathbb{E}[(\varkappa^3/2-v\varkappa)1_-]|=|\Upsilon'''(0)\mathbb{E}[(\varkappa^3/2-v\varkappa)1_+ ]|\le6r^{-3}\mathbb{E}[(1+|\varkappa|)^4 1_+]. ∣ Υ ′′′ ( 0 ) E [( ϰ 3 /2 − v ϰ ) 1 − ] ∣ = ∣ Υ ′′′ ( 0 ) E [( ϰ 3 /2 − v ϰ ) 1 + ] ∣ ≤ 6 r − 3 E [( 1 + ∣ ϰ ∣ ) 4 1 + ] . The remainder term is now bounded by
∣ E [ ( ϰ ρ 3 − v ρ 4 ) 1 − ] ∣ ≤ 4 r − 4 E ϰ 4 + 12 r − 4 v 2 ≤ 16 r − 4 E ϰ 4 , |\mathbb{E}[(\varkappa\rho_3-v\rho_4)1_-]|\le4r^{-4}\mathbb{E}\varkappa^4+12r^{-4}v^2\le16r^{-4}\mathbb{E}\varkappa^4, ∣ E [( ϰ ρ 3 − v ρ 4 ) 1 − ] ∣ ≤ 4 r − 4 E ϰ 4 + 12 r − 4 v 2 ≤ 16 r − 4 E ϰ 4 , where we used v 2 ≤ E ϰ 4 v^2\le\mathbb{E}\varkappa^4 v 2 ≤ E ϰ 4
( 3 r − 3 + 6 r − 3 + K 0 ) E [ ( 1 + ∣ ϰ ∣ ) 4 1 + ] ≤ t . (3r^{-3}+6r^{-3}+K_0)\mathbb{E}[(1+|\varkappa|)^4 1_+]\le t. ( 3 r − 3 + 6 r − 3 + K 0 ) E [( 1 + ∣ ϰ ∣ ) 4 1 + ] ≤ t . This gives (169) .
Finally, (166) with n = 2 n=2 n = 2 (167) give
∣ E Υ ′ ′ ( ϰ ) ∣ ≤ 2 r − 2 + K 0 E [ ( 1 + ∣ ϰ ∣ ) 2 1 + ] ≤ 2 r − 2 + t , |\mathbb{E}\Upsilon''(\varkappa)|\le2r^{-2}+K_0\mathbb{E}[(1+|\varkappa|)^2 1_+]\le2r^{-2}+t, ∣ E Υ ′′ ( ϰ ) ∣ ≤ 2 r − 2 + K 0 E [( 1 + ∣ ϰ ∣ ) 2 1 + ] ≤ 2 r − 2 + t , which is (170) .
Proposition 5.9. Let ε > 0 \varepsilon>0 ε > 0 α + 5 ε < 1 / 2 \alpha+5\varepsilon<1/2 α + 5 ε < 1/2 F F F Definition 5.4 , with p = max j ∈ J p j p=\max_{j\in\mathcal{J}}p_j p = max j ∈ J p j r N = c p N − α − 4 ε r_N=c_pN^{-\alpha-4\varepsilon} r N = c p N − α − 4 ε C > 0 C>0 C > 0 D > 0 D>0 D > 0 C D > 0 C_D>0 C D > 0 N ≥ 2 N\ge2 N ≥ 2 t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ]
∣ E F ( H t ) − E F ( H ) ∣ ≤ C t ( N 1 / 2 r N − 3 + r N − 2 ) + C D N − D . (172) |\mathbb{E}F(H_t)-\mathbb{E}F(H)|\le Ct(N^{1/2}r_N^{-3}+r_N^{-2})+C_DN^{-D}. \tag*{(172)} ∣ E F ( H t ) − E F ( H ) ∣ ≤ C t ( N 1/2 r N − 3 + r N − 2 ) + C D N − D . (172) If E X 3 = 0 \mathbb{E}X^3=0 E X 3 = 0 (172) holds with N 1 / 2 r N − 3 N^{1/2}r_N^{-3} N 1/2 r N − 3 r N − 4 r_N^{-4} r N − 4 C C C α \alpha α ε \varepsilon ε p p p X X X Z Z Z C D C_D C D D D D J \mathcal{J} J φ j \varphi_j φ j θ j \theta_j θ j
The proof has four steps. We first express E F ( H t ) − E F ( H ) \mathbb{E}F(H_t)-\mathbb{E}F(H) E F ( H t ) − E F ( H ) Lemma 5.7 . We then show that for every time t ′ ∈ ( 0 , 1 ] t'\in(0,1] t ′ ∈ ( 0 , 1 ] ζ ↦ F ( H t ′ [ a b ] + ζ V a b ) \zeta\mapsto F(H_{t'}^{[ab]}+\zeta V^{ab}) ζ ↦ F ( H t ′ [ ab ] + ζ V ab ) 4 r N 4r_N 4 r N [ − r N , r N ] [-r_N,r_N] [ − r N , r N ] Lemma 5.2 . Next, we Taylor expand in the coordinate x a b ( t ′ ) x_{ab}(t') x ab ( t ′ ) Lemma 5.8 . Finally, we sum over the coordinates and integrate in time.
Proof. We abbreviate R = R α , p \mathfrak{R}=\mathfrak{R}_{\alpha,p} R = R α , p N 0 ≥ 1 N_0\ge1 N 0 ≥ 1 N ≥ N 0 N\ge N_0 N ≥ N 0 N − 1 / 2 ≤ r N N^{-1/2}\le r_N N − 1/2 ≤ r N max ( 1 , σ Z 2 , 2 ) ≤ N \max(1,\sigma_Z^2,2)\le N max ( 1 , σ Z 2 , 2 ) ≤ N N ≥ N 0 N\ge N_0 N ≥ N 0 N 0 N_0 N 0 α + 4 ε < 1 / 2 \alpha+4\varepsilon<1/2 α + 4 ε < 1/2 N < N 0 N<N_0 N < N 0 2 ≤ 2 N 0 D N − D 2\le2N_0^D N^{-D} 2 ≤ 2 N 0 D N − D N ≥ N 0 N\ge N_0 N ≥ N 0 α + 5 ε < 1 / 2 \alpha+5\varepsilon<1/2 α + 5 ε < 1/2
N r N 2 = c p 2 N 1 − 2 α − 8 ε ≥ c p 2 N 2 ε , ℓ α h N 2 = N − 6 / 5 − α ≥ N − 2 . (173) N r_N^2=c_p^2N^{1-2\alpha-8\varepsilon}\ge c_p^2N^{2\varepsilon},\qquad\ell_\alpha h_N^2=N^{-6/5-\alpha}\ge N^{-2}. \tag*{(173)} N r N 2 = c p 2 N 1 − 2 α − 8 ε ≥ c p 2 N 2 ε , ℓ α h N 2 = N − 6/5 − α ≥ N − 2 . (173) Step 1. The derivative formula. By part (4) of Lemma 5.5, the function F F F F F F U α , p U_{\alpha,p} U α , p ( x a b ) a ≤ b ↦ F ( ∑ a ≤ b x a b V a b ) (x_{ab})_{a\le b}\mapsto F\left(\sum_{a\le b}x_{ab}V^{ab}\right) ( x ab ) a ≤ b ↦ F ( ∑ a ≤ b x ab V ab ) C N ( N + 1 ) / 2 \mathbb{C}^{N(N+1)/2} C N ( N + 1 ) /2 R N ( N + 1 ) / 2 \mathbb{R}^{N(N+1)/2} R N ( N + 1 ) /2 F F F A A A a ≤ b a\le b a ≤ b
∣ ∂ a b F ( A ) ∣ ≤ 2 ω p N 2 ( 1 + ∥ A ∥ ) , ∣ ∂ a b 2 F ( A ) ∣ ≤ 8 ω p 2 N 4 ( 1 + ∥ A ∥ ) 2 . (174) \left|\partial_{ab}F(A)\right|\le\frac{2}{\omega_p}N^2(1+\|A\|),\qquad\left|\partial_{ab}^2F(A)\right|\le\frac{8}{\omega_p^2}N^4(1+\|A\|)^2. \tag*{(174)} ∣ ∂ ab F ( A ) ∣ ≤ ω p 2 N 2 ( 1 + ∥ A ∥ ) , ∂ ab 2 F ( A ) ≤ ω p 2 8 N 4 ( 1 + ∥ A ∥ ) 2 . (174) Hence Lemma 5.7 applies with Q = F Q=F Q = F t ′ ∈ [ 0 , 1 ] t'\in[0,1] t ′ ∈ [ 0 , 1 ] a ≤ b a\le b a ≤ b
Q a b ( t ′ ) = E [ v a b G ∂ a b 2 F ( H t ′ ) − x a b ( t ′ ) ∂ a b F ( H t ′ ) ] , Q_{ab}(t')=\mathbb{E}\left[v_{ab}^G\partial_{ab}^2F(H_{t'})-x_{ab}(t')\partial_{ab}F(H_{t'})\right], Q ab ( t ′ ) = E [ v ab G ∂ ab 2 F ( H t ′ ) − x ab ( t ′ ) ∂ ab F ( H t ′ ) ] , and (5.36) reads
E F ( H t ) − E F ( H ) = 1 2 ∫ 0 t ∑ a ≤ b Q a b ( t ′ ) d t ′ . (175) \mathbb{E}F(H_t)-\mathbb{E}F(H)=\frac{1}{2}\int_0^t\sum_{a\le b}Q_{ab}(t')\,\mathrm{d}t'. \tag*{(175)} E F ( H t ) − E F ( H ) = 2 1 ∫ 0 t a ≤ b ∑ Q ab ( t ′ ) d t ′ . (175) Step 2. A holomorphic bound on a disk. Fix t ′ ∈ ( 0 , 1 ] t'\in(0,1] t ′ ∈ ( 0 , 1 ] a ≤ b a\le b a ≤ b E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ )
Spec ( H t ′ [ a b ] + ζ V a b ) ⊂ R for all ζ ∈ C with ∣ ζ ∣ ≤ 4 r N . (176) \operatorname{Spec}\left(H_{t'}^{[ab]}+\zeta V^{ab}\right)\subset\mathfrak{R}\quad\text{for all }\zeta\in\mathbb{C}\text{ with }|\zeta|\le4r_N. \tag*{(176)} Spec ( H t ′ [ ab ] + ζ V ab ) ⊂ R for all ζ ∈ C with ∣ ζ ∣ ≤ 4 r N . (176) The set of real symmetric matrices B B B Spec ( B + ζ V a b ) ⊂ R \operatorname{Spec}(B+\zeta V^{ab})\subset\mathfrak{R} Spec ( B + ζ V ab ) ⊂ R ∣ ζ ∣ ≤ 4 r N |\zeta|\le4r_N ∣ ζ ∣ ≤ 4 r N B n → B B_n\to B B n → B B n + ζ n V a b B_n+\zeta_nV^{ab} B n + ζ n V ab R \mathfrak{R} R ∣ ζ n ∣ ≤ 4 r N |\zeta_n|\le4r_N ∣ ζ n ∣ ≤ 4 r N ζ \zeta ζ ( ζ n ) (\zeta_n) ( ζ n ) B + ζ V a b B+\zeta V^{ab} B + ζ V ab R \mathfrak{R} R O = R O=\mathfrak{R} O = R E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ]
Next, we bound the probability of the complement of E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) H t ′ ∈ G ε H_{t'}\in G_\varepsilon H t ′ ∈ G ε ∣ x a b ( t ′ ) ∣ ≤ r N |x_{ab}(t')|\le r_N ∣ x ab ( t ′ ) ∣ ≤ r N ∣ ζ ∣ ≤ 4 r N |\zeta|\le4r_N ∣ ζ ∣ ≤ 4 r N
H t ′ [ a b ] + ζ V a b = H t ′ + ( ζ − x a b ( t ′ ) ) V a b , ∣ ζ − x a b ( t ′ ) ∣ ≤ 5 r N ≤ 8 c p N − α − 4 ε , H_{t'}^{[ab]}+\zeta V^{ab}=H_{t'}+(\zeta-x_{ab}(t'))V^{ab},\qquad|\zeta-x_{ab}(t')|\le5r_N\le8c_pN^{-\alpha-4\varepsilon}, H t ′ [ ab ] + ζ V ab = H t ′ + ( ζ − x ab ( t ′ )) V ab , ∣ ζ − x ab ( t ′ ) ∣ ≤ 5 r N ≤ 8 c p N − α − 4 ε , so Lemma 5.2 shows that Spec ( H t ′ [ a b ] + ζ V a b ) ⊂ R \operatorname{Spec}(H_{t'}^{[ab]}+\zeta V^{ab})\subset\mathfrak{R} Spec ( H t ′ [ ab ] + ζ V ab ) ⊂ R
{ H t ′ ∈ G ε } ∩ { ∣ x a b ( t ′ ) ∣ ≤ r N } ⊂ E a b ( t ′ ) . \{H_{t'}\in G_\varepsilon\}\cap\{|x_{ab}(t')|\le r_N\}\subset\mathcal{E}_{ab}(t'). { H t ′ ∈ G ε } ∩ { ∣ x ab ( t ′ ) ∣ ≤ r N } ⊂ E ab ( t ′ ) . Using Proposition 3.2 , (157) , and (173) , we obtain, for every D > 0 D > 0 D > 0
P ( E a b ( t ′ ) c ) ≤ P ( H t ′ ∉ G ε ) + P ( ∣ x a b ( t ′ ) ∣ > r N ) ≤ C D N − D , (177) \mathbb{P}\left(\mathcal{E}_{ab}(t')^{c}\right) \leq\mathbb{P}\left(H_{t'} \notin\mathcal{G}_{\varepsilon}\right) + \mathbb{P}\left(\left|x_{ab}(t')\right| > r_N\right) \leq C_D N^{-D}, \tag*{(177)} P ( E ab ( t ′ ) c ) ≤ P ( H t ′ ∈ / G ε ) + P ( ∣ x ab ( t ′ ) ∣ > r N ) ≤ C D N − D , (177) uniformly in t ′ ∈ ( 0 , 1 ] t' \in(0,1] t ′ ∈ ( 0 , 1 ] a ≤ b a \leq b a ≤ b
We now turn to the derivatives in the coordinate x a b ( t ′ ) x_{ab}(t') x ab ( t ′ )
Y ( x ) = F ( H t ′ [ a b ] + x V a b ) , x ∈ R , (178) \mathsf{Y}(x) = F\left(H_{t'}^{[ab]} + xV^{ab}\right), \qquad x \in\mathbb{R}, \tag*{(178)} Y ( x ) = F ( H t ′ [ ab ] + x V ab ) , x ∈ R , (178) which is determined by H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] Lemma 5.6 . Observe that (153) gives
∂ a b F ( H t ′ ) = Y ′ ( x a b ( t ′ ) ) , ∂ a b 2 F ( H t ′ ) = Y ′ ′ ( x a b ( t ′ ) ) . \partial_{ab}F(H_{t'}) = \mathsf{Y}'\left(x_{ab}(t')\right), \qquad\partial_{ab}^{2}F(H_{t'}) = \mathsf{Y}''\left(x_{ab}(t')\right). ∂ ab F ( H t ′ ) = Y ′ ( x ab ( t ′ ) ) , ∂ ab 2 F ( H t ′ ) = Y ′′ ( x ab ( t ′ ) ) . On the event E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) ζ ∈ C \zeta\in\mathbb{C} ζ ∈ C H t ′ [ a b ] + ζ V a b ∈ U α , p H_{t'}^{[ab]} + \zeta V^{ab} \in\mathcal{U}_{\alpha,p} H t ′ [ ab ] + ζ V ab ∈ U α , p { ∣ ζ ∣ ≤ 4 r N } \{|\zeta| \leq4r_N\} { ∣ ζ ∣ ≤ 4 r N } Lemma 5.5 shows that ζ ↦ F ( H t ′ [ a b ] + ζ V a b ) \zeta\mapsto F(H_{t'}^{[ab]} + \zeta V^{ab}) ζ ↦ F ( H t ′ [ ab ] + ζ V ab ) x x x ∣ x ∣ ≤ r N |x| \leq r_N ∣ x ∣ ≤ r N 3 r N 3r_N 3 r N x x x { ∣ ζ ∣ ≤ 4 r N } \{|\zeta| \leq4r_N\} { ∣ ζ ∣ ≤ 4 r N }
∣ Y ( n ) ( x ) ∣ ≤ n ! ( 3 r N ) − n ≤ n ! r N − n for ∣ x ∣ ≤ r N and n ≥ 1 , (179) |\mathsf{Y}^{(n)}(x)| \leq n!(3r_N)^{-n} \leq n!r_N^{-n} \qquad\text{for } |x| \leq r_N \text{ and } n \geq1, \tag*{(179)} ∣ Y ( n ) ( x ) ∣ ≤ n ! ( 3 r N ) − n ≤ n ! r N − n for ∣ x ∣ ≤ r N and n ≥ 1 , (179) on the event E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ )
Without restriction to E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) (174) gives
∣ Y ′ ( x ) ∣ + ∣ Y ′ ′ ( x ) ∣ ≤ 10 ω p 2 N 4 ( 1 + ∥ H t ′ [ a b ] ∥ ) 2 ( 1 + ∣ x ∣ ) 2 , x ∈ R , (180) |\mathsf{Y}'(x)| + |\mathsf{Y}''(x)| \leq\frac{10}{\omega_p^2}N^4\left(1+\|H_{t'}^{[ab]}\|\right)^2(1+|x|)^2, \qquad x \in\mathbb{R}, \tag*{(180)} ∣ Y ′ ( x ) ∣ + ∣ Y ′′ ( x ) ∣ ≤ ω p 2 10 N 4 ( 1 + ∥ H t ′ [ ab ] ∥ ) 2 ( 1 + ∣ x ∣ ) 2 , x ∈ R , (180) where we used ω p ≤ 1 \omega_p \leq1 ω p ≤ 1
1 + ∥ H t ′ [ a b ] + x V a b ∥ ≤ ( 1 + ∥ H t ′ [ a b ] ∥ ) ( 1 + ∣ x ∣ ) . 1+\|H_{t'}^{[ab]}+xV^{ab}\| \leq\left(1+\|H_{t'}^{[ab]}\|\right)(1+|x|). 1 + ∥ H t ′ [ ab ] + x V ab ∥ ≤ ( 1 + ∥ H t ′ [ ab ] ∥ ) ( 1 + ∣ x ∣ ) . Step 3. Taylor expansion in one coordinate. We keep t ′ t' t ′ a ≤ b a \leq b a ≤ b ϰ = x a b ( t ′ ) \varkappa=x_{ab}(t') ϰ = x ab ( t ′ ) E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ]
Q a b ( t ′ ) = E [ 1 E a b ( t ′ ) E [ v a b G Y ′ ′ ( ϰ ) − ϰ Y ′ ( ϰ ) ∣ H t ′ [ a b ] ] ] + E [ 1 E a b ( t ′ ) c ( v a b G Y ′ ′ ( ϰ ) − ϰ Y ′ ( ϰ ) ) ] . (181) \begin{aligned}
Q_{ab}(t')={}&\mathbb{E}\left[\mathbf{1}_{\mathcal{E}_{ab}(t')}\mathbb{E}\left[\mathsf{v}_{ab}^{G}\mathsf{Y}''(\varkappa)-\varkappa\mathsf{Y}'(\varkappa)\mid H_{t'}^{[ab]}\right]\right] \\
&+\mathbb{E}\left[\mathbf{1}_{\mathcal{E}_{ab}(t')^{c}}\left(\mathsf{v}_{ab}^{G}\mathsf{Y}''(\varkappa)-\varkappa\mathsf{Y}'(\varkappa)\right)\right]. \tag*{(181)}
\end{aligned} Q ab ( t ′ ) = E [ 1 E ab ( t ′ ) E [ v ab G Y ′′ ( ϰ ) − ϰ Y ′ ( ϰ ) ∣ H t ′ [ ab ] ] ] + E [ 1 E ab ( t ′ ) c ( v ab G Y ′′ ( ϰ ) − ϰ Y ′ ( ϰ ) ) ] . (181) We begin by bounding the second term. Using (174) , we have
∣ v a b G Y ′ ′ ( ϰ ) − ϰ Y ′ ( ϰ ) ∣ ≤ C N 4 ( 1 + ∥ H t ′ ∥ ) 2 ( N − 1 + ∣ ϰ ∣ ) . \left|\mathsf{v}_{ab}^{G}\mathsf{Y}''(\varkappa)-\varkappa\mathsf{Y}'(\varkappa)\right| \leq C N^4\left(1+\|H_{t'}\|\right)^2\left(N^{-1}+|\varkappa|\right). v ab G Y ′′ ( ϰ ) − ϰ Y ′ ( ϰ ) ≤ C N 4 ( 1 + ∥ H t ′ ∥ ) 2 ( N − 1 + ∣ ϰ ∣ ) . Combining this bound with the Cauchy–Schwarz inequality, Proposition 3.2 , (158) , and (177) , we find that the second term in (181) is bounded in absolute value by
C N 4 ( E [ ( 1 + ∥ H t ′ ∥ ) 4 ( N − 1 + ∣ ϰ ∣ ) 2 ] ) 1 / 2 P ( E a b ( t ′ ) c ) 1 / 2 ≤ C D N − D . C N^4\left(\mathbb{E}\left[(1+\|H_{t'}\|)^4(N^{-1}+|\varkappa|)^2\right]\right)^{1/2}\mathbb{P}\left(\mathcal{E}_{ab}(t')^{c}\right)^{1/2} \leq C_DN^{-D}. C N 4 ( E [ ( 1 + ∥ H t ′ ∥ ) 4 ( N − 1 + ∣ ϰ ∣ ) 2 ] ) 1/2 P ( E ab ( t ′ ) c ) 1/2 ≤ C D N − D . To bound the first term in (5.56), we apply Lemma 5.8 conditionally on H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] Υ \Upsilon Υ ϰ \varkappa ϰ H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] E ϰ = 0 \mathbb{E}\varkappa= 0 E ϰ = 0 N 0 N_0 N 0
E ϰ 2 ≤ max ( 1 , σ Z 2 , 2 ) N − 1 ≤ 1. \mathbb{E}\varkappa^2 \le\max(1,\sigma_Z^2,2)N^{-1} \le1. E ϰ 2 ≤ max ( 1 , σ Z 2 , 2 ) N − 1 ≤ 1. On the event E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) Υ \Upsilon Υ r = r N ≤ 1 r = r_N \le1 r = r N ≤ 1
K 0 = 10 ω p 2 N 4 ( 1 + ∥ H t ′ [ a b ] ∥ ) 2 K_0 = \frac{10}{\omega_p^2}N^4\left(1+\left\lVert H_{t'}^{[ab]}\right\rVert\right)^2 K 0 = ω p 2 10 N 4 ( 1 + H t ′ [ ab ] ) 2 by (5.55). By the Cauchy–Schwarz inequality, (5.32), (5.33), and (5.48), we have
E [ ( 1 + ∣ ϰ ∣ ) 4 1 ∣ ϰ ∣ > r N ] ≤ ( E ( 1 + ∣ ϰ ∣ ) 8 ) 1 / 2 P ( ∣ ϰ ∣ > r N ) 1 / 2 ≤ C exp ( − c N 2 ε ) . \mathbb{E}\left[(1+|\varkappa|)^4\mathbf{1}_{|\varkappa|>r_N}\right]
\le\left(\mathbb{E}(1+|\varkappa|)^8\right)^{1/2}\mathbb{P}(|\varkappa|>r_N)^{1/2}
\le C\exp\left(-cN^{2\varepsilon}\right). E [ ( 1 + ∣ ϰ ∣ ) 4 1 ∣ ϰ ∣ > r N ] ≤ ( E ( 1 + ∣ ϰ ∣ ) 8 ) 1/2 P ( ∣ ϰ ∣ > r N ) 1/2 ≤ C exp ( − c N 2 ε ) . for some c > 0 c>0 c > 0 c 0 c_0 c 0 p p p r N − 3 ≤ N 3 / 2 r_N^{-3}\le N^{3/2} r N − 3 ≤ N 3/2
τ = ( 9 r N − 3 + 20 ω p 2 N 4 ( 1 + ∥ H t ′ [ a b ] ∥ ) 2 ) E [ ( 1 + ∣ ϰ ∣ ) 4 1 ∣ ϰ ∣ > r N ] ≤ C D N − D ( 1 + ∥ H t ′ [ a b ] ∥ ) 2 . (182) \begin{aligned}
\tau
&=\left(9r_N^{-3}+\frac{20}{\omega_p^2}N^4\left(1+\left\lVert H_{t'}^{[ab]}\right\rVert\right)^2\right)
\mathbb{E}\left[(1+|\varkappa|)^4\mathbf{1}_{|\varkappa|>r_N}\right] \\
&\le C_DN^{-D}\left(1+\left\lVert H_{t'}^{[ab]}\right\rVert\right)^2. \tag*{(182)}
\end{aligned} τ = ( 9 r N − 3 + ω p 2 20 N 4 ( 1 + H t ′ [ ab ] ) 2 ) E [ ( 1 + ∣ ϰ ∣ ) 4 1 ∣ ϰ ∣ > r N ] ≤ C D N − D ( 1 + H t ′ [ ab ] ) 2 . (182) We now decompose the integrand of the first term in (5.56) as
v a b G Υ ′ ′ ( ϰ ) − ϰ Υ ′ ( ϰ ) = ( v a b G − E ϰ 2 ) Υ ′ ′ ( ϰ ) + ( E [ ϰ 2 ] Υ ′ ′ ( ϰ ) − ϰ Υ ′ ( ϰ ) ) . v_{ab}^{G}\Upsilon''(\varkappa)-\varkappa\Upsilon'(\varkappa)
=\left(v_{ab}^{G}-\mathbb{E}\varkappa^2\right)\Upsilon''(\varkappa)
+\left(\mathbb{E}[\varkappa^2]\Upsilon''(\varkappa)-\varkappa\Upsilon'(\varkappa)\right). v ab G Υ ′′ ( ϰ ) − ϰ Υ ′ ( ϰ ) = ( v ab G − E ϰ 2 ) Υ ′′ ( ϰ ) + ( E [ ϰ 2 ] Υ ′′ ( ϰ ) − ϰ Υ ′ ( ϰ ) ) . We bound the conditional expectations of the two terms on the right using (5.45) and (5.43), respectively. Then we take expectations, using (5.33) and the bound
E ( 1 + ∥ H t ′ [ a b ] ∥ ) 2 ≤ C . \mathbb{E}\left(1+\left\lVert H_{t'}^{[ab]}\right\rVert\right)^2\le C. E ( 1 + H t ′ [ ab ] ) 2 ≤ C . which follows from Proposition 3.2 and ∥ H t ′ [ a b ] ∥ ≤ ∥ H t ′ ∥ + ∣ ϰ ∣ \left\lVert H_{t'}^{[ab]}\right\rVert\le\left\lVert H_{t'}\right\rVert+|\varkappa| H t ′ [ ab ] ≤ ∥ H t ′ ∥ + ∣ ϰ ∣
2 ∣ v a b G − E ϰ 2 ∣ r N − 2 + 9 r N − 3 E ∣ ϰ ∣ 3 + C D N − D ≤ 2 ∣ v a b G − E ϰ 2 ∣ r N − 2 + C N − 3 / 2 r N − 3 + C D N − D . 2|v_{ab}^{G}-\mathbb{E}\varkappa^2|r_N^{-2}+9r_N^{-3}\mathbb{E}|\varkappa|^3+C_DN^{-D}
\le2|v_{ab}^{G}-\mathbb{E}\varkappa^2|r_N^{-2}+CN^{-3/2}r_N^{-3}+C_DN^{-D}. 2∣ v ab G − E ϰ 2 ∣ r N − 2 + 9 r N − 3 E ∣ ϰ ∣ 3 + C D N − D ≤ 2∣ v ab G − E ϰ 2 ∣ r N − 2 + C N − 3/2 r N − 3 + C D N − D . When E X 3 = 0 \mathbb{E}X^3=0 E X 3 = 0 a < b a<b a < b E ϰ 3 = 0 \mathbb{E}\varkappa^3=0 E ϰ 3 = 0 C N − 3 / 2 r N − 3 CN^{-3/2}r_N^{-3} C N − 3/2 r N − 3
16 r N − 4 E ϰ 4 ≤ C N − 2 r N − 4 . 16r_N^{-4}\mathbb{E}\varkappa^4\le CN^{-2}r_N^{-4}. 16 r N − 4 E ϰ 4 ≤ C N − 2 r N − 4 . Recall from (5.34) that
v a b G = E x a b ( t ′ ) 2 ( a < b ) , ∣ v a a G − E x a a ( t ′ ) 2 ∣ ≤ ∣ σ Z 2 − 2 ∣ N − 1 . v_{ab}^{G}=\mathbb{E}x_{ab}(t')^2\quad(a<b),\qquad
\left|v_{aa}^{G}-\mathbb{E}x_{aa}(t')^2\right|\le|\sigma_Z^2-2|N^{-1}. v ab G = E x ab ( t ′ ) 2 ( a < b ) , v aa G − E x aa ( t ′ ) 2 ≤ ∣ σ Z 2 − 2∣ N − 1 . Combining these facts with the bounds on the two terms in (181) , we obtain
∣ Q a b ( t ′ ) ∣ ≤ C N − 3 / 2 r N − 3 + C D N − D , a < b , ∣ Q a a ( t ′ ) ∣ ≤ C N − 1 r N − 2 + C N − 3 / 2 r N − 3 + C D N − D , (183) \begin{aligned}
\lvert Q_{ab}(t')\rvert&\le CN^{-3/2}r_N^{-3}+C_DN^{-D}, \qquad a<b,\\
\lvert Q_{aa}(t')\rvert&\le CN^{-1}r_N^{-2}+CN^{-3/2}r_N^{-3}+C_DN^{-D}, \tag*{(183)}
\end{aligned} ∣ Q ab ( t ′ )∣ ∣ Q aa ( t ′ )∣ ≤ C N − 3/2 r N − 3 + C D N − D , a < b , ≤ C N − 1 r N − 2 + C N − 3/2 r N − 3 + C D N − D , (183) uniformly in t ′ ∈ ( 0 , 1 ] t' \in(0,1] t ′ ∈ ( 0 , 1 ] E X 3 = 0 \mathbb{E}X^3=0 E X 3 = 0 a < b a<b a < b C N − 3 / 2 r N − 3 CN^{-3/2}r_N^{-3} C N − 3/2 r N − 3 C N − 2 r N − 4 CN^{-2}r_N^{-4} C N − 2 r N − 4
Step 4. Summation of the generator. There are at most N 2 N^2 N 2 a < b a<b a < b N N N N 0 N_0 N 0 N − 1 / 2 r N − 3 ≤ r N − 2 N^{-1/2}r_N^{-3}\le r_N^{-2} N − 1/2 r N − 3 ≤ r N − 2 (183) implies that for every t ′ ∈ ( 0 , 1 ] t' \in(0,1] t ′ ∈ ( 0 , 1 ]
∑ a ≤ b ∣ Q a b ( t ′ ) ∣ ≤ C N 1 / 2 r N − 3 + C r N − 2 + C N − 1 / 2 r N − 3 + C D N 2 − D ≤ C ( N 1 / 2 r N − 3 + r N − 2 ) + C D N 2 − D . \begin{aligned}
\sum_{a\le b}\lvert Q_{ab}(t')\rvert
&\le CN^{1/2}r_N^{-3}+Cr_N^{-2}+CN^{-1/2}r_N^{-3}+C_DN^{2-D}\\
&\le C\left(N^{1/2}r_N^{-3}+r_N^{-2}\right)+C_DN^{2-D}.
\end{aligned} a ≤ b ∑ ∣ Q ab ( t ′ )∣ ≤ C N 1/2 r N − 3 + C r N − 2 + C N − 1/2 r N − 3 + C D N 2 − D ≤ C ( N 1/2 r N − 3 + r N − 2 ) + C D N 2 − D . Inserting this bound into (175) and replacing D D D D + 2 D+2 D + 2 E X 3 = 0 \mathbb{E}X^3=0 E X 3 = 0 C r N − 4 Cr_N^{-4} C r N − 4 C N 1 / 2 r N − 3 CN^{1/2}r_N^{-3} C N 1/2 r N − 3 (172) with N 1 / 2 r N − 3 N^{1/2}r_N^{-3} N 1/2 r N − 3 r N − 4 r_N^{-4} r N − 4 α \alpha α ε \varepsilon ε p p p D D D X X X Z Z Z Lemma 5.2 , Proposition 3.2 , (157) , and (158) . This completes the proof.
Remark 5.10. The proof of Proposition 5.9 uses the specific form of F F F F F F
∣ ∂ a b F ( A ) ∣ + ∣ ∂ a b 2 F ( A ) ∣ ≤ C N C ( 1 + ∥ A ∥ ) 2 \lvert\partial_{ab}F(A)\rvert+\lvert\partial_{ab}^{2}F(A)\rvert\le CN^{C}(1+\lVert A\rVert)^2 ∣ ∂ ab F ( A )∣ + ∣ ∂ ab 2 F ( A )∣ ≤ C N C ( 1 + ∥ A ∥ ) 2 for some C > 0 C>0 C > 0 A A A a ≤ b a\le b a ≤ b t ′ ∈ [ 0 , 1 ] t'\in[0,1] t ′ ∈ [ 0 , 1 ] a ≤ b a\le b a ≤ b H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] C D N − D C_DN^{-D} C D N − D D > 0 D>0 D > 0 t ′ t' t ′ ( a , b ) (a,b) ( a , b ) ζ ↦ F ( H t ′ [ a b ] + ζ V a b ) \zeta\mapsto F(H_{t'}^{[ab]}+\zeta V^{ab}) ζ ↦ F ( H t ′ [ ab ] + ζ V ab ) { ∣ ζ ∣ ≤ 4 r N } \{\lvert\zeta\rvert\le4r_N\} {∣ ζ ∣ ≤ 4 r N } (172) and its variant for E X 3 = 0 \mathbb{E}X^3=0 E X 3 = 0
Pair statistics at the scales N − 1 − α N^{-1-\alpha} N − 1 − α
In this section, we prove Theorem 1.3 and Proposition 4.1 . These results follow from a law of large numbers for the pair statistics of H H H ℓ α = N − 1 − α \ell_\alpha=N^{-1-\alpha} ℓ α = N − 1 − α α \alpha α ℓ α \ell_\alpha ℓ α M α = N 1 − 2 α = N 3 ℓ α 2 M_\alpha=N^{1-2\alpha}=N^3\ell_\alpha^2 M α = N 1 − 2 α = N 3 ℓ α 2 f p f_p f p X α ( A ; φ , f ) \mathcal{X}_\alpha(A;\varphi,f) X α ( A ; φ , f ) φ N \varphi_N φ N φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) h N = N − 1 / 10 h_N=N^{-1/10} h N = N − 1/10 p ∗ = 61 p_*=61 p ∗ = 61
We use the same notation for ordered sequences. For x = ( x 1 , … , x N ) ∈ R N x=(x_1,\ldots,x_N)\in\mathbb{R}^N x = ( x 1 , … , x N ) ∈ R N x 1 ≤ ⋯ ≤ x N x_1\leq\cdots\leq x_N x 1 ≤ ⋯ ≤ x N X α ( x ; φ , f ) \mathcal{X}_\alpha(x;\varphi,f) X α ( x ; φ , f ) λ i ( A ) \lambda_i(A) λ i ( A ) x i x_i x i i ∈ [ N ] i\in[N] i ∈ [ N ] p ≥ 2 p\geq2 p ≥ 2
∫ 0 ∞ y f p ( y ) d y = 1 2 ( p − 1 ) . (184) \int_0^\infty y f_p(y)\,\mathrm{d}y=\frac{1}{2(p-1)}. \tag*{(184)} ∫ 0 ∞ y f p ( y ) d y = 2 ( p − 1 ) 1 . (184) Definition 6.1. Let α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) P ( α ) \mathcal{P}(\alpha) P ( α ) φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) p ≥ p ∗ p\geq p_* p ≥ p ∗
1 M α X α ( H ; φ N , f p ) ⟶ 1 2 ( p − 1 ) ∫ − 2 2 φ ( E ) c ( E ) d E \frac{1}{M_\alpha}\mathcal{X}_\alpha(H;\varphi_N,f_p)\longrightarrow\frac{1}{2(p-1)}\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E M α 1 X α ( H ; φ N , f p ) ⟶ 2 ( p − 1 ) 1 ∫ − 2 2 φ ( E ) c ( E ) d E in probability as $N\to\infty.
The argument is outlined in § 3.5 . We prove P ( α ) \mathcal{P}(\alpha) P ( α ) α ∈ ( 0 , 1 / 8 ) \alpha\in(0,1/8) α ∈ ( 0 , 1/8 ) § 8.1 , extend P ( α ) \mathcal{P}(\alpha) P ( α ) § 8.2 , show in § 8.3 that P ( α ) \mathcal{P}(\alpha) P ( α ) P ( α ′ ) \mathcal{P}(\alpha') P ( α ′ ) α ′ \alpha' α ′ α \alpha α § 8.4 to deduce Theorem 1.3 and Proposition 4.1 .
We will use the following two lemmas, the first in § 8.1 and § 8.2 , and the second in § 8.1 and § 8.3 . The first compares pair statistics with smoothed and unsmoothed energy weights.
Lemma 6.2. Let α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) δ > 0 \delta>0 δ > 0 E ∈ R E\in\mathbb{R} E ∈ R dist ( E , supp φ ) ≤ δ \operatorname{dist}(E,\operatorname{supp}\varphi)\leq\delta dist ( E , supp φ ) ≤ δ ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) χ ∈ C c ∞ ( ( − 2 , 2 ) ) \chi\in C^\infty_c((-2,2)) χ ∈ C c ∞ (( − 2 , 2 )) χ ( E ) = 2 \chi(E)=2 χ ( E ) = 2 E ∈ R E\in\mathbb{R} E ∈ R dist ( E , supp φ ) ≤ δ \operatorname{dist}(E,\operatorname{supp}\varphi)\leq\delta dist ( E , supp φ ) ≤ δ
and N 0 ≥ 1 N_0 \ge1 N 0 ≥ 1 N ≥ N 0 N \ge N_0 N ≥ N 0 N × N N \times N N × N A A A ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3 f : [ 0 , ∞ ) → R f : [0,\infty) \to\mathbb{R} f : [ 0 , ∞ ) → R ∣ f ∣ ≤ f p ∗ \lvert f\rvert\le f_{p_*} ∣ f ∣ ≤ f p ∗
∣ X α ( A ; φ N , f ) − X α ( A ; φ , f ) ∣ ≤ C h N min { X α ( A ; χ , f p ∗ ) , X α ( A ; χ N , f p ∗ ) } + C N 2 exp ( − δ 2 4 h N 2 ) . (185) \left\lvert\mathcal{X}_{\alpha}(A;\varphi_N,f)-\mathcal{X}_{\alpha}(A;\varphi,f)\right\rvert
\le Ch_N\min\left\{\mathcal{X}_{\alpha}(A;\chi,f_{p_*}),\mathcal{X}_{\alpha}(A;\chi_N,f_{p_*})\right\}+CN^2\exp\left(-\frac{\delta^2}{4h_N^2}\right). \tag*{(185)} ∣ X α ( A ; φ N , f ) − X α ( A ; φ , f ) ∣ ≤ C h N min { X α ( A ; χ , f p ∗ ) , X α ( A ; χ N , f p ∗ ) } + C N 2 exp ( − 4 h N 2 δ 2 ) . (185) Proof. We begin by recalling two consequences of parts (4) and (5) of Lemma 5.1 . There exists C > 0 C>0 C > 0
∥ φ N − φ ∥ ∞ + ∥ χ N − χ ∥ ∞ ≤ C h N , sup E ∈ [ − 3 , 3 ] : dist ( E , supp φ ) ≥ δ φ N ( E ) ≤ C exp ( − δ 2 4 h N 2 ) . (186) \lVert\varphi_N-\varphi\rVert_{\infty}+\lVert\chi_N-\chi\rVert_{\infty}\le Ch_N,\qquad\sup_{\substack{E\in[-3,3]:\\ \operatorname{dist}(E,\operatorname{supp}\varphi)\ge\delta}}\varphi_N(E)\le C\exp\left(-\frac{\delta^2}{4h_N^2}\right). \tag*{(186)} ∥ φ N − φ ∥ ∞ + ∥ χ N − χ ∥ ∞ ≤ C h N , E ∈ [ − 3 , 3 ] : dist ( E , supp φ ) ≥ δ sup φ N ( E ) ≤ C exp ( − 4 h N 2 δ 2 ) . (186) Let N 0 N_0 N 0 C h N ≤ 1 Ch_N\le1 C h N ≤ 1 N ≥ N 0 N\ge N_0 N ≥ N 0 χ N ( E ) ≥ 1 \chi_N(E)\ge1 χ N ( E ) ≥ 1 E E E dist ( E , supp φ ) ≤ δ \operatorname{dist}(E,\operatorname{supp}\varphi)\le\delta dist ( E , supp φ ) ≤ δ
Fix A A A f f f i < j i<j i < j E = ( λ i ( A ) + λ j ( A ) ) / 2 E=(\lambda_i(A)+\lambda_j(A))/2 E = ( λ i ( A ) + λ j ( A )) /2 [ − 3 , 3 ] [-3,3] [ − 3 , 3 ] ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3
∣ φ N ( E ) − φ ( E ) ∣ ≤ C h N min { χ ( E ) , χ N ( E ) } + C exp ( − δ 2 4 h N 2 ) . (187) \lvert\varphi_N(E)-\varphi(E)\rvert\le Ch_N\min\{\chi(E),\chi_N(E)\}+C\exp\left(-\frac{\delta^2}{4h_N^2}\right). \tag*{(187)} ∣ φ N ( E ) − φ ( E )∣ ≤ C h N min { χ ( E ) , χ N ( E )} + C exp ( − 4 h N 2 δ 2 ) . (187) When dist ( E , supp φ ) ≤ δ \operatorname{dist}(E,\operatorname{supp}\varphi)\le\delta dist ( E , supp φ ) ≤ δ (186) and the fact that χ ( E ) \chi(E) χ ( E ) χ N ( E ) \chi_N(E) χ N ( E ) φ ( E ) = 0 \varphi(E)=0 φ ( E ) = 0 (187) follows from the second bound in (186) .
Next, we multiply (187) by ∣ f ( ( λ j ( A ) − λ i ( A ) ) / ℓ α ) ∣ \left\lvert f((\lambda_j(A)-\lambda_i(A))/\ell_\alpha)\right\rvert ∣ f (( λ j ( A ) − λ i ( A )) / ℓ α ) ∣ f p ∗ ( ( λ j ( A ) − λ i ( A ) ) / ℓ α ) ≤ 1 f_{p_*}((\lambda_j(A)-\lambda_i(A))/\ell_\alpha)\le1 f p ∗ (( λ j ( A ) − λ i ( A )) / ℓ α ) ≤ 1 N 2 N^2 N 2 i < j i<j i < j (185) . □
The second lemma transfers a law of large numbers for the pair statistic from H t H_t H t H H H
Lemma 6.3. Let α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) p ≥ 1 p\ge1 p ≥ 1 φ ∈ C ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty((-2,2)) φ ∈ C ∞ (( − 2 , 2 )) L ≥ 0 L\ge0 L ≥ 0 ε > 0 \varepsilon>0 ε > 0 α + 5 ε < 1 / 2 \alpha+5\varepsilon<1/2 α + 5 ε < 1/2 ( t N ) N ≥ 1 (t_N)_{N\ge1} ( t N ) N ≥ 1 [ 0 , 1 ] [0,1] [ 0 , 1 ]
lim N → ∞ t N N 1 / 2 + 3 α + 12 ε = 0. \lim_{N\to\infty}t_NN^{1/2+3\alpha+12\varepsilon}=0. N → ∞ lim t N N 1/2 + 3 α + 12 ε = 0. If M α − 1 X α ( H t N ; φ N , f p ) → L M_\alpha^{-1}\mathcal{X}_{\alpha}(H_{t_N};\varphi_N,f_p)\to L M α − 1 X α ( H t N ; φ N , f p ) → L M α − 1 X α ( H ; φ N , f p ) → L M_\alpha^{-1}\mathcal{X}_{\alpha}(H;\varphi_N,f_p)\to L M α − 1 X α ( H ; φ N , f p ) → L
Proof. For every real symmetric matrix A A A X α ( A ; φ N , f p ) ≥ 0 \mathcal{X}_{\alpha}(A;\varphi_N,f_p)\ge0 X α ( A ; φ N , f p ) ≥ 0 φ N ≥ 0 \varphi_N\ge0 φ N ≥ 0 R \mathbb{R} R f p > 0 f_p>0 f p > 0 θ > 0 \theta>0 θ > 0 exp ( − θ M α − 1 X α ( H t N ; φ N , f p ) ) \exp(-\theta M_\alpha^{-1}\mathcal{X}_{\alpha}(H_{t_N};\varphi_N,f_p)) exp ( − θ M α − 1 X α ( H t N ; φ N , f p )) ( 0 , 1 ] (0,1] ( 0 , 1 ] e − θ L e^{-\theta L} e − θ L
lim N → ∞ E exp ( − θ M α − 1 X α ( H t N ; φ N , f p ) ) = e − θ L . \lim_{N\to\infty}\mathbb{E}\exp\left(-\theta M_\alpha^{-1}\mathcal{X}_{\alpha}(H_{t_N};\varphi_N,f_p)\right)=e^{-\theta L}. N → ∞ lim E exp ( − θ M α − 1 X α ( H t N ; φ N , f p ) ) = e − θ L . Let r N = c p N − α − 4 ε r_N=c_pN^{-\alpha-4\varepsilon} r N = c p N − α − 4 ε
t N ( N 1 / 2 r N − 3 + r N − 2 ) = c p − 3 t N N 1 / 2 + 3 α + 12 ε + c p − 2 t N N 2 α + 8 ε ≤ ( c p − 3 + c p − 2 ) t N N 1 / 2 + 3 α + 12 ε , t_N\left(N^{1/2}r_N^{-3}+r_N^{-2}\right)=c_p^{-3}t_NN^{1/2+3\alpha+12\varepsilon}+c_p^{-2}t_NN^{2\alpha+8\varepsilon}\leq\left(c_p^{-3}+c_p^{-2}\right)t_NN^{1/2+3\alpha+12\varepsilon}, t N ( N 1/2 r N − 3 + r N − 2 ) = c p − 3 t N N 1/2 + 3 α + 12 ε + c p − 2 t N N 2 α + 8 ε ≤ ( c p − 3 + c p − 2 ) t N N 1/2 + 3 α + 12 ε , which tends to zero. We apply Proposition 5.9 with the index set { 1 } \{1\} { 1 } φ 1 = φ \varphi_1=\varphi φ 1 = φ p 1 = p p_1=p p 1 = p θ 1 = θ / M α \theta_1=\theta/M_\alpha θ 1 = θ / M α D = 1 D=1 D = 1 t = t N t=t_N t = t N F F F
F ( A ) = exp ( − θ M α − 1 X α ( A ; φ N , f p ) ) F(A)=\exp\left(-\theta M_\alpha^{-1}\mathcal{X}_\alpha(A;\varphi_N,f_p)\right) F ( A ) = exp ( − θ M α − 1 X α ( A ; φ N , f p ) ) for every real symmetric matrix A A A
lim N → ∞ E exp ( − θ M α − 1 X α ( H ; φ N , f p ) ) = e − θ L (188) \lim_{N\to\infty}\mathbb{E}\exp\left(-\theta M_\alpha^{-1}\mathcal{X}_\alpha(H;\varphi_N,f_p)\right)=e^{-\theta L} \tag*{(188)} N → ∞ lim E exp ( − θ M α − 1 X α ( H ; φ N , f p ) ) = e − θ L (188) for all θ > 0 \theta>0 θ > 0
We now apply (188) with θ = 1 \theta=1 θ = 1 θ = 2 \theta=2 θ = 2
E [ ( e − M α − 1 X α ( H ; φ N , f p ) − e − L ) 2 ] = E e − 2 M α − 1 X α ( H ; φ N , f p ) − 2 e − L E e − M α − 1 X α ( H ; φ N , f p ) + e − 2 L → 0. \mathbb{E}\left[\left(e^{-M_\alpha^{-1}\mathcal{X}_\alpha(H;\varphi_N,f_p)}-e^{-L}\right)^2\right]=\mathbb{E}e^{-2M_\alpha^{-1}\mathcal{X}_\alpha(H;\varphi_N,f_p)}-2e^{-L}\mathbb{E}e^{-M_\alpha^{-1}\mathcal{X}_\alpha(H;\varphi_N,f_p)}+e^{-2L}\to0. E [ ( e − M α − 1 X α ( H ; φ N , f p ) − e − L ) 2 ] = E e − 2 M α − 1 X α ( H ; φ N , f p ) − 2 e − L E e − M α − 1 X α ( H ; φ N , f p ) + e − 2 L → 0. Hence exp ( − M α − 1 X α ( H ; φ N , f p ) ) → e − L \exp(-M_\alpha^{-1}\mathcal{X}_\alpha(H;\varphi_N,f_p))\to e^{-L} exp ( − M α − 1 X α ( H ; φ N , f p )) → e − L ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) e − L > 0 e^{-L}>0 e − L > 0 M α − 1 X α ( H ; φ N , f p ) → L M_\alpha^{-1}\mathcal{X}_\alpha(H;\varphi_N,f_p)\to L M α − 1 X α ( H ; φ N , f p ) → L
The base case
We begin with a deterministic lemma. It shows that the pair statistic changes by a small relative error when the bulk gaps are perturbed by much less than ℓ α \ell_\alpha ℓ α γ i \gamma_i γ i
Lemma 6.4. Let α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) ε ∈ ( 0 , 1 / 10 ] \varepsilon\in(0,1/10] ε ∈ ( 0 , 1/10 ] p ≥ 1 p\geq1 p ≥ 1 φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) supp φ ⊂ [ − 2 + 3 δ , 2 − 3 δ ] \operatorname{supp}\varphi\subset[-2+3\delta,2-3\delta] supp φ ⊂ [ − 2 + 3 δ , 2 − 3 δ ]
c ∗ = ∫ − 2 − 2 + δ / 4 ρ s c ( E ) d E , J = { i ∈ [ N ] : c ∗ N ≤ i ≤ ( 1 − c ∗ ) N } . c_*=\int_{-2}^{-2+\delta/4}\rho_{\mathrm{sc}}(E)\,\mathrm{d}E,\qquad\mathcal{J}=\{i\in[N]:c_*N\leq i\leq(1-c_*)N\}. c ∗ = ∫ − 2 − 2 + δ /4 ρ sc ( E ) d E , J = { i ∈ [ N ] : c ∗ N ≤ i ≤ ( 1 − c ∗ ) N } . Then for every ϖ > 0 \varpi>0 ϖ > 0 N 0 N_0 N 0 N ≥ N 0 N\geq N_0 N ≥ N 0 x , x ′ ∈ [ − 3 , 3 ] N x,x'\in[-3,3]^N x , x ′ ∈ [ − 3 , 3 ] N
(a) # { i ∈ [ N ] : x i ∈ [ E , E + N − 1 ] } ≤ N ε \#\{i\in[N]:x_i\in[E,E+N^{-1}]\}\leq N^\varepsilon # { i ∈ [ N ] : x i ∈ [ E , E + N − 1 ]} ≤ N ε E ∈ R E\in\mathbb{R} E ∈ R
(b) ∣ x i − γ i ∣ ≤ N − 1 + ε |x_i-\gamma_i|\leq N^{-1+\varepsilon} ∣ x i − γ i ∣ ≤ N − 1 + ε ∣ x i ′ − γ i ∣ ≤ N − 1 + ε |x'_i-\gamma_i|\leq N^{-1+\varepsilon} ∣ x i ′ − γ i ∣ ≤ N − 1 + ε i ∈ J i\in\mathcal{J} i ∈ J
(c) ∣ ( x i + 1 − x i ) − ( x i + 1 ′ − x i ′ ) ∣ ≤ N − 2 ε ℓ α |(x_{i+1}-x_i)-(x'_{i+1}-x'_i)|\leq N^{-2\varepsilon}\ell_\alpha ∣ ( x i + 1 − x i ) − ( x i + 1 ′ − x i ′ ) ∣ ≤ N − 2 ε ℓ α i i i i , i + 1 ∈ J i,i+1\in\mathcal{J} i , i + 1 ∈ J
Then
∣ X α ( x ; φ N , f p ) − X α ( x ′ ; φ N , f p ) ∣ ≤ ϖ ( X α ( x ′ ; φ N , f p ) + M α ) . (189) \left|\mathcal{X}_\alpha(x;\varphi_N,f_p)-\mathcal{X}_\alpha(x';\varphi_N,f_p)\right|\leq\varpi\left(\mathcal{X}_\alpha(x';\varphi_N,f_p)+M_\alpha\right). \tag*{(189)} ∣ X α ( x ; φ N , f p ) − X α ( x ′ ; φ N , f p ) ∣ ≤ ϖ ( X α ( x ′ ; φ N , f p ) + M α ) . (189) Proof. We may assume that φ \varphi φ C C C α \alpha α ε \varepsilon ε p p p φ \varphi φ δ \delta δ N 0 N_0 N 0 0 < c ∗ < 1 / 2 0 < c_* < 1/2 0 < c ∗ < 1/2 δ / 4 < 2 \delta/4 < 2 δ /4 < 2 i − = ⌈ c ∗ N ⌉ i_- = \lceil c_*N\rceil i − = ⌈ c ∗ N ⌉ i + = ⌊ ( 1 − c ∗ ) N ⌋ i_+ = \lfloor(1-c_*)N\rfloor i + = ⌊( 1 − c ∗ ) N ⌋
∫ − 2 γ i − ρ s c ( E ) d E = i − N ≤ c ∗ + 1 N < ∫ − 2 − 2 + δ / 2 ρ s c ( E ) d E \int_{-2}^{\gamma_{i_-}} \rho_{\mathrm{sc}}(E)\,\mathrm{d}E = \frac{i_-}{N} \le c_* + \frac{1}{N} < \int_{-2}^{-2+\delta/2} \rho_{\mathrm{sc}}(E)\,\mathrm{d}E ∫ − 2 γ i − ρ sc ( E ) d E = N i − ≤ c ∗ + N 1 < ∫ − 2 − 2 + δ /2 ρ sc ( E ) d E for N ≥ N 0 N \ge N_0 N ≥ N 0 ρ s c \rho_{\mathrm{sc}} ρ sc ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) ρ s c ( − E ) = ρ s c ( E ) \rho_{\mathrm{sc}}(-E)=\rho_{\mathrm{sc}}(E) ρ sc ( − E ) = ρ sc ( E ) N ≥ N 0 N \ge N_0 N ≥ N 0 i − , i + ∈ J i_-,i_+ \in\mathcal{J} i − , i + ∈ J
γ i − ≤ − 2 + δ 2 , γ i + ≥ 2 − δ 2 . (190) \gamma_{i_-} \le-2+\frac{\delta}{2}, \qquad\gamma_{i_+} \ge2-\frac{\delta}{2}. \tag*{(190)} γ i − ≤ − 2 + 2 δ , γ i + ≥ 2 − 2 δ . (190) We first consider the pairs with a label outside J \mathcal{J} J x x x x ′ x' x ′ x x x (190) , and the monotonicity of x x x N ≥ N 0 N \ge N_0 N ≥ N 0
x i ≤ x i − ≤ − 2 + δ 2 + N − 1 + ε ≤ − 2 + δ for i < c ∗ N , x i ≥ 2 − δ for i > ( 1 − c ∗ ) N . (191) x_i \le x_{i_-} \le-2+\frac{\delta}{2}+N^{-1+\varepsilon}\le-2+\delta\quad\text{for } i<c_*N,\qquad x_i\ge2-\delta\quad\text{for } i>(1-c_*)N. \tag*{(191)} x i ≤ x i − ≤ − 2 + 2 δ + N − 1 + ε ≤ − 2 + δ for i < c ∗ N , x i ≥ 2 − δ for i > ( 1 − c ∗ ) N . (191) Let i < j i<j i < j i ∉ J i\notin\mathcal{J} i ∈ / J j ∉ J j\notin\mathcal{J} j ∈ / J E = ( x i + x j ) / 2 E=(x_i+x_j)/2 E = ( x i + x j ) /2 dist ( E , supp φ ) ≤ δ \operatorname{dist}(E,\operatorname{supp}\varphi)\le\delta dist ( E , supp φ ) ≤ δ E ∈ [ − 2 + 2 δ , 2 − 2 δ ] E\in[-2+2\delta,2-2\delta] E ∈ [ − 2 + 2 δ , 2 − 2 δ ] i i i ( 1 − c ∗ ) N (1-c_*)N ( 1 − c ∗ ) N j > ( 1 − c ∗ ) N j>(1-c_*)N j > ( 1 − c ∗ ) N E ≥ 2 − δ E\ge2-\delta E ≥ 2 − δ i < c ∗ N i<c_*N i < c ∗ N i ∈ J i\in\mathcal{J} i ∈ J j > ( 1 − c ∗ ) N j>(1-c_*)N j > ( 1 − c ∗ ) N E ∈ [ − 2 + 2 δ , 2 − 2 δ ] E\in[-2+2\delta,2-2\delta] E ∈ [ − 2 + 2 δ , 2 − 2 δ ]
x j = 2 E − x i ≥ − 2 + 3 δ ≥ x i + 2 δ if i < c ∗ N , x_j=2E-x_i\ge-2+3\delta\ge x_i+2\delta\quad\text{if } i<c_*N, x j = 2 E − x i ≥ − 2 + 3 δ ≥ x i + 2 δ if i < c ∗ N , x i = 2 E − x j ≤ 2 − 3 δ ≤ x j − 2 δ if i ∈ J . x_i=2E-x_j\le2-3\delta\le x_j-2\delta\quad\text{if } i\in\mathcal{J}. x i = 2 E − x j ≤ 2 − 3 δ ≤ x j − 2 δ if i ∈ J . Thus x j − x i ≥ 2 δ x_j-x_i\ge2\delta x j − x i ≥ 2 δ
φ N ( E ) f p ( x j − x i ℓ α ) ≤ ∥ φ ∥ ∞ ( ℓ α 2 δ ) 2 p ≤ C N − 2 − 2 α , \varphi_N(E)f_p\left(\frac{x_j-x_i}{\ell_\alpha}\right)\le\|\varphi\|_\infty\left(\frac{\ell_\alpha}{2\delta}\right)^{2p}\le CN^{-2-2\alpha}, φ N ( E ) f p ( ℓ α x j − x i ) ≤ ∥ φ ∥ ∞ ( 2 δ ℓ α ) 2 p ≤ C N − 2 − 2 α , where we used the bound φ N ≤ ∥ φ ∥ ∞ \varphi_N\le\|\varphi\|_\infty φ N ≤ ∥ φ ∥ ∞ Lemma 5.1 and the inequality 2 p ( 1 + α ) ≥ 2 + 2 α 2p(1+\alpha)\ge2+2\alpha 2 p ( 1 + α ) ≥ 2 + 2 α
Suppose next that dist ( E , supp φ ) > δ \operatorname{dist}(E,\operatorname{supp}\varphi)>\delta dist ( E , supp φ ) > δ Lemma 5.1 gives
φ N ( E ) f p ( x j − x i ℓ α ) ≤ φ N ( E ) ≤ C exp ( − δ 2 4 ℓ N 2 ) = C exp ( − δ 2 N 1 / 5 4 ) . \varphi_N(E)f_p\left(\frac{x_j-x_i}{\ell_\alpha}\right)\le\varphi_N(E)\le C\exp\left(-\frac{\delta^2}{4\ell_N^2}\right)=C\exp\left(-\frac{\delta^2N^{1/5}}{4}\right). φ N ( E ) f p ( ℓ α x j − x i ) ≤ φ N ( E ) ≤ C exp ( − 4 ℓ N 2 δ 2 ) = C exp ( − 4 δ 2 N 1/5 ) . Summing over the at most N 2 N^2 N 2
∑ 1 ≤ i < j ≤ N { i , j } ⊄ J φ N ( x i + x j 2 ) f p ( x j − x i ℓ α ) ≤ C N − 2 α + C N 2 exp ( − δ 2 N 1 / 5 4 ) ≤ ω 4 M α . (192) \sum_{\substack{1\le i<j\le N\\ \{i,j\}\not\subset\mathcal{J}}}\varphi_N\left(\frac{x_i+x_j}{2}\right)f_p\left(\frac{x_j-x_i}{\ell_\alpha}\right)\le CN^{-2\alpha}+CN^2\exp\left(-\frac{\delta^2N^{1/5}}{4}\right)\le\frac{\omega}{4}M_\alpha. \tag*{(192)} 1 ≤ i < j ≤ N { i , j } ⊂ J ∑ φ N ( 2 x i + x j ) f p ( ℓ α x j − x i ) ≤ C N − 2 α + C N 2 exp ( − 4 δ 2 N 1/5 ) ≤ 4 ω M α . (192) for N ≥ N 0 N \ge N_0 N ≥ N 0 x ′ x' x ′ x x x
We now compare the terms with i , j ∈ J i,j \in\mathcal{J} i , j ∈ J i < j i<j i < j x i , … , x j x_i,\ldots,x_j x i , … , x j [ x i , x j ] [x_i,x_j] [ x i , x j ] ⌊ N ( x j − x i ) ⌋ + 1 \lfloor N(x_j-x_i)\rfloor+1 ⌊ N ( x j − x i )⌋ + 1 N − 1 N^{-1} N − 1
j − i + 1 ≤ N ε ( N ( x j − x i ) + 1 ) . j-i+1 \le N^{\varepsilon}\bigl(N(x_j-x_i)+1\bigr). j − i + 1 ≤ N ε ( N ( x j − x i ) + 1 ) . The set J \mathcal{J} J k k k i ≤ k < j i\le k<j i ≤ k < j k , k + 1 ∈ J k,k+1\in\mathcal{J} k , k + 1 ∈ J
∣ ( x j ′ − x i ′ ) − ( x j − x i ) ∣ ≤ ( j − i ) N − 2 ε ℓ α ≤ N − ε ( N ( x j − x i ) + 1 ) ℓ α ≤ N − ε ( x j − x i + ℓ α ) , (193) \begin{aligned}
\left|(x'_j-x'_i)-(x_j-x_i)\right| &\le(j-i)N^{-2\varepsilon}\ell_{\alpha} \\
&\le N^{-\varepsilon}\bigl(N(x_j-x_i)+1\bigr)\ell_{\alpha}\le N^{-\varepsilon}(x_j-x_i+\ell_{\alpha}), \tag*{(193)}
\end{aligned} ( x j ′ − x i ′ ) − ( x j − x i ) ≤ ( j − i ) N − 2 ε ℓ α ≤ N − ε ( N ( x j − x i ) + 1 ) ℓ α ≤ N − ε ( x j − x i + ℓ α ) , (193) where in the last inequality we used that N ℓ α = N − α ≤ 1 N\ell_{\alpha}=N^{-\alpha}\le1 N ℓ α = N − α ≤ 1
Next, observe that for y ≥ 0 y\ge0 y ≥ 0
∣ d d y log f p ( y ) ∣ = 2 p y 1 + y 2 ≤ 4 p 1 + y , (194) \left|\frac{\mathrm{d}}{\mathrm{d}y}\log f_p(y)\right|=\frac{2py}{1+y^2}\le\frac{4p}{1+y}, \tag*{(194)} d y d log f p ( y ) = 1 + y 2 2 p y ≤ 1 + y 4 p , (194) since y + y 2 ≤ 2 + 2 y 2 y+y^2\le2+2y^2 y + y 2 ≤ 2 + 2 y 2
y = x j − x i ℓ α , y ′ = x j ′ − x i ′ ℓ α . y=\frac{x_j-x_i}{\ell_{\alpha}},\qquad y'=\frac{x'_j-x'_i}{\ell_{\alpha}}. y = ℓ α x j − x i , y ′ = ℓ α x j ′ − x i ′ . By (6.10), we have ∣ y − y ′ ∣ ≤ N − ε ( 1 + y ) |y-y'|\le N^{-\varepsilon}(1+y) ∣ y − y ′ ∣ ≤ N − ε ( 1 + y ) y ′ ′ y'' y ′′ y y y y ′ y' y ′
1 + y ′ ′ ≥ ( 1 − N − ε ) ( 1 + y ) ≥ 1 + y 2 1+y''\ge(1-N^{-\varepsilon})(1+y)\ge\frac{1+y}{2} 1 + y ′′ ≥ ( 1 − N − ε ) ( 1 + y ) ≥ 2 1 + y for N ≥ N 0 N\ge N_0 N ≥ N 0 y y y y ′ y' y ′
∣ log f p ( x j − x i ℓ α ) − log f p ( x j ′ − x i ′ ℓ α ) ∣ ≤ 8 p ∣ y − y ′ ∣ 1 + y ≤ 8 p N − ε . (195) \left|\log f_p\left(\frac{x_j-x_i}{\ell_{\alpha}}\right)-\log f_p\left(\frac{x'_j-x'_i}{\ell_{\alpha}}\right)\right|\le\frac{8p|y-y'|}{1+y}\le8pN^{-\varepsilon}. \tag*{(195)} log f p ( ℓ α x j − x i ) − log f p ( ℓ α x j ′ − x i ′ ) ≤ 1 + y 8 p ∣ y − y ′ ∣ ≤ 8 p N − ε . (195) It remains to compare the energy weights. The midpoints E = ( x i + x j ) / 2 E=(x_i+x_j)/2 E = ( x i + x j ) /2 E ′ = ( x i ′ + x j ′ ) / 2 E'=(x'_i+x'_j)/2 E ′ = ( x i ′ + x j ′ ) /2 [ − 3 , 3 ] [-3,3] [ − 3 , 3 ]
∣ E − E ′ ∣ ≤ 2 N − 1 + ε ≤ 1. |E-E'|\le2N^{-1+\varepsilon}\le1. ∣ E − E ′ ∣ ≤ 2 N − 1 + ε ≤ 1. By part (2) of Lemma 5.1, we have
∣ φ N ( E ) − φ N ( E ′ ) ∣ ≤ ( exp ( 16 N − 1 + ε h N − 2 ) − 1 ) φ N ( E ′ ) = ( exp ( 16 N − 4 / 5 + ε ) − 1 ) φ N ( E ′ ) , |\varphi_N(E)-\varphi_N(E')|\le\bigl(\exp(16N^{-1+\varepsilon}h_N^{-2})-1\bigr)\varphi_N(E')=\bigl(\exp(16N^{-4/5+\varepsilon})-1\bigr)\varphi_N(E'), ∣ φ N ( E ) − φ N ( E ′ ) ∣ ≤ ( exp ( 16 N − 1 + ε h N − 2 ) − 1 ) φ N ( E ′ ) = ( exp ( 16 N − 4/5 + ε ) − 1 ) φ N ( E ′ ) , and the factor in parentheses tends to zero, as ε ≤ 1 / 10 \varepsilon\le1/10 ε ≤ 1/10 N ≥ N 0 N\ge N_0 N ≥ N 0 i < j i<j i < j J \mathcal{J} J
∣ φ N ( E ) f p ( x j − x i ℓ α ) − φ N ( E ′ ) f p ( x j ′ − x i ′ ℓ α ) ∣ ≤ ω 2 φ N ( E ′ ) f p ( x j ′ − x i ′ ℓ α ) . \left|\varphi_N(E)f_p\left(\frac{x_j-x_i}{\ell_{\alpha}}\right)-\varphi_N(E')f_p\left(\frac{x'_j-x'_i}{\ell_{\alpha}}\right)\right|\le\frac{\omega}{2}\varphi_N(E')f_p\left(\frac{x'_j-x'_i}{\ell_{\alpha}}\right). φ N ( E ) f p ( ℓ α x j − x i ) − φ N ( E ′ ) f p ( ℓ α x j ′ − x i ′ ) ≤ 2 ω φ N ( E ′ ) f p ( ℓ α x j ′ − x i ′ ) . We sum over these pairs and bound the resulting sum of nonnegative terms on the right by X α ( x ′ ; φ N , f p ) \mathcal{X}_{\alpha}(x';\varphi_N,f_p) X α ( x ′ ; φ N , f p ) J \mathcal{J} J X α ( x ; φ N , f p ) \mathcal{X}_{\alpha}(x;\varphi_N,f_p) X α ( x ; φ N , f p ) X α ( x ′ ; φ N , f p ) \mathcal{X}_{\alpha}(x';\varphi_N,f_p) X α ( x ′ ; φ N , f p ) ( ϖ / 2 ) X α ( x ′ ; φ N , f p ) (\varpi/2)\mathcal{X}_{\alpha}(x';\varphi_N,f_p) ( ϖ /2 ) X α ( x ′ ; φ N , f p ) x x x x ′ x' x ′
Proposition 6.5. For every α ∈ ( 0 , 1 / 8 ) \alpha\in(0,1/8) α ∈ ( 0 , 1/8 ) P ( α ) \mathcal{P}(\alpha) P ( α )
Proof. Fix α ∈ ( 0 , 1 / 8 ) \alpha\in(0,1/8) α ∈ ( 0 , 1/8 ) p ≥ p ∗ p \ge p_* p ≥ p ∗ φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_{\mathrm{c}}((-2,2)) φ ∈ C c ∞ (( − 2 , 2 ))
L = 1 2 ( p − 1 ) ∫ − 2 2 φ ( E ) c ( E ) d E , T = N − 3 / 4 − α , ε = 1 100 ( 1 4 − 2 α ) . (196) L = \frac{1}{2(p-1)}\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E,\qquad T = N^{-3/4-\alpha},\qquad\varepsilon= \frac{1}{100}\left(\frac{1}{4}-2\alpha\right). \tag*{(196)} L = 2 ( p − 1 ) 1 ∫ − 2 2 φ ( E ) c ( E ) d E , T = N − 3/4 − α , ε = 100 1 ( 4 1 − 2 α ) . (196) Then ε ∈ ( 0 , 1 / 400 ] \varepsilon\in(0,1/400] ε ∈ ( 0 , 1/400 ]
T ≥ N − 1 + ε , α + 5 ε < 1 2 , T \ge N^{-1+\varepsilon},\qquad\alpha+5\varepsilon< \frac{1}{2}, T ≥ N − 1 + ε , α + 5 ε < 2 1 , N ε N 2 T = N − 5 / 4 + α + ε ≤ N − 2 ℓ α , T N 1 / 2 + 3 α + 12 ε = N − 1 / 4 + 2 α + 12 ε . (197) \frac{N^\varepsilon}{N^2T}=N^{-5/4+\alpha+\varepsilon}\le N^{-2\ell_\alpha},\qquad TN^{1/2+3\alpha+12\varepsilon}=N^{-1/4+2\alpha+12\varepsilon}. \tag*{(197)} N 2 T N ε = N − 5/4 + α + ε ≤ N − 2 ℓ α , T N 1/2 + 3 α + 12 ε = N − 1/4 + 2 α + 12 ε . (197) Indeed, the first inequality is equivalent to ε ≤ 1 / 4 − α \varepsilon\le1/4-\alpha ε ≤ 1/4 − α 3 ε ≤ 1 / 4 − 2 α 3\varepsilon\le1/4-2\alpha 3 ε ≤ 1/4 − 2 α − ( 88 / 100 ) ( 1 / 4 − 2 α ) < 0 -(88/100)(1/4-2\alpha)<0 − ( 88/100 ) ( 1/4 − 2 α ) < 0 t N = T t_N=T t N = T
1 M α X α ( H T ; φ N , f p ) → L in probability . (198) \frac{1}{M_\alpha}\mathcal{X}_{\alpha}(H_T;\varphi_N,f_p)\to L\qquad\text{in probability}. \tag*{(198)} M α 1 X α ( H T ; φ N , f p ) → L in probability . (198) We first consider the GOE matrix G G G δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) supp φ ⊂ [ − 2 + 3 δ , 2 − 3 δ ] \operatorname{supp}\varphi\subset[-2+3\delta,2-3\delta] supp φ ⊂ [ − 2 + 3 δ , 2 − 3 δ ] χ ∈ C c ∞ ( ( − 2 , 2 ) ) \chi\in C^\infty_{\mathrm{c}}((-2,2)) χ ∈ C c ∞ (( − 2 , 2 )) 0 ≤ χ ≤ 2 0\le\chi\le2 0 ≤ χ ≤ 2 χ = 2 \chi=2 χ = 2 [ − 2 + 2 δ , 2 − 2 δ ] [-2+2\delta,2-2\delta] [ − 2 + 2 δ , 2 − 2 δ ] [ − 2 + 2 δ , 2 − 2 δ ] [-2+2\delta,2-2\delta] [ − 2 + 2 δ , 2 − 2 δ ] δ \delta δ supp φ \operatorname{supp}\varphi supp φ
The functions ( r , E ) ↦ φ ( E ) f p ( r ) (r,E)\mapsto\varphi(E)f_p(r) ( r , E ) ↦ φ ( E ) f p ( r ) ( r , E ) ↦ χ ( E ) f p ∗ ( r ) (r,E)\mapsto\chi(E)f_{p_*}(r) ( r , E ) ↦ χ ( E ) f p ∗ ( r ) E E E ( 1 + r 2 ) − 2 (1+r^2)^{-2} ( 1 + r 2 ) − 2
1 M α X α ( G ; φ , f p ) → L , 1 M α X α ( G ; χ , f p ∗ ) → 1 2 ( p ∗ − 1 ) ∫ − 2 2 χ ( E ) c ( E ) d E (199) \frac{1}{M_\alpha}\mathcal{X}_{\alpha}(G;\varphi,f_p)\to L,\qquad\frac{1}{M_\alpha}\mathcal{X}_{\alpha}(G;\chi,f_{p_*})\to\frac{1}{2(p_*-1)}\int_{-2}^{2}\chi(E)c(E)\,\mathrm{d}E \tag*{(199)} M α 1 X α ( G ; φ , f p ) → L , M α 1 X α ( G ; χ , f p ∗ ) → 2 ( p ∗ − 1 ) 1 ∫ − 2 2 χ ( E ) c ( E ) d E (199) in probability. Since p ≥ p ∗ p\ge p_* p ≥ p ∗ f p ≤ f p ∗ f_p\le f_{p_*} f p ≤ f p ∗ A = G A=G A = G f = f p f=f_p f = f p { ∥ G ∥ ≤ 3 } \{\lVert G\rVert\le3\} {∥ G ∥ ≤ 3 } N ≥ N 0 N\ge N_0 N ≥ N 0
1 M α ∣ X α ( G ; φ N , f p ) − X α ( G ; φ , f p ) ∣ ≤ C ℏ N M α X α ( G ; χ , f p ∗ ) + C N 2 M α exp ( − δ 2 N 1 / 5 4 ) . \frac{1}{M_\alpha}\left|\mathcal{X}_{\alpha}(G;\varphi_N,f_p)-\mathcal{X}_{\alpha}(G;\varphi,f_p)\right|\le\frac{C\hbar_N}{M_\alpha}\mathcal{X}_{\alpha}(G;\chi,f_{p_*})+\frac{CN^2}{M_\alpha}\exp\left(-\frac{\delta^2N^{1/5}}{4}\right). M α 1 ∣ X α ( G ; φ N , f p ) − X α ( G ; φ , f p ) ∣ ≤ M α C ℏ N X α ( G ; χ , f p ∗ ) + M α C N 2 exp ( − 4 δ 2 N 1/5 ) . By (6.16), the right side tends to zero in probability, and the event { ∥ G ∥ ≤ 3 } \{\lVert G\rVert\le3\} {∥ G ∥ ≤ 3 }
1 M α X α ( G ; φ N , f p ) → L in probability . (200) \frac{1}{M_\alpha}\mathcal{X}_{\alpha}(G;\varphi_N,f_p)\to L\qquad\text{in probability}. \tag*{(200)} M α 1 X α ( G ; φ N , f p ) → L in probability . (200) We now use the coupling. Let ( H t d y n ) t ≥ 0 (H_t^{\mathrm{dyn}})_{t\geq0} ( H t dyn ) t ≥ 0 ( μ ( t ) ) t ≥ 0 (\mu(t))_{t\geq0} ( μ ( t ) ) t ≥ 0 Proposition 3.5 , and set
λ ( T ) = ( λ 1 ( H T d y n ) , … , λ N ( H T d y n ) ) , μ ( T ) = ( μ 1 ( T ) , … , μ N ( T ) ) . \lambda(T) = \left(\lambda_1(H_T^{\mathrm{dyn}}),\ldots,\lambda_N(H_T^{\mathrm{dyn}})\right), \qquad\mu(T) = \left(\mu_1(T),\ldots,\mu_N(T)\right). λ ( T ) = ( λ 1 ( H T dyn ) , … , λ N ( H T dyn ) ) , μ ( T ) = ( μ 1 ( T ) , … , μ N ( T ) ) . Recall that H T d y n H_T^{\mathrm{dyn}} H T dyn H T H_T H T μ ( T ) \mu(T) μ ( T ) G G G
Let c ∗ c_* c ∗ Lemma 6.4 , defined with our choice of δ \delta δ H T d y n ∈ G ε H_T^{\mathrm{dyn}}\in\mathcal{G}_\varepsilon H T dyn ∈ G ε max i ∈ [ N ] ∣ μ i ( T ) ∣ ≤ 3 \max_{i\in[N]}|\mu_i(T)|\leq3 max i ∈ [ N ] ∣ μ i ( T ) ∣ ≤ 3
max c ∗ N ≤ i ≤ ( 1 − c ∗ ) N max { ∣ λ i ( H T d y n ) − γ i ∣ , ∣ μ i ( T ) − γ i ∣ } ≤ N − 1 + ε , \max_{c_*N\leq i\leq(1-c_*)N}\max\left\{|\lambda_i(H_T^{\mathrm{dyn}})-\gamma_i|,\ |\mu_i(T)-\gamma_i|\right\}\leq N^{-1+\varepsilon}, c ∗ N ≤ i ≤ ( 1 − c ∗ ) N max max { ∣ λ i ( H T dyn ) − γ i ∣ , ∣ μ i ( T ) − γ i ∣ } ≤ N − 1 + ε , max c ∗ N ≤ i ≤ ( 1 − c ∗ ) N ∣ ( λ i + 1 ( H T d y n ) − λ i ( H T d y n ) ) − ( μ i + 1 ( T ) − μ i ( T ) ) ∣ ≤ N ε N 2 T . (201) \max_{c_*N\leq i\leq(1-c_*)N}\left|(\lambda_{i+1}(H_T^{\mathrm{dyn}})-\lambda_i(H_T^{\mathrm{dyn}}))-(\mu_{i+1}(T)-\mu_i(T))\right|\leq\frac{N^\varepsilon}{N^2T}. \tag*{(201)} c ∗ N ≤ i ≤ ( 1 − c ∗ ) N max ( λ i + 1 ( H T dyn ) − λ i ( H T dyn )) − ( μ i + 1 ( T ) − μ i ( T )) ≤ N 2 T N ε . (201) By Proposition 3.2 , applied to H T H_T H T G G G μ ( T ) \mu(T) μ ( T ) (201) hold with overwhelming probability. We have N − 1 + ε ≤ T < 1 N^{-1+\varepsilon}\leq T<1 N − 1 + ε ≤ T < 1 (197) , so Proposition 3.5 shows that the second inequality in Eq. (201) holds with probability at least 1 − N − 1 1-N^{-1} 1 − N − 1 N ≥ N 0 N\geq N_0 N ≥ N 0
The sequences x = λ ( T ) x=\lambda(T) x = λ ( T ) x ′ = μ ( T ) x'=\mu(T) x ′ = μ ( T ) Lemma 6.4 on this event, where for (c) we used the third relation in Eq. (197) . Hence, for every ϖ > 0 \varpi>0 ϖ > 0
∣ 1 M α X α ( λ ( T ) ; φ N , f p ) − L ∣ ≤ ∣ 1 M α X α ( μ ( T ) ; φ N , f p ) − L ∣ + ϖ ( 1 M α X α ( μ ( T ) ; φ N , f p ) + 1 ) . \left|\frac{1}{M_\alpha}\mathcal{X}_\alpha(\lambda(T);\varphi_N,f_p)-L\right|
\leq\left|\frac{1}{M_\alpha}\mathcal{X}_\alpha(\mu(T);\varphi_N,f_p)-L\right|+\varpi\left(\frac{1}{M_\alpha}\mathcal{X}_\alpha(\mu(T);\varphi_N,f_p)+1\right). M α 1 X α ( λ ( T ) ; φ N , f p ) − L ≤ M α 1 X α ( μ ( T ) ; φ N , f p ) − L + ϖ ( M α 1 X α ( μ ( T ) ; φ N , f p ) + 1 ) . By Eq. (200) , we have M α − 1 X α ( μ ( T ) ; φ N , f p ) → L M_\alpha^{-1}\mathcal{X}_\alpha(\mu(T);\varphi_N,f_p)\to L M α − 1 X α ( μ ( T ) ; φ N , f p ) → L ϖ > 0 \varpi>0 ϖ > 0 M α − 1 X α ( λ ( T ) ; φ N , f p ) → L M_\alpha^{-1}\mathcal{X}_\alpha(\lambda(T);\varphi_N,f_p)\to L M α − 1 X α ( λ ( T ) ; φ N , f p ) → L λ ( T ) \lambda(T) λ ( T ) H T H_T H T (198) . □
General test functions
We now show that P ( α ) \mathcal{P}(\alpha) P ( α ) f f f ℓ α \ell_\alpha ℓ α
Lemma 6.6. Let α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) P ( α ) \mathcal{P}(\alpha) P ( α ) φ ∈ C ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty((-2,2)) φ ∈ C ∞ (( − 2 , 2 )) f : [ 0 , ∞ ) → R f:[0,\infty)\to\mathbb{R} f : [ 0 , ∞ ) → R lim y → ∞ ( 1 + y 2 ) p f ( y ) \lim_{y\to\infty}(1+y^2)^p f(y) lim y → ∞ ( 1 + y 2 ) p f ( y ) R \mathbb{R} R
1 M α X α ( H ; φ N , f ) → ∫ − 2 2 φ ( E ) c ( E ) d E ∫ 0 ∞ y f ( y ) d y (202) \frac{1}{M_\alpha}\mathcal{X}_\alpha(H;\varphi_N,f)\to\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E\int_{0}^{\infty}y f(y)\,\mathrm{d}y \tag*{(202)} M α 1 X α ( H ; φ N , f ) → ∫ − 2 2 φ ( E ) c ( E ) d E ∫ 0 ∞ y f ( y ) d y (202) :::`
in probability, and the same holds with φ N \varphi_N φ N φ \varphi φ I I I C I > 0 C_I > 0 C I > 0
lim N → ∞ P ( ∑ 1 ≤ i < j ≤ N ( λ i + λ j ) / 2 ∈ I f p ∗ ( λ j − λ i ℓ α ) ≤ C I M α ) = 1. (203) \lim_{N\to\infty} \mathbb{P}\left(\sum_{\substack{1\leq i<j\leq N\\(\lambda_i+\lambda_j)/2\in I}} f_{p_*}\left(\frac{\lambda_j-\lambda_i}{\ell_\alpha}\right)\leq C_I M_\alpha\right)=1. \tag*{(203)} N → ∞ lim P 1 ≤ i < j ≤ N ( λ i + λ j ) /2 ∈ I ∑ f p ∗ ( ℓ α λ j − λ i ) ≤ C I M α = 1. (203) Proof. We set
C f = sup y ≥ 0 ( 1 + y 2 ) p ∗ ∣ f ( y ) ∣ , c φ = ∫ − 2 2 φ ( E ) c ( E ) d E . C_f=\sup_{y\geq0}(1+y^2)^{p_*}|f(y)|,\qquad c_\varphi=\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E. C f = y ≥ 0 sup ( 1 + y 2 ) p ∗ ∣ f ( y ) ∣ , c φ = ∫ − 2 2 φ ( E ) c ( E ) d E . The constant C f C_f C f f f f y ↦ e − y 2 / 8 y\mapsto e^{-y^2/8} y ↦ e − y 2 /8 ∣ f ∣ ≤ C f f p ∗ |f|\leq C_f f_{p_*} ∣ f ∣ ≤ C f f p ∗ ∫ 0 ∞ y f ( y ) d y \int_0^\infty y f(y)\,\mathrm{d}y ∫ 0 ∞ y f ( y ) d y
We begin with (202) , and we first approximate f f f f p f_p f p Θ f : [ 0 , 1 ] → R \Theta_f:[0,1]\to\mathbb{R} Θ f : [ 0 , 1 ] → R
Θ f ( x ) = x − p ∗ f ( ( x − 1 − 1 ) 1 / 2 ) for x ∈ ( 0 , 1 ] , Θ f ( 0 ) = lim y → ∞ ( 1 + y 2 ) p ∗ f ( y ) . \Theta_f(x)=x^{-p_*}f\left((x^{-1}-1)^{1/2}\right)\quad\text{for }x\in(0,1],\qquad\Theta_f(0)=\lim_{y\to\infty}(1+y^2)^{p_*}f(y). Θ f ( x ) = x − p ∗ f ( ( x − 1 − 1 ) 1/2 ) for x ∈ ( 0 , 1 ] , Θ f ( 0 ) = y → ∞ lim ( 1 + y 2 ) p ∗ f ( y ) . For y ≥ 0 y\geq0 y ≥ 0 x = ( 1 + y 2 ) − 1 x=(1+y^2)^{-1} x = ( 1 + y 2 ) − 1 ( x − 1 − 1 ) 1 / 2 = y (x^{-1}-1)^{1/2}=y ( x − 1 − 1 ) 1/2 = y x − p ∗ = ( 1 + y 2 ) p ∗ x^{-p_*}=(1+y^2)^{p_*} x − p ∗ = ( 1 + y 2 ) p ∗ Θ f \Theta_f Θ f [ 0 , 1 ] [0,1] [ 0 , 1 ]
f ( y ) = f p ∗ ( y ) Θ f ( ( 1 + y 2 ) − 1 ) for all y ≥ 0. f(y)=f_{p_*}(y)\Theta_f\left((1+y^2)^{-1}\right)\quad\text{for all }y\geq0. f ( y ) = f p ∗ ( y ) Θ f ( ( 1 + y 2 ) − 1 ) for all y ≥ 0. Let ϖ ∈ ( 0 , 1 ) \varpi\in(0,1) ϖ ∈ ( 0 , 1 ) p 1 ≥ p ∗ p_1\geq p_* p 1 ≥ p ∗ β p ∗ , … , β p 1 \beta_{p_*},\ldots,\beta_{p_1} β p ∗ , … , β p 1
sup x ∈ [ 0 , 1 ] ∣ Θ f ( x ) − ∑ p = p ∗ p 1 β p x p − p ∗ ∣ ≤ ϖ . \sup_{x\in[0,1]}\left|\Theta_f(x)-\sum_{p=p_*}^{p_1}\beta_p x^{p-p_*}\right|\leq\varpi. x ∈ [ 0 , 1 ] sup Θ f ( x ) − p = p ∗ ∑ p 1 β p x p − p ∗ ≤ ϖ . Note that f p ∗ ( y ) ( 1 + y 2 ) − ( p − p ∗ ) = f p ( y ) f_{p_*}(y)(1+y^2)^{-(p-p_*)}=f_p(y) f p ∗ ( y ) ( 1 + y 2 ) − ( p − p ∗ ) = f p ( y )
f ˉ = ∑ p = p ∗ p 1 β p f p \bar f=\sum_{p=p_*}^{p_1}\beta_p f_p f ˉ = p = p ∗ ∑ p 1 β p f p satisfies
∣ f ( y ) − f ˉ ( y ) ∣ ≤ ϖ f p ∗ ( y ) for all y ≥ 0. (204) |f(y)-\bar f(y)|\leq\varpi f_{p_*}(y)\quad\text{for all }y\geq0. \tag*{(204)} ∣ f ( y ) − f ˉ ( y ) ∣ ≤ ϖ f p ∗ ( y ) for all y ≥ 0. (204) The statistic X α ( H ; φ N , f ) \mathcal{X}_\alpha(H;\varphi_N,f) X α ( H ; φ N , f ) f f f P ( α ) \mathcal{P}(\alpha) P ( α ) p ∈ { p ∗ , … , p 1 } p\in\{p_*,\ldots,p_1\} p ∈ { p ∗ , … , p 1 } (184) , we obtain
1 M α X α ( H ; φ N , f ˉ ) → c φ ∑ p = p ∗ p 1 β p 2 ( p − 1 ) = c φ ∫ 0 ∞ y f ˉ ( y ) d y in probability . \frac{1}{M_\alpha}\mathcal{X}_\alpha(H;\varphi_N,\bar f)\to c_\varphi\sum_{p=p_*}^{p_1}\frac{\beta_p}{2(p-1)}=c_\varphi\int_0^\infty y\bar f(y)\,\mathrm{d}y\quad\text{in probability}. M α 1 X α ( H ; φ N , f ˉ ) → c φ p = p ∗ ∑ p 1 2 ( p − 1 ) β p = c φ ∫ 0 ∞ y f ˉ ( y ) d y in probability . Since φ N ≥ 0 \varphi_N \ge0 φ N ≥ 0
∣ X α ( H ; φ N , f ) − X α ( H ; φ N , f ˉ ) ∣ ≤ ϖ X α ( H ; φ N , f p ∗ ) , ∣ ∫ 0 ∞ y ( f − f ˉ ) ( y ) d y ∣ ≤ ϖ , \left|\mathcal{X}_{\alpha}(H;\varphi_N,f)-\mathcal{X}_{\alpha}(H;\varphi_N,\bar{f})\right|\le\varpi\mathcal{X}_{\alpha}(H;\varphi_N,f_{p_*}),\qquad\left|\int_0^\infty y(f-\bar{f})(y)\,\mathrm{d}y\right|\le\varpi, X α ( H ; φ N , f ) − X α ( H ; φ N , f ˉ ) ≤ ϖ X α ( H ; φ N , f p ∗ ) , ∫ 0 ∞ y ( f − f ˉ ) ( y ) d y ≤ ϖ , where for the second bound we also used (6.1). Hence
∣ 1 M α X α ( H ; φ N , f ) − c φ ∫ 0 ∞ y f ( y ) d y ∣ ≤ ∣ 1 M α X α ( H ; φ N , f ˉ ) − c φ ∫ 0 ∞ y f ˉ ( y ) d y ∣ + ϖ M α X α ( H ; φ N , f p ∗ ) + ϖ c φ . \left|\frac{1}{M_\alpha}\mathcal{X}_{\alpha}(H;\varphi_N,f)-c_\varphi\int_0^\infty yf(y)\,\mathrm{d}y\right|
\le\left|\frac{1}{M_\alpha}\mathcal{X}_{\alpha}(H;\varphi_N,\bar{f})-c_\varphi\int_0^\infty y\bar{f}(y)\,\mathrm{d}y\right|+\frac{\varpi}{M_\alpha}\mathcal{X}_{\alpha}(H;\varphi_N,f_{p_*})+\varpi c_\varphi. M α 1 X α ( H ; φ N , f ) − c φ ∫ 0 ∞ y f ( y ) d y ≤ M α 1 X α ( H ; φ N , f ˉ ) − c φ ∫ 0 ∞ y f ˉ ( y ) d y + M α ϖ X α ( H ; φ N , f p ∗ ) + ϖ c φ . By P ( α ) \mathcal{P}(\alpha) P ( α ) p = p ∗ p=p_* p = p ∗ ϖ c φ / ( 2 ( p ∗ − 1 ) ) ≤ ϖ c φ \varpi c_\varphi/(2(p_*-1))\le\varpi c_\varphi ϖ c φ / ( 2 ( p ∗ − 1 )) ≤ ϖ c φ 3 ϖ ( c φ + 1 ) 3\varpi(c_\varphi+1) 3 ϖ ( c φ + 1 ) ϖ \varpi ϖ
Next, we remove the smoothing, and we may assume that C f > 0 C_f>0 C f > 0 δ > 0 \delta>0 δ > 0 E E E dist ( E , supp φ ) ≤ δ \operatorname{dist}(E,\operatorname{supp}\varphi)\le\delta dist ( E , supp φ ) ≤ δ ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) χ ∈ C c ∞ ( ( − 2 , 2 ) ) \chi\in C_c^\infty((-2,2)) χ ∈ C c ∞ (( − 2 , 2 )) 0 ≤ χ ≤ 2 0\le\chi\le2 0 ≤ χ ≤ 2 χ ( E ) = 2 \chi(E)=2 χ ( E ) = 2 E E E f / C f f/C_f f / C f { ∥ H ∥ ≤ 3 } \{\lVert H\rVert\le3\} {∥ H ∥ ≤ 3 } N ≥ N 0 N\ge N_0 N ≥ N 0
1 M α ∣ X α ( H ; φ N , f ) − X α ( H ; φ , f ) ∣ ≤ C C f h N M α X α ( H ; χ N , f p ∗ ) + C C f N 2 M α exp ( − δ 2 N 1 / 5 4 ) . \frac{1}{M_\alpha}\left|\mathcal{X}_{\alpha}(H;\varphi_N,f)-\mathcal{X}_{\alpha}(H;\varphi,f)\right|
\le\frac{CC_fh_N}{M_\alpha}\mathcal{X}_{\alpha}(H;\chi_N,f_{p_*})+\frac{C C_fN^2}{M_\alpha}\exp\left(-\frac{\delta^2N^{1/5}}{4}\right). M α 1 ∣ X α ( H ; φ N , f ) − X α ( H ; φ , f ) ∣ ≤ M α C C f h N X α ( H ; χ N , f p ∗ ) + M α C C f N 2 exp ( − 4 δ 2 N 1/5 ) . By P ( α ) \mathcal{P}(\alpha) P ( α ) χ \chi χ φ \varphi φ p = p ∗ p=p_* p = p ∗ M α − 1 X α ( H ; χ N , f p ∗ ) M_\alpha^{-1}\mathcal{X}_{\alpha}(H;\chi_N,f_{p_*}) M α − 1 X α ( H ; χ N , f p ∗ ) { ∥ H ∥ ≤ 3 } \{\lVert H\rVert\le3\} {∥ H ∥ ≤ 3 } φ \varphi φ φ N \varphi_N φ N
Finally, we prove (6.20). Let I I I χ \chi χ C c ∞ ( ( − 2 , 2 ) ) C_c^\infty((-2,2)) C c ∞ (( − 2 , 2 )) 0 ≤ χ ≤ 2 0\le\chi\le2 0 ≤ χ ≤ 2 χ = 2 \chi=2 χ = 2 I I I
∥ χ N − χ ∥ ∞ ≤ C h N ≤ 1 \lVert\chi_N-\chi\rVert_\infty\le Ch_N\le1 ∥ χ N − χ ∥ ∞ ≤ C h N ≤ 1 for N ≥ N 0 N\ge N_0 N ≥ N 0 χ N ≥ 1 \chi_N\ge1 χ N ≥ 1 I I I X α ( H ; χ N , f p ∗ ) \mathcal{X}_{\alpha}(H;\chi_N,f_{p_*}) X α ( H ; χ N , f p ∗ ) X α ( H ; χ N , f p ∗ ) \mathcal{X}_{\alpha}(H;\chi_N,f_{p_*}) X α ( H ; χ N , f p ∗ ) P ( α ) \mathcal{P}(\alpha) P ( α )
1 M α X α ( H ; χ N , f p ∗ ) ⟶ 1 2 ( p ∗ − 1 ) ∫ − 2 2 χ ( E ) c ( E ) d E \frac{1}{M_\alpha}\mathcal{X}_{\alpha}(H;\chi_N,f_{p_*})\longrightarrow\frac{1}{2(p_*-1)}\int_{-2}^{2}\chi(E)c(E)\,\mathrm{d}E M α 1 X α ( H ; χ N , f p ∗ ) ⟶ 2 ( p ∗ − 1 ) 1 ∫ − 2 2 χ ( E ) c ( E ) d E in probability. Hence (6.20) holds with
C I = 1 + ∫ − 2 2 χ ( E ) c ( E ) d E . C_I=1+\int_{-2}^{2}\chi(E)c(E)\,\mathrm{d}E. C I = 1 + ∫ − 2 2 χ ( E ) c ( E ) d E . The induction step
Proposition 6.7. Let α , α ′ \alpha,\alpha' α , α ′
1 10 ≤ α < α ′ < 1 2 , α ′ < 6 α 5 , α ′ < 1 6 + 2 α 3 . (205) \frac{1}{10} \le\alpha< \alpha' < \frac{1}{2}, \qquad\alpha' < \frac{6\alpha}{5}, \qquad\alpha' < \frac{1}{6} + \frac{2\alpha}{3}. \tag*{(205)} 10 1 ≤ α < α ′ < 2 1 , α ′ < 5 6 α , α ′ < 6 1 + 3 2 α . (205) If P ( α ) \mathcal{P}(\alpha) P ( α ) P ( α ′ ) \mathcal{P}(\alpha') P ( α ′ )
To prove Proposition 6.7, we use Lemma 6.3 to replace H H H H t α H_{t_\alpha} H t α t α t_\alpha t α (206) below, and we approximate X α ′ ( H t α ; φ N , f p ) \mathcal{X}_{\alpha'}(H_{t_\alpha};\varphi_N,f_p) X α ′ ( H t α ; φ N , f p ) 2 × 2 2 \times2 2 × 2 Lemma 3.10 and determined by P ( α ) \mathcal{P}(\alpha) P ( α ) Lemma 6.8 collects the properties of the clusters that we need, Lemma 6.9 bounds the error of the approximation, Lemma 6.10 computes the conditional mean of the sum, and Lemma 6.11 shows that the sum concentrates around this mean. We prove Proposition 6.7 after the proofs of these lemmas. The main technical point is the approximation, which relies on Corollary 3.9 and on the first three constraints in (208) .
Parameters. Throughout the rest of § 8.3 , we fix α \alpha α α ′ \alpha' α ′ (205) . The parameters and the notation introduced in this paragraph and the next two are used in the statements and the proofs of all lemmas below. We set
κ = α − α ′ 6 , q = N − 1 − κ , t α = N ℓ α 2 = N − 1 − 2 α . (206) \kappa= \alpha- \frac{\alpha'}{6}, \qquad q = N^{-1-\kappa}, \qquad t_\alpha= N\ell_\alpha^2 = N^{-1-2\alpha}. \tag*{(206)} κ = α − 6 α ′ , q = N − 1 − κ , t α = N ℓ α 2 = N − 1 − 2 α . (206) Then (205) implies that
2 α ′ 3 < κ < 2 α − α ′ < α , α ′ − κ = 7 α ′ 6 − α > α 6 ≥ 1 60 . (207) \frac{2\alpha'}{3} < \kappa< 2\alpha-\alpha' < \alpha, \qquad\alpha' - \kappa= \frac{7\alpha'}{6} - \alpha> \frac{\alpha}{6} \ge\frac{1}{60}. \tag*{(207)} 3 2 α ′ < κ < 2 α − α ′ < α , α ′ − κ = 6 7 α ′ − α > 6 α ≥ 60 1 . (207) Indeed, the first two inequalities are equivalent to α ′ < 6 α / 5 \alpha' < 6\alpha/5 α ′ < 6 α /5 α < α ′ \alpha< \alpha' α < α ′ α < α ′ \alpha< \alpha' α < α ′ α ≥ 1 / 10 \alpha\ge1/10 α ≥ 1/10
We fix ε > 0 \varepsilon> 0 ε > 0
100 ε < min { α ′ 6 , α − 5 α ′ 6 , 3 α − 5 α ′ 2 , 1 2 + 2 α − 3 α ′ , 1 2 − α ′ } . (208) 100\varepsilon< \min\left\{\frac{\alpha'}{6},\alpha-\frac{5\alpha'}{6},3\alpha-\frac{5\alpha'}{2},\frac{1}{2}+2\alpha-3\alpha',\frac{1}{2}-\alpha'\right\}. \tag*{(208)} 100 ε < min { 6 α ′ , α − 6 5 α ′ , 3 α − 2 5 α ′ , 2 1 + 2 α − 3 α ′ , 2 1 − α ′ } . (208) Every number in this minimum is positive. For the second and third numbers, this follows from α ′ < 6 α / 5 \alpha' < 6\alpha/5 α ′ < 6 α /5 α ′ < 1 / 6 + 2 α / 3 \alpha' < 1/6 + 2\alpha/3 α ′ < 1/6 + 2 α /3 α ′ < 1 / 2 \alpha' < 1/2 α ′ < 1/2 κ \kappa κ α − κ \alpha-\kappa α − κ 2 α − κ − α ′ 2\alpha-\kappa-\alpha' 2 α − κ − α ′ 3 κ − 2 α ′ 3\kappa-2\alpha' 3 κ − 2 α ′
ε < α ′ 6 < κ , \varepsilon< \frac{\alpha'}{6} < \kappa, ε < 6 α ′ < κ , where the last inequality is equivalent to α ′ < 3 α \alpha' < 3\alpha α ′ < 3 α
For an integer p ≥ p ∗ p \ge p_* p ≥ p ∗ φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 ))
L = 1 2 ( p − 1 ) ∫ − 2 2 φ ( E ) c ( E ) d E . (209) L = \frac{1}{2(p-1)} \int_{-2}^{2} \varphi(E)c(E)\,\mathrm{d}E. \tag*{(209)} L = 2 ( p − 1 ) 1 ∫ − 2 2 φ ( E ) c ( E ) d E . (209) In the statements and the proofs of the lemmas below, constants may depend on α \alpha α α ′ \alpha' α ′ ε \varepsilon ε p p p φ \varphi φ N 0 N_0 N 0
Gaussian blocks. Let U = ( u 1 , … , u N ) U=(u_1,\ldots,u_N) U = ( u 1 , … , u N ) H H H
y i = e − t α / 2 λ i ( i ∈ [ N ] ) , D y = diag ( y 1 , … , y N ) , W = ( 1 − e − t α ) 1 / 2 U T G U . (210) y_i=e^{-t_\alpha/2}\lambda_i\quad(i\in[N]),\qquad D_y=\operatorname{diag}(y_1,\ldots,y_N),\qquad W=(1-e^{-t_\alpha})^{1/2}U^{T}GU. \tag*{(210)} y i = e − t α /2 λ i ( i ∈ [ N ]) , D y = diag ( y 1 , … , y N ) , W = ( 1 − e − t α ) 1/2 U T G U . (210) Then
U T H t α U = D y + W , so λ i ( H t α ) = λ i ( D y + W ) for all i ∈ [ N ] . (211) U^{T}H_{t_\alpha}U=D_y+W,\qquad\text{so}\qquad\lambda_i(H_{t_\alpha})=\lambda_i(D_y+W)\quad\text{for all }i\in[N]. \tag*{(211)} U T H t α U = D y + W , so λ i ( H t α ) = λ i ( D y + W ) for all i ∈ [ N ] . (211) The matrix G G G H H H G G G U U U H H H U T G U U^{T}GU U T G U H H H G G G W W W H H H W i j W_{ij} W ij i ≤ j i\le j i ≤ j
E W i j 2 = 1 − e − t α N ( i < j ) , E W i i 2 = 2 ( 1 − e − t α ) N . (212) \mathbb{E}W_{ij}^{2}=\frac{1-e^{-t_\alpha}}{N}\quad(i<j),\qquad\mathbb{E}W_{ii}^{2}=\frac{2(1-e^{-t_\alpha})}{N}. \tag*{(212)} E W ij 2 = N 1 − e − t α ( i < j ) , E W ii 2 = N 2 ( 1 − e − t α ) . (212) We next bound the entries of W W W (212) , we have for all i ≤ j i\le j i ≤ j
E W i j 2 ≤ 2 ( 1 − e − t α ) N ≤ 2 t α N = 2 ℓ α 2 . \mathbb{E}W_{ij}^{2}\le\frac{2(1-e^{-t_\alpha})}{N}\le\frac{2t_\alpha}{N}=2\ell_\alpha^{2}. E W ij 2 ≤ N 2 ( 1 − e − t α ) ≤ N 2 t α = 2 ℓ α 2 . We set
w 0 = N ε ℓ α . (213) w_0=N^\varepsilon\ell_\alpha. \tag*{(213)} w 0 = N ε ℓ α . (213) The standard Gaussian tail bound and a union bound over the N ( N + 1 ) / 2 N(N+1)/2 N ( N + 1 ) /2
P ( max i ≤ j ∣ W i j ∣ > w 0 ) ≤ 2 N 2 exp ( − N 2 ε 4 ) → 0. (214) \mathbb{P}\left(\max_{i\le j}|W_{ij}|>w_0\right)\le2N^2\exp\left(-\frac{N^{2\varepsilon}}{4}\right)\to0. \tag*{(214)} P ( i ≤ j max ∣ W ij ∣ > w 0 ) ≤ 2 N 2 exp ( − 4 N 2 ε ) → 0. (214) Clusters. Let K \mathcal{K} K y 1 ≤ ⋯ ≤ y N y_1\le\cdots\le y_N y 1 ≤ ⋯ ≤ y N i i i i + 1 i+1 i + 1 y i + 1 − y i < q y_{i+1}-y_i<q y i + 1 − y i < q q q q ℓ α \ell_\alpha ℓ α κ < α \kappa<\alpha κ < α N − 1 N^{-1} N − 1 K \mathcal{K} K H H H
Let K , K ′ ∈ K K,K'\in\mathcal{K} K , K ′ ∈ K max K < min K ′ \max K<\min K' max K < min K ′ max K \max K max K max K + 1 \max K+1 max K + 1 y y y
y min K ′ − y max K ≥ y max K + 1 − y max K ≥ q . (215) y_{\min K'}-y_{\max K}\ge y_{\max K+1}-y_{\max K}\ge q. \tag*{(215)} y m i n K ′ − y m a x K ≥ y m a x K + 1 − y m a x K ≥ q . (215) Let P 2 \mathcal{P}_2 P 2 K ∈ P 2 \mathcal{K} \in\mathcal{P}_2 K ∈ P 2 K = { i , i + 1 } \mathcal{K} = \{i,i+1\} K = { i , i + 1 } g K = g i g_{\mathcal{K}} = g_i g K = g i E K = E i E_{\mathcal{K}} = E_i E K = E i
B K = ( y i + W i i W i , i + 1 W i , i + 1 y i + 1 + W i + 1 , i + 1 ) , g ^ K = λ 2 ( B K ) − λ 1 ( B K ) . B_{\mathcal{K}} =
\begin{pmatrix}
y_i + W_{ii} & W_{i,i+1} \\
W_{i,i+1} & y_{i+1} + W_{i+1,i+1}
\end{pmatrix},
\qquad
\widehat{g}_{\mathcal{K}} = \lambda_2(B_{\mathcal{K}}) - \lambda_1(B_{\mathcal{K}}). B K = ( y i + W ii W i , i + 1 W i , i + 1 y i + 1 + W i + 1 , i + 1 ) , g K = λ 2 ( B K ) − λ 1 ( B K ) . Thus B K = ( D y + W ) K K B_{\mathcal{K}} = (D_y + W)_{\mathcal{K}\mathcal{K}} B K = ( D y + W ) KK p ≥ p ∗ p \ge p_* p ≥ p ∗ φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 ))
X ^ α ′ ( φ , p ) = ∑ K ∈ P 2 φ N ( E K ) f p ( g ^ K ℓ α ′ ) . (216) \widehat{\mathcal{X}}_{\alpha'}(\varphi,p)
=
\sum_{\mathcal{K}\in\mathcal{P}_2}
\varphi_N(E_{\mathcal{K}})
f_p\left(\frac{\widehat{g}_{\mathcal{K}}}{\ell_{\alpha'} }\right). \tag*{(216)} X α ′ ( φ , p ) = K ∈ P 2 ∑ φ N ( E K ) f p ( ℓ α ′ g K ) . (216) Lemma 6.8. There exists N 0 ≥ 1 N_0 \ge1 N 0 ≥ 1 N ≥ N 0 N \ge N_0 N ≥ N 0 H ∈ G ε / 2 H \in\mathcal{G}_{\varepsilon/2} H ∈ G ε /2 2 N ε q < N − 1 2N^\varepsilon q < N^{-1} 2 N ε q < N − 1 y 1 ≤ ⋯ ≤ y N y_1 \le\cdots\le y_N y 1 ≤ ⋯ ≤ y N [ − 3 , 3 ] [-3,3] [ − 3 , 3 ] N − 1 N^{-1} N − 1 N ε N^\varepsilon N ε R \mathcal{R} R N ε N^\varepsilon N ε i < j i<j i < j K ∈ R \mathcal{K}\in\mathcal{R} K ∈ R ∣ K ∣ ≥ 3 |\mathcal{K}|\ge3 ∣ K ∣ ≥ 3 2 N ε N 3 ( 4 q ) 2N^\varepsilon\mathcal{N}_3(4q) 2 N ε N 3 ( 4 q ) N 3 ( r ) \mathcal{N}_3(r) N 3 ( r )
lim N → ∞ 1 N α ′ E [ N ε N 3 ( 4 q ) ] = 0. \lim_{N\to\infty}\frac{1}{N^{\alpha'}}\mathbb{E}\left[N^\varepsilon\mathcal{N}_3(4q)\right]=0. N → ∞ lim N α ′ 1 E [ N ε N 3 ( 4 q ) ] = 0. Proof. Suppose that H ∈ G ε / 2 H \in\mathcal{G}_{\varepsilon/2} H ∈ G ε /2 y 1 ≤ ⋯ ≤ y N y_1 \le\cdots\le y_N y 1 ≤ ⋯ ≤ y N q q q ∥ H ∥ ≤ 3 \lVert H\rVert\le3 ∥ H ∥ ≤ 3 y i ∈ [ − 3 , 3 ] y_i \in[-3,3] y i ∈ [ − 3 , 3 ] i i i λ i \lambda_i λ i y i ∈ [ E , E + N − 1 ] y_i \in[E,E+N^{-1}] y i ∈ [ E , E + N − 1 ] e t α / 2 N − 1 ≤ 2 N − 1 e^{t\alpha/2}N^{-1} \le2N^{-1} e t α /2 N − 1 ≤ 2 N − 1 N − 1 N^{-1} N − 1 H ∈ G ε / 2 H \in\mathcal{G}_{\varepsilon/2} H ∈ G ε /2 N − 1 N^{-1} N − 1 2 N ε / 2 ≤ N ε 2N^{\varepsilon/2} \le N^\varepsilon 2 N ε /2 ≤ N ε y i y_i y i N ≥ N 0 N \ge N_0 N ≥ N 0 ε < κ \varepsilon< \kappa ε < κ
2 N ε q = 2 N ε − κ N − 1 < N − 1 2N^\varepsilon q = 2N^{\varepsilon-\kappa}N^{-1} < N^{-1} 2 N ε q = 2 N ε − κ N − 1 < N − 1 for N ≥ N 0 N \ge N_0 N ≥ N 0 N ε N^\varepsilon N ε
We next count the pairs i < j i<j i < j K \mathcal{K} K ∣ K ∣ ≥ 3 |\mathcal{K}|\ge3 ∣ K ∣ ≥ 3 N 3 ( r ) \mathcal{N}_3(r) N 3 ( r ) i ∈ [ N − 2 ] i\in[N-2] i ∈ [ N − 2 ] λ i + 2 − λ i ≤ r \lambda_{i+2}-\lambda_i\le r λ i + 2 − λ i ≤ r K ∈ R \mathcal{K}\in\mathcal{R} K ∈ R ∣ K ∣ ≥ 3 |\mathcal{K}|\ge3 ∣ K ∣ ≥ 3 ∣ K ∣ − 2 |\mathcal{K}|-2 ∣ K ∣ − 2 i i i i , i + 2 ∈ K i,i+2\in\mathcal{K} i , i + 2 ∈ K
λ i + 2 − λ i = e t α / 2 ( y i + 2 − y i ) < 2 e t α / 2 q ≤ 4 q . \lambda_{i+2}-\lambda_i
=
e^{t\alpha/2}(y_{i+2}-y_i)
<
2e^{t\alpha/2}q
\le4q. λ i + 2 − λ i = e t α /2 ( y i + 2 − y i ) < 2 e t α /2 q ≤ 4 q . These ∣ K ∣ − 2 |\mathcal{K}|-2 ∣ K ∣ − 2
∑ K ∈ R : ∣ K ∣ ≥ 3 ( ∣ K ∣ − 2 ) ≤ N 3 ( 4 q ) . \sum_{\substack{\mathcal{K}\in\mathcal{R}:\\|\mathcal{K}|\ge3}}
\left(|\mathcal{K}|-2\right)
\le\mathcal{N}_3(4q). K ∈ R : ∣ K ∣ ≥ 3 ∑ ( ∣ K ∣ − 2 ) ≤ N 3 ( 4 q ) . Recall that ∣ K ∣ ≤ N ε |\mathcal{K}|\le N^\varepsilon ∣ K ∣ ≤ N ε ∣ K ∣ ≤ 3 ( ∣ K ∣ − 2 ) |\mathcal{K}|\le3(|\mathcal{K}|-2) ∣ K ∣ ≤ 3 ( ∣ K ∣ − 2 ) ∣ K ∣ ≥ 3 |\mathcal{K}|\ge3 ∣ K ∣ ≥ 3
∑ K ∈ R : ∣ K ∣ ≥ 3 ∣ K ∣ 2 2 ≤ N ε 2 ∑ K ∈ R : ∣ K ∣ ≥ 3 3 ( ∣ K ∣ − 2 ) ≤ 2 N ε N 3 ( 4 q ) . (217) \sum_{\substack{\mathcal{K}\in\mathcal{R}:\\|\mathcal{K}|\ge3}}
\frac{|\mathcal{K}|^2}{2}
\le
\frac{N^\varepsilon}{2}
\sum_{\substack{\mathcal{K}\in\mathcal{R}:\\|\mathcal{K}|\ge3}}
3(|\mathcal{K}|-2)
\le2N^\varepsilon\mathcal{N}_3(4q). \tag*{(217)} K ∈ R : ∣ K ∣ ≥ 3 ∑ 2 ∣ K ∣ 2 ≤ 2 N ε K ∈ R : ∣ K ∣ ≥ 3 ∑ 3 ( ∣ K ∣ − 2 ) ≤ 2 N ε N 3 ( 4 q ) . (217) Finally, using Proposition 3.4 with r = 4 q ≥ N − 2 r = 4q \ge N^{-2} r = 4 q ≥ N − 2
1 M α ′ E [ N ε N 3 ( 4 q ) ] ≤ C N ε M α ′ N ( 4 N q ) 3 ≤ C N ε + 2 α ′ − 3 κ = C N ε − ( 3 α − 5 α ′ / 2 ) → 0 , (218) \frac{1}{M_{\alpha'}}\mathbb{E}\left[N^{\varepsilon}\mathcal{N}_3(4q)\right] \le\frac{CN^{\varepsilon}}{M_{\alpha'}}N(4Nq)^3 \le CN^{\varepsilon+2\alpha'-3\kappa}=CN^{\varepsilon-(3\alpha-5\alpha'/2)}\to0, \tag*{(218)} M α ′ 1 E [ N ε N 3 ( 4 q ) ] ≤ M α ′ C N ε N ( 4 N q ) 3 ≤ C N ε + 2 α ′ − 3 κ = C N ε − ( 3 α − 5 α ′ /2 ) → 0 , (218) where the last step follows from (6.25).
Lemma 6.9. Let p ≥ p ∗ p \ge p_* p ≥ p ∗ φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) X ^ α ′ ( φ , p ) \widehat{\mathcal{X}}_{\alpha'}(\varphi,p) X α ′ ( φ , p ) N 0 ≥ 1 N_0 \ge1 N 0 ≥ 1 o ( 1 ) o(1) o ( 1 ) N ≥ N 0 N \ge N_0 N ≥ N 0 H ∈ G ε / 2 H \in\mathcal{G}_{\varepsilon/2} H ∈ G ε /2 max i ≤ j ∣ W i j ∣ ≤ w 0 \max_{i\le j}|W_{ij}| \le w_0 max i ≤ j ∣ W ij ∣ ≤ w 0 W W W w 0 w_0 w 0
∣ X α ′ ( H t α ; φ N , f p ) − X ^ α ′ ( φ , p ) ∣ ≤ o ( 1 ) X ^ α ′ ( φ , p ) + 2 ∥ φ ∥ ∞ N ε N 3 ( 4 q ) + 4 p ∥ φ ∥ ∞ N 2 ( ℓ α ′ q ) 2 p , (219) \left|\mathcal{X}_{\alpha'}(H_{t_\alpha};\varphi_N,f_p)-\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)\right|
\le o(1)\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)+2\|\varphi\|_\infty N^\varepsilon\mathcal{N}_3(4q)+4^p\|\varphi\|_\infty N^2\left(\frac{\ell_{\alpha'}}{q}\right)^{2p}, \tag*{(219)} X α ′ ( H t α ; φ N , f p ) − X α ′ ( φ , p ) ≤ o ( 1 ) X α ′ ( φ , p ) + 2∥ φ ∥ ∞ N ε N 3 ( 4 q ) + 4 p ∥ φ ∥ ∞ N 2 ( q ℓ α ′ ) 2 p , (219) where N 3 ( r ) \mathcal{N}_3(r) N 3 ( r )
Proof. Suppose that the event holds and that N ≥ N 0 N \ge N_0 N ≥ N 0 y 1 ≤ ⋯ ≤ y N y_1 \le\cdots\le y_N y 1 ≤ ⋯ ≤ y N q q q W W W W W W w 0 = N ε ℓ α w_0=N^\varepsilon\ell_\alpha w 0 = N ε ℓ α N log N ≤ q − 1 N\log N\le q^{-1} N log N ≤ q − 1 N ≥ N 0 N\ge N_0 N ≥ N 0 α − κ = α ′ / 6 \alpha-\kappa=\alpha'/6 α − κ = α ′ /6
N ε w 0 ( q − 1 + N log N ) ≤ 2 N 2 ε ℓ α q − 1 = 2 N 2 ε − α ′ / 6 ≤ 1 48 N^\varepsilon w_0(q^{-1}+N\log N)\le2N^{2\varepsilon}\ell_\alpha q^{-1}=2N^{2\varepsilon-\alpha'/6}\le\frac{1}{48} N ε w 0 ( q − 1 + N log N ) ≤ 2 N 2 ε ℓ α q − 1 = 2 N 2 ε − α ′ /6 ≤ 48 1 for N ≥ N 0 N\ge N_0 N ≥ N 0 2 ε < α ′ / 6 2\varepsilon<\alpha'/6 2 ε < α ′ /6 y 1 ≤ ⋯ ≤ y N y_1\le\cdots\le y_N y 1 ≤ ⋯ ≤ y N q q q W W W
λ i + 1 ( H t α ) − λ i ( H t α ) ≥ q 2 (220) \lambda_{i+1}(H_{t_\alpha})-\lambda_i(H_{t_\alpha})\ge\frac{q}{2} \tag*{(220)} λ i + 1 ( H t α ) − λ i ( H t α ) ≥ 2 q (220) whenever i i i i + 1 i+1 i + 1 K = { i 0 + 1 , … , i 0 + ∣ K ∣ } \mathcal{K}=\{i_0+1,\ldots,i_0+|\mathcal{K}|\} K = { i 0 + 1 , … , i 0 + ∣ K ∣ } i ∈ [ ∣ K ∣ ] i\in[|\mathcal{K}|] i ∈ [ ∣ K ∣ ]
∣ λ i 0 + i ( H t α ) − λ i ( ( D y + W ) K K ) ∣ ≤ 24 N 2 ε w 0 2 ( q − 1 + N log N ) ≤ 48 N 4 ε ℓ α 2 q − 1 = 48 N 4 ε − α + 5 α ′ / 6 ℓ α ′ ≤ N − ε ℓ α ′ (221) \begin{aligned}
\left|\lambda_{i_0+i}(H_{t_\alpha})-\lambda_i\left((D_y+W)_{\mathcal{K}\mathcal{K}}\right)\right|
\le24N^{2\varepsilon}w_0^2(q^{-1}+N\log N)\le48N^{4\varepsilon}\ell_\alpha^2q^{-1} \\
&=48N^{4\varepsilon-\alpha+5\alpha'/6}\ell_{\alpha'}\le N^{-\varepsilon}\ell_{\alpha'} \tag*{(221)}
\end{aligned} ∣ λ i 0 + i ( H t α ) − λ i ( ( D y + W ) KK ) ∣ ≤ 24 N 2 ε w 0 2 ( q − 1 + N log N ) ≤ 48 N 4 ε ℓ α 2 q − 1 = 48 N 4 ε − α + 5 α ′ /6 ℓ α ′ ≤ N − ε ℓ α ′ (221) for N ≥ N 0 N\ge N_0 N ≥ N 0
ℓ α 2 q − 1 = N − 1 − 2 α + κ , − 1 − 2 α + κ = − 1 − α ′ + ( 5 α ′ / 6 − α ) , \ell_\alpha^2q^{-1}=N^{-1-2\alpha+\kappa},\qquad-1-2\alpha+\kappa=-1-\alpha'+(5\alpha'/6-\alpha), ℓ α 2 q − 1 = N − 1 − 2 α + κ , − 1 − 2 α + κ = − 1 − α ′ + ( 5 α ′ /6 − α ) , and in the last inequality, we used 5 ε < α − 5 α ′ / 6 5\varepsilon<\alpha-5\alpha'/6 5 ε < α − 5 α ′ /6
We now sort the pairs i < j i<j i < j X α ′ ( H t α ; φ N , f p ) \mathcal{X}_{\alpha'}(H_{t_\alpha};\varphi_N,f_p) X α ′ ( H t α ; φ N , f p ) i ∈ K i\in\mathcal{K} i ∈ K j ∈ K ′ j\in\mathcal{K}' j ∈ K ′ K \mathcal{K} K K ′ \mathcal{K}' K ′ max K < min K ′ \max\mathcal{K}<\min\mathcal{K}' max K < min K ′ max K \max\mathcal{K} max K max K + 1 \max\mathcal{K}+1 max K + 1
λ j ( H t α ) − λ i ( H t α ) ≥ λ max K + 1 ( H t α ) − λ max K ( H t α ) ≥ q 2 . \lambda_j(H_{t_\alpha})-\lambda_i(H_{t_\alpha}) \ge\lambda_{\max\mathcal{K}+1}(H_{t_\alpha})-\lambda_{\max\mathcal{K}}(H_{t_\alpha}) \ge\frac{q}{2}. λ j ( H t α ) − λ i ( H t α ) ≥ λ m a x K + 1 ( H t α ) − λ m a x K ( H t α ) ≥ 2 q . Hence
f p ( λ j ( H t α ) − λ i ( H t α ) ℓ α ′ ) ≤ ( 1 + q 2 4 ℓ α ′ 2 ) − p ≤ 4 p ( ℓ α ′ q ) 2 p . f_p\left(\frac{\lambda_j(H_{t_\alpha})-\lambda_i(H_{t_\alpha})}{\ell_{\alpha'}}\right) \le\left(1+\frac{q^2}{4\ell_{\alpha'}^2}\right)^{-p} \le4^p\left(\frac{\ell_{\alpha'}}{q}\right)^{2p}. f p ( ℓ α ′ λ j ( H t α ) − λ i ( H t α ) ) ≤ ( 1 + 4 ℓ α ′ 2 q 2 ) − p ≤ 4 p ( q ℓ α ′ ) 2 p . Using φ N ≤ ∥ φ ∥ ∞ \varphi_N \le\|\varphi\|_\infty φ N ≤ ∥ φ ∥ ∞ 4 p ∥ φ ∥ ∞ N 2 ( ℓ α ′ / q ) 2 p 4^p\|\varphi\|_\infty N^2(\ell_{\alpha'}/q)^{2p} 4 p ∥ φ ∥ ∞ N 2 ( ℓ α ′ / q ) 2 p
The second type consists of the pairs i < j i<j i < j K \mathcal{K} K ∣ K ∣ ≥ 3 |\mathcal{K}|\ge3 ∣ K ∣ ≥ 3 2 N ε N 3 ( 4 q ) 2N^\varepsilon\mathcal{N}_3(4q) 2 N ε N 3 ( 4 q ) ∥ φ ∥ ∞ \|\varphi\|_\infty ∥ φ ∥ ∞ f p ≤ 1 f_p\le1 f p ≤ 1
The third type consists of the pairs ( i , i + 1 ) (i,i+1) ( i , i + 1 ) { i , i + 1 } ∈ P 2 \{i,i+1\}\in\mathfrak{P}_2 { i , i + 1 } ∈ P 2 K = { i , i + 1 } \mathcal{K}=\{i,i+1\} K = { i , i + 1 }
∣ ( λ i + 1 ( H t α ) − λ i ( H t α ) ) − g ^ K ∣ ≤ 2 N − ε ℓ α ′ . \left|(\lambda_{i+1}(H_{t_\alpha})-\lambda_i(H_{t_\alpha}))-\widehat{g}_{\mathcal{K}}\right|\le2N^{-\varepsilon}\ell_{\alpha'}. ∣ ( λ i + 1 ( H t α ) − λ i ( H t α )) − g K ∣ ≤ 2 N − ε ℓ α ′ . The right side of (6.11) is at most 4 p 4p 4 p
∣ log f p ( λ i + 1 ( H t α ) − λ i ( H t α ) ℓ α ′ ) − log f p ( g ^ K ℓ α ′ ) ∣ ≤ 8 p N − ε . \left|\log f_p\left(\frac{\lambda_{i+1}(H_{t_\alpha})-\lambda_i(H_{t_\alpha})}{\ell_{\alpha'}}\right)-\log f_p\left(\frac{\widehat{g}_{\mathcal{K}}}{\ell_{\alpha'}}\right)\right|\le8pN^{-\varepsilon}. log f p ( ℓ α ′ λ i + 1 ( H t α ) − λ i ( H t α ) ) − log f p ( ℓ α ′ g K ) ≤ 8 p N − ε . We now turn to the midpoints. Observe that
λ 1 ( B K ) + λ 2 ( B K ) = Tr B K = y i + y i + 1 + W i i + W i + 1 , i + 1 . \lambda_1(B_{\mathcal{K}})+\lambda_2(B_{\mathcal{K}})=\operatorname{Tr} B_{\mathcal{K}}=y_i+y_{i+1}+W_{ii}+W_{i+1,i+1}. λ 1 ( B K ) + λ 2 ( B K ) = Tr B K = y i + y i + 1 + W ii + W i + 1 , i + 1 . Using this identity, (6.38), and the bound ∣ E K ∣ ≤ 3 |E_{\mathcal{K}}|\le3 ∣ E K ∣ ≤ 3
∣ λ i ( H t α ) + λ i + 1 ( H t α ) 2 − E K ∣ ≤ N − ε ℓ α ′ + w 0 + 3 ( 1 − e − t α / 2 ) ≤ 5 N ε ℓ α , \left|\frac{\lambda_i(H_{t_\alpha})+\lambda_{i+1}(H_{t_\alpha})}{2}-E_{\mathcal{K}}\right|\le N^{-\varepsilon}\ell_{\alpha'}+w_0+3(1-e^{-t_\alpha/2})\le5N^\varepsilon\ell_\alpha, 2 λ i ( H t α ) + λ i + 1 ( H t α ) − E K ≤ N − ε ℓ α ′ + w 0 + 3 ( 1 − e − t α /2 ) ≤ 5 N ε ℓ α , where in the last inequality we used ℓ α ′ ≤ ℓ α \ell_{\alpha'}\le\ell_\alpha ℓ α ′ ≤ ℓ α
1 − e − t α / 2 ≤ t α 2 ≤ ℓ α . 1-e^{-t_\alpha/2}\le\frac{t_\alpha}{2}\le\ell_\alpha. 1 − e − t α /2 ≤ 2 t α ≤ ℓ α . Part (2) of Lemma 5.1, applied at the point E K ∈ [ − 3 , 3 ] E_{\mathcal{K}}\in[-3,3] E K ∈ [ − 3 , 3 ] φ N \varphi_N φ N φ N ( E K ) \varphi_N(E_{\mathcal{K}}) φ N ( E K )
( exp ( 40 N ε ℓ α h N − 2 ) − 1 ) φ N ( E K ) = ( exp ( 40 N ε − 4 / 5 − α ) − 1 ) φ N ( E K ) . \left(\exp(40N^\varepsilon\ell_\alpha h_N^{-2})-1\right)\varphi_N(E_{\mathcal{K}})=\left(\exp(40N^{\varepsilon-4/5-\alpha})-1\right)\varphi_N(E_{\mathcal{K}}). ( exp ( 40 N ε ℓ α h N − 2 ) − 1 ) φ N ( E K ) = ( exp ( 40 N ε − 4/5 − α ) − 1 ) φ N ( E K ) . (This bound holds at every point of [ − 3 , 3 ] [-3,3] [ − 3 , 3 ] φ \varphi φ
Combining the last display with the bound on log f p \log f_p log f p
∣ φ N ( λ i ( H t α ) + λ i + 1 ( H t α ) 2 ) f p ( λ i + 1 ( H t α ) − λ i ( H t α ) ℓ α ′ ) − φ N ( E K ) f p ( g ^ K ℓ α ′ ) ∣ ≤ o ( 1 ) φ N ( E K ) f p ( g ^ K ℓ α ′ ) , \left|\varphi_N\left(\frac{\lambda_i(H_{t_\alpha})+\lambda_{i+1}(H_{t_\alpha})}{2}\right)f_p\left(\frac{\lambda_{i+1}(H_{t_\alpha})-\lambda_i(H_{t_\alpha})}{\ell_{\alpha'}}\right)-\varphi_N(E_{\mathcal K})f_p\left(\frac{\widehat{g}_{\mathcal K}}{\ell_{\alpha'}}\right)\right|
\le o(1)\,\varphi_N(E_{\mathcal K})f_p\left(\frac{\widehat{g}_{\mathcal K}}{\ell_{\alpha'}}\right), φ N ( 2 λ i ( H t α ) + λ i + 1 ( H t α ) ) f p ( ℓ α ′ λ i + 1 ( H t α ) − λ i ( H t α ) ) − φ N ( E K ) f p ( ℓ α ′ g K ) ≤ o ( 1 ) φ N ( E K ) f p ( ℓ α ′ g K ) , where the o ( 1 ) o(1) o ( 1 ) K \mathcal K K (219) .
Lemma 6.10. Suppose that P ( α ) \mathcal{P}(\alpha) P ( α ) p ≥ p ∗ p \ge p_* p ≥ p ∗ φ ∈ C ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty((-2,2)) φ ∈ C ∞ (( − 2 , 2 )) L L L (209) . Let X ^ α ′ ( φ , p ) \widehat{\mathcal X}_{\alpha'}(\varphi,p) X α ′ ( φ , p ) (216) . Then
1 M α ′ E [ X ^ α ′ ( φ , p ) ∣ H ] ⟶ L (222) \frac{1}{M_{\alpha'}}\mathbb{E}\left[\widehat{\mathcal X}_{\alpha'}(\varphi,p)\mid H\right]\longrightarrow L \tag*{(222)} M α ′ 1 E [ X α ′ ( φ , p ) ∣ H ] ⟶ L (222) in probability.
Proof. Let K ∈ P 2 \mathcal K \in\mathcal P_2 K ∈ P 2 K = { i , i + 1 } \mathcal K=\{i,i+1\} K = { i , i + 1 } g ^ K \widehat{g}_{\mathcal K} g K y i − y i + 1 = − e − t α / 2 g K y_i-y_{i+1}=-e^{-t_\alpha/2}g_{\mathcal K} y i − y i + 1 = − e − t α /2 g K (83) for the gap of a real symmetric 2 × 2 2\times2 2 × 2
g ^ K ℓ α = ∣ − e − t α / 2 g K ℓ α + 2 ℓ α ⋅ W i i − W i + 1 , i + 1 2 , 2 ℓ α W i , i + 1 ∣ . (223) \frac{\widehat{g}_{\mathcal K}}{\ell_\alpha}
=
\left|
-\frac{e^{-t_\alpha/2}g_{\mathcal K}}{\ell_\alpha}
+\frac{2}{\ell_\alpha}\cdot\frac{W_{ii}-W_{i+1,i+1}}{2},
\frac{2}{\ell_\alpha}W_{i,i+1}
\right|. \tag*{(223)} ℓ α g K = − ℓ α e − t α /2 g K + ℓ α 2 ⋅ 2 W ii − W i + 1 , i + 1 , ℓ α 2 W i , i + 1 . (223) By (212) , conditionally on H H H ( W i i − W i + 1 , i + 1 ) / 2 (W_{ii}-W_{i+1,i+1})/2 ( W ii − W i + 1 , i + 1 ) /2 W i , i + 1 W_{i,i+1} W i , i + 1 ( 1 − e − t α ) / N (1-e^{-t_\alpha})/N ( 1 − e − t α ) / N (223) are independent centered Gaussian random variables with variance v N v_N v N
v N = 4 ( 1 − e − t α ) N ℓ α 2 = 4 ( 1 − e − t α ) t α ∈ [ 4 − 2 t α , 4 ] . (224) v_N=\frac{4(1-e^{-t_\alpha})}{N\ell_\alpha^2}
=\frac{4(1-e^{-t_\alpha})}{t_\alpha}
\in[4-2t_\alpha,4]. \tag*{(224)} v N = N ℓ α 2 4 ( 1 − e − t α ) = t α 4 ( 1 − e − t α ) ∈ [ 4 − 2 t α , 4 ] . (224) Here we used that
1 − t 2 ≤ 1 − e − t t ≤ 1 for t ∈ ( 0 , 1 ] . 1-\frac{t}{2}\le\frac{1-e^{-t}}{t}\le1
\qquad\text{for }t\in(0,1]. 1 − 2 t ≤ t 1 − e − t ≤ 1 for t ∈ ( 0 , 1 ] . The law of a centered Gaussian vector in R 2 \mathbb R^2 R 2 v N I d 2 v_N I d_2 v N I d 2 H H H
g ^ K ℓ α = d ∣ G x ∣ , (225) \frac{\widehat{g}_{\mathcal K}}{\ell_\alpha}\mathrel{\mathop=^{\mathrm d}}|G_x|, \tag*{(225)} ℓ α g K = d ∣ G x ∣ , (225) where G x G_x G x Lemma 3.10 , with n = 2 n=2 n = 2
x = g K ℓ α , ϱ = e − t α / 2 , v = v N . x=\frac{g_{\mathcal K}}{\ell_\alpha},
\qquad
\varrho=e^{-t_\alpha/2},
\qquad
v=v_N. x = ℓ α g K , ϱ = e − t α /2 , v = v N . We now compute the conditional mean. Recall that
ℓ α ′ = N α − α ′ ℓ α , M α ′ = N 2 ( α − α ′ ) M α . (226) \ell_{\alpha'}=N^{\alpha-\alpha'}\ell_\alpha,
\qquad
M_{\alpha'}=N^{2(\alpha-\alpha')}M_\alpha. \tag*{(226)} ℓ α ′ = N α − α ′ ℓ α , M α ′ = N 2 ( α − α ′ ) M α . (226) Using the first relation and (225) , we find
E [ f p ( g ^ K ℓ α ′ ) | H ] = E f p ( ∣ G x ∣ N α − α ′ ) . \mathbb{E}\left[f_p\left(\frac{\widehat{g}_{\mathcal{K}}}{\ell_{\alpha'}}\right)\middle|H\right]=\mathbb{E}f_p\left(\frac{|G_x|}{N^{\alpha-\alpha'}}\right). E [ f p ( ℓ α ′ g K ) H ] = E f p ( N α − α ′ ∣ G x ∣ ) . Since ϱ ∈ [ 1 / 2 , 2 ] \varrho\in[1/2,2] ϱ ∈ [ 1/2 , 2 ] v N ∈ [ 2 , 4 ] v_N \in[2,4] v N ∈ [ 2 , 4 ] Lemma 3.10 with n = 2 n=2 n = 2 p > 3 / 2 p>3/2 p > 3/2 v 0 = 4 v_0=4 v 0 = 4 N α − α ′ ≤ 1 / 2 N^{\alpha-\alpha'}\leq1/2 N α − α ′ ≤ 1/2 δ \delta δ g 4 g_4 g 4 R 2 \mathbb{R}^2 R 2
g 4 ( x , 0 ) = ( 8 π ) − 1 e − x 2 / 8 , ∫ R 2 f p ( ∣ y ∣ ) d y = 2 π ∫ 0 ∞ r f p ( r ) d r = π p − 1 , g_4(x,0)=(8\pi)^{-1}e^{-x^2/8},\qquad\int_{\mathbb{R}^2}f_p(|y|)\,\mathrm{d}y=2\pi\int_0^\infty r f_p(r)\,\mathrm{d}r=\frac{\pi}{p-1}, g 4 ( x , 0 ) = ( 8 π ) − 1 e − x 2 /8 , ∫ R 2 f p ( ∣ y ∣ ) d y = 2 π ∫ 0 ∞ r f p ( r ) d r = p − 1 π , where the last equality follows from (184) . Therefore Lemma 3.10 gives
∣ N 2 ( α ′ − α ) E [ f p ( g ^ K ℓ α ′ ) | H ] − e − g K 2 / ( 8 ℓ α ′ 2 ) 8 ( p − 1 ) ∣ ≤ C ( N α − α ′ + ∣ e t α / 2 − 1 ∣ + ∣ v N − 4 ∣ ) f p ( g K ℓ α ) , \left|N^{2(\alpha'-\alpha)}\mathbb{E}\left[f_p\left(\frac{\widehat{g}_{\mathcal{K}}}{\ell_{\alpha'}}\right)\middle|H\right]-\frac{e^{-g_{\mathcal{K}}^2/(8\ell_{\alpha'}^2)}}{8(p-1)}\right|
\leq C\left(N^{\alpha-\alpha'}+\left|e^{t_\alpha/2}-1\right|+|v_N-4|\right)f_p\left(\frac{g_{\mathcal{K}}}{\ell_\alpha}\right), N 2 ( α ′ − α ) E [ f p ( ℓ α ′ g K ) H ] − 8 ( p − 1 ) e − g K 2 / ( 8 ℓ α ′ 2 ) ≤ C ( N α − α ′ + e t α /2 − 1 + ∣ v N − 4∣ ) f p ( ℓ α g K ) , and by (224) , the factor in parentheses on the right is o ( 1 ) o(1) o ( 1 ) K \mathcal{K} K
We multiply by φ N ( E K ) \varphi_N(E_{\mathcal{K}}) φ N ( E K ) K ∈ P 2 \mathcal{K}\in\mathcal{P}_2 K ∈ P 2 K ∈ P 2 \mathcal{K}\in\mathcal{P}_2 K ∈ P 2 X α ( H ; φ N , f p ) X_\alpha(H;\varphi_N,f_p) X α ( H ; φ N , f p ) (226) gives
∣ 1 M α ′ E [ I α ′ ( φ , p ) | H ] − 1 8 ( p − 1 ) M α ∑ K ∈ P 2 φ N ( E K ) e − g K 2 / ( 8 ℓ α 2 ) ∣ ≤ o ( 1 ) M α X α ( H ; φ N , f p ) . (227) \left|\frac{1}{M_{\alpha'}}\mathbb{E}\left[\mathcal{I}_{\alpha'}(\varphi,p)\middle|H\right]-\frac{1}{8(p-1)M_\alpha}\sum_{\mathcal{K}\in\mathcal{P}_2}\varphi_N(E_{\mathcal{K}})e^{-g_{\mathcal{K}}^2/(8\ell_\alpha^2)}\right|\leq\frac{o(1)}{M_\alpha}X_\alpha(H;\varphi_N,f_p). \tag*{(227)} M α ′ 1 E [ I α ′ ( φ , p ) ∣ H ] − 8 ( p − 1 ) M α 1 K ∈ P 2 ∑ φ N ( E K ) e − g K 2 / ( 8 ℓ α 2 ) ≤ M α o ( 1 ) X α ( H ; φ N , f p ) . (227) By P ( α ) \mathcal{P}(\alpha) P ( α )
We now evaluate the sum in (227) . The function y ↦ e − y 2 / 8 y\mapsto e^{-y^2/8} y ↦ e − y 2 /8
∫ 0 ∞ y e − y 2 / 8 d y = 4. \int_0^\infty ye^{-y^2/8}\,\mathrm{d}y=4. ∫ 0 ∞ y e − y 2 /8 d y = 4. So P ( α ) \mathcal{P}(\alpha) P ( α )
1 M α ∑ 1 ≤ i < j ≤ N φ N ( λ i + λ j 2 ) exp ( − ( λ j − λ i ) 2 8 ℓ α 2 ) ⟶ 4 ∫ − 2 2 φ ( E ) c ( E ) d E in probability . \frac{1}{M_\alpha}\sum_{1\leq i<j\leq N}\varphi_N\left(\frac{\lambda_i+\lambda_j}{2}\right)\exp\left(-\frac{(\lambda_j-\lambda_i)^2}{8\ell_\alpha^2}\right)\longrightarrow4\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E\quad\text{in probability}. M α 1 1 ≤ i < j ≤ N ∑ φ N ( 2 λ i + λ j ) exp ( − 8 ℓ α 2 ( λ j − λ i ) 2 ) ⟶ 4 ∫ − 2 2 φ ( E ) c ( E ) d E in probability . The sum on the left differs from the sum in (227) by the terms of the pairs i < j i<j i < j P 2 \mathcal{P}_2 P 2
Suppose that H ∈ G ε / 2 H\in\mathcal{G}_{\varepsilon/2} H ∈ G ε /2 N ≥ N 0 N\geq N_0 N ≥ N 0 P 2 \mathcal{P}_2 P 2 K \mathcal{K} K K ′ \mathcal{K}' K ′ i ∈ K i\in\mathcal{K} i ∈ K j ∈ K ′ j\in\mathcal{K}' j ∈ K ′ Lemma 6.8 , there are at most 2 N ε N 3 ( 4 q ) 2N^\varepsilon N_3(4q) 2 N ε N 3 ( 4 q ) (215) gives
λ j − λ i ≥ e t α / 2 ( y min K ′ − y max K ) ≥ q = N α ′ / 6 ℓ α ′ , \lambda_j-\lambda_i\geq e^{t_\alpha/2}(y_{\min\mathcal{K}'}-y_{\max\mathcal{K}})\geq q=N^{\alpha'}/6\ell_{\alpha'}, λ j − λ i ≥ e t α /2 ( y m i n K ′ − y m a x K ) ≥ q = N α ′ /6 ℓ α ′ , where we used that α − κ = α ′ / 6 \alpha-\kappa=\alpha'/6 α − κ = α ′ /6
2 ∥ φ ∥ ∞ N ε N 3 ( 4 q ) + ∥ φ ∥ ∞ N 2 exp ( − q 2 8 ℓ α 2 ) = 2 ∥ φ ∥ ∞ N ε N 3 ( 4 q ) + ∥ φ ∥ ∞ N 2 exp ( − N α ′ / 3 8 ) . 2\lVert\varphi\rVert_\infty N^\varepsilon\mathcal{N}_3(4q)+\lVert\varphi\rVert_\infty N^2\exp\left(-\frac{q^2}{8\ell_\alpha^2}\right)=2\lVert\varphi\rVert_\infty N^\varepsilon\mathcal{N}_3(4q)+\lVert\varphi\rVert_\infty N^2\exp\left(-\frac{N^{\alpha'/3}}{8}\right). 2 ∥ φ ∥ ∞ N ε N 3 ( 4 q ) + ∥ φ ∥ ∞ N 2 exp ( − 8 ℓ α 2 q 2 ) = 2 ∥ φ ∥ ∞ N ε N 3 ( 4 q ) + ∥ φ ∥ ∞ N 2 exp ( − 8 N α ′ /3 ) . Since M α ≥ M α ′ M_\alpha\ge M_{\alpha'} M α ≥ M α ′ M α M_{\alpha} M α (227) , and recalling the definition (209) of L L L (222) .
Lemma 6.11. Assume the hypotheses of Lemma 6.10. Then
1 M α ′ X ^ α ′ ( φ , p ) → L in probability . (228) \frac{1}{M_{\alpha'}}\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)\to L\qquad\text{in probability}. \tag*{(228)} M α ′ 1 X α ′ ( φ , p ) → L in probability . (228) Proof. Conditionally on H H H B K B_K B K K ∈ P 2 K\in\mathcal{P}_2 K ∈ P 2 P 2 \mathcal{P}_2 P 2 y y y H H H B K B_K B K W W W W i i W_{ii} W ii W i + 1 , i + 1 W_{i+1,i+1} W i + 1 , i + 1 W i , i + 1 W_{i,i+1} W i , i + 1 W W W H H H X ^ α ′ ( φ , p ) \widehat{\mathcal{X}}_{\alpha'}(\varphi,p) X α ′ ( φ , p ) 0 ≤ φ N ≤ ∥ φ ∥ ∞ 0\le\varphi_N\le\lVert\varphi\rVert_\infty 0 ≤ φ N ≤ ∥ φ ∥ ∞ 0 ≤ f p 2 ≤ f p ≤ 1 0\le f_p^2\le f_p\le1 0 ≤ f p 2 ≤ f p ≤ 1
Var ( X ^ α ′ ( φ , p ) ∣ H ) ≤ ∑ K ∈ P 2 φ N ( E K ) 2 E [ f p ( g ^ K ℓ α ) 2 | H ] ≤ ∥ φ ∥ ∞ E [ X ^ α ′ ( φ , p ) ∣ H ] . \operatorname{Var}\left(\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)\mid H\right)\le\sum_{K\in\mathcal{P}_2}\varphi_N(E_K)^2\mathbb{E}\left[f_p\left(\frac{\widehat{g}_K}{\ell_\alpha}\right)^2\middle|H\right]\le\lVert\varphi\rVert_\infty\mathbb{E}\left[\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)\mid H\right]. Var ( X α ′ ( φ , p ) ∣ H ) ≤ K ∈ P 2 ∑ φ N ( E K ) 2 E [ f p ( ℓ α g K ) 2 H ] ≤ ∥ φ ∥ ∞ E [ X α ′ ( φ , p ) ∣ H ] . Let ϖ > 0 \varpi>0 ϖ > 0
P ( ∣ X ^ α ′ ( φ , p ) − E [ X ^ α ′ ( φ , p ) ∣ H ] ∣ > ϖ M α ′ | H ) ≤ min { 1 , ∥ φ ∥ ∞ ϖ 2 M α ′ ⋅ 1 M α ′ E [ X ^ α ′ ( φ , p ) ∣ H ] } . \mathbb{P}\left(\left|\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)-\mathbb{E}\left[\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)\mid H\right]\right|>\varpi M_{\alpha'}\middle|H\right)\le\min\left\{1,\frac{\lVert\varphi\rVert_\infty}{\varpi^2M_{\alpha'}}\cdot\frac{1}{M_{\alpha'}}\mathbb{E}\left[\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)\mid H\right]\right\}. P ( X α ′ ( φ , p ) − E [ X α ′ ( φ , p ) ∣ H ] > ϖ M α ′ H ) ≤ min { 1 , ϖ 2 M α ′ ∥ φ ∥ ∞ ⋅ M α ′ 1 E [ X α ′ ( φ , p ) ∣ H ] } . By Lemma 6.10 and M α ′ → ∞ M_{\alpha'}\to\infty M α ′ → ∞ (222) , this implies (228) .
Proof of Proposition 6.7. Fix an integer p ≥ p ∗ p\ge p_* p ≥ p ∗ φ ∈ C ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty((-2,2)) φ ∈ C ∞ (( − 2 , 2 )) L L L (209) . By (208) , we have α ′ + 5 ε < 1 / 2 \alpha'+5\varepsilon<1/2 α ′ + 5 ε < 1/2
t α N 1 / 2 + 3 α ′ + 12 ε = N − 1 / 2 − 2 α + 3 α ′ + 12 ε → 0. t_{\alpha}N^{1/2+3\alpha'+12\varepsilon}=N^{-1/2-2\alpha+3\alpha'+12\varepsilon}\to0. t α N 1/2 + 3 α ′ + 12 ε = N − 1/2 − 2 α + 3 α ′ + 12 ε → 0. Hence Lemma 6.3, applied with α ′ \alpha' α ′ α \alpha α t N = t α t_N=t_{\alpha} t N = t α
1 M α ′ X α ′ ( H t α ; φ N , f p ) → L in probability . (229) \frac{1}{M_{\alpha'}}\mathcal{X}_{\alpha'}(H_{t_{\alpha}};\varphi_N,f_p)\to L\qquad\text{in probability}. \tag*{(229)} M α ′ 1 X α ′ ( H t α ; φ N , f p ) → L in probability . (229) The event that H ∈ G ε / 2 H\in\mathcal{G}_{\varepsilon/2} H ∈ G ε /2 max i ≤ j ∣ W i j ∣ ≤ w 0 \max_{i\le j}|W_{ij}|\le w_0 max i ≤ j ∣ W ij ∣ ≤ w 0 (214) . On this event, Lemma 6.9 gives
∣ 1 M α ′ X α ′ ( H t α ; φ N , f p ) − L ∣ ≤ ∣ 1 M α ′ X ^ α ′ ( φ , p ) − L ∣ + o ( 1 ) M α ′ X ^ α ′ ( φ , p ) + 2 ∥ φ ∥ ∞ N ε N 3 ( 4 q ) M α ′ + 4 p ∥ φ ∥ ∞ N 2 M α ′ ( ℓ α ′ q ) 2 p . \left|\frac{1}{M_{\alpha'}}\mathcal{X}_{\alpha'}(H_{t_{\alpha}};\varphi_N,f_p)-L\right|\le\left|\frac{1}{M_{\alpha'}}\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)-L\right|+\frac{o(1)}{M_{\alpha'}}\widehat{\mathcal{X}}_{\alpha'}(\varphi,p)+\frac{2\lVert\varphi\rVert_\infty N^\varepsilon\mathcal{N}_3(4q)}{M_{\alpha'}}+\frac{4^p\lVert\varphi\rVert_\infty N^2}{M_{\alpha'}}\left(\frac{\ell_{\alpha'}}{q}\right)^{2p}. M α ′ 1 X α ′ ( H t α ; φ N , f p ) − L ≤ M α ′ 1 X α ′ ( φ , p ) − L + M α ′ o ( 1 ) X α ′ ( φ , p ) + M α ′ 2 ∥ φ ∥ ∞ N ε N 3 ( 4 q ) + M α ′ 4 p ∥ φ ∥ ∞ N 2 ( q ℓ α ′ ) 2 p . We now bound the last term on the right side. Since M α ′ = N 1 − 2 α ′ M_{\alpha'} = N^{1-2\alpha'} M α ′ = N 1 − 2 α ′ ℓ α ′ / q = N κ − α ′ \ell_{\alpha'}/q = N^{\kappa-\alpha'} ℓ α ′ / q = N κ − α ′ N 1 + 2 α ′ − 2 p ( α ′ − κ ) N^{1+2\alpha'-2p(\alpha'-\kappa)} N 1 + 2 α ′ − 2 p ( α ′ − κ ) 2 p ( α ′ − κ ) > 1 + 2 α ′ 2p(\alpha'-\kappa) > 1+2\alpha' 2 p ( α ′ − κ ) > 1 + 2 α ′ Theorem 1.3 that uses the value p ∗ = 61 p_* = 61 p ∗ = 61 α ≥ 1 / 10 \alpha\ge1/10 α ≥ 1/10 α ′ − κ > α / 6 \alpha'-\kappa> \alpha/6 α ′ − κ > α /6
2 p ( α ′ − κ ) > p α 3 ≥ p 30 ≥ 61 30 > 1 + 2 α ′ , 2p(\alpha'-\kappa) > \frac{p\alpha}{3} \ge\frac{p}{30} \ge\frac{61}{30} > 1+2\alpha', 2 p ( α ′ − κ ) > 3 p α ≥ 30 p ≥ 30 61 > 1 + 2 α ′ , where we used α ≥ 1 / 10 \alpha\ge1/10 α ≥ 1/10 p ≥ p ∗ = 61 p \ge p_* = 61 p ≥ p ∗ = 61 α ′ < 1 / 2 \alpha' < 1/2 α ′ < 1/2
1 M α ′ N 2 ( ℓ α ′ q ) 2 p = N 1 + 2 α ′ − 2 p ( α ′ − κ ) ≤ N 1 + 2 α ′ − p / 30 ≤ N 2 − 61 / 30 → 0. (230) \frac{1}{M_{\alpha'}}N^2\left(\frac{\ell_{\alpha'}}{q}\right)^{2p}
= N^{1+2\alpha'-2p(\alpha'-\kappa)}
\le N^{1+2\alpha'-p/30}
\le N^{2-61/30} \to0. \tag*{(230)} M α ′ 1 N 2 ( q ℓ α ′ ) 2 p = N 1 + 2 α ′ − 2 p ( α ′ − κ ) ≤ N 1 + 2 α ′ − p /30 ≤ N 2 − 61/30 → 0. (230) By Lemma 6.8 and Markov’s inequality, the third term on the right side tends to zero in probability. Since P ( α ) \mathcal{P}(\alpha) P ( α )
Iteration and conclusion
We now iterate Proposition 6.7, starting from Proposition 6.5.
Corollary 6.12. For every α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) P ( α ) \mathcal{P}(\alpha) P ( α )
Proof. We define α ( 0 ) = 1 / 10 \alpha^{(0)} = 1/10 α ( 0 ) = 1/10
α ( n + 1 ) = α ( n ) + 1 10 min { α ( n ) , 1 2 − α ( n ) } for n ≥ 0. (231) \alpha^{(n+1)} = \alpha^{(n)} + \frac{1}{10}\min\left\{\alpha^{(n)},\frac{1}{2}-\alpha^{(n)}\right\}\qquad\text{for }n \ge0. \tag*{(231)} α ( n + 1 ) = α ( n ) + 10 1 min { α ( n ) , 2 1 − α ( n ) } for n ≥ 0. (231) By induction on n n n α ( n ) ∈ [ 1 / 10 , 1 / 2 ) \alpha^{(n)} \in[1/10,1/2) α ( n ) ∈ [ 1/10 , 1/2 ) n n n α ( n ) ∈ [ 1 / 10 , 1 / 2 ) \alpha^{(n)} \in[1/10,1/2) α ( n ) ∈ [ 1/10 , 1/2 ) α ( n + 1 ) > α ( n ) \alpha^{(n+1)} > \alpha^{(n)} α ( n + 1 ) > α ( n )
α ( n + 1 ) ≤ α ( n ) + 1 10 ( 1 2 − α ( n ) ) < 1 2 . \alpha^{(n+1)} \le\alpha^{(n)}+\frac{1}{10}\left(\frac{1}{2}-\alpha^{(n)}\right)<\frac{1}{2}. α ( n + 1 ) ≤ α ( n ) + 10 1 ( 2 1 − α ( n ) ) < 2 1 . The sequence ( α ( n ) ) (\alpha^{(n)}) ( α ( n ) ) 1 / 2 1/2 1/2 α ( ∞ ) ∈ ( 1 / 10 , 1 / 2 ] \alpha^{(\infty)} \in(1/10,1/2] α ( ∞ ) ∈ ( 1/10 , 1/2 ] min { α ( ∞ ) , 1 / 2 − α ( ∞ ) } = 0 \min\{\alpha^{(\infty)},1/2-\alpha^{(\infty)}\}=0 min { α ( ∞ ) , 1/2 − α ( ∞ ) } = 0 α ( n ) → 1 / 2 \alpha^{(n)}\to1/2 α ( n ) → 1/2
We claim that for all n ≥ 0 n\ge0 n ≥ 0 α ′ ∈ ( α ( n ) , α ( n + 1 ) ] \alpha'\in(\alpha^{(n)},\alpha^{(n+1)}] α ′ ∈ ( α ( n ) , α ( n + 1 ) ] ( α ( n ) , α ′ ) (\alpha^{(n)},\alpha') ( α ( n ) , α ′ ) 1 / 10 ≤ α ( n ) < α ′ ≤ α ( n + 1 ) < 1 / 2 1/10\le\alpha^{(n)}<\alpha'\le\alpha^{(n+1)}<1/2 1/10 ≤ α ( n ) < α ′ ≤ α ( n + 1 ) < 1/2
α ′ − α ( n ) ≤ α ( n ) 10 < α ( n ) 5 , α ′ − α ( n ) ≤ 1 10 ( 1 2 − α ( n ) ) < 1 3 ( 1 2 − α ( n ) ) , \alpha'-\alpha^{(n)}\le\frac{\alpha^{(n)}}{10}<\frac{\alpha^{(n)}}{5},\qquad
\alpha'-\alpha^{(n)}\le\frac{1}{10}\left(\frac{1}{2}-\alpha^{(n)}\right)<\frac{1}{3}\left(\frac{1}{2}-\alpha^{(n)}\right), α ′ − α ( n ) ≤ 10 α ( n ) < 5 α ( n ) , α ′ − α ( n ) ≤ 10 1 ( 2 1 − α ( n ) ) < 3 1 ( 2 1 − α ( n ) ) , so that
α ′ < 6 α ( n ) 5 , α ′ < α ( n ) + 1 6 − α ( n ) 3 = 1 6 + 2 α ( n ) 3 , \alpha'<\frac{6\alpha^{(n)}}{5},\qquad
\alpha'<\alpha^{(n)}+\frac{1}{6}-\frac{\alpha^{(n)}}{3}
=\frac{1}{6}+\frac{2\alpha^{(n)}}{3}, α ′ < 5 6 α ( n ) , α ′ < α ( n ) + 6 1 − 3 α ( n ) = 6 1 + 3 2 α ( n ) , as claimed.
We have α ( 0 ) < 1 / 8 \alpha^{(0)} < 1/8 α ( 0 ) < 1/8 P ( α ( 0 ) ) \mathcal{P}(\alpha^{(0)}) P ( α ( 0 ) ) P ( α ( n ) ) \mathcal{P}(\alpha^{(n)}) P ( α ( n ) ) ( α , α ′ ) = ( α ( n ) , α ( n + 1 ) ) (\alpha,\alpha') = (\alpha^{(n)},\alpha^{(n+1)}) ( α , α ′ ) = ( α ( n ) , α ( n + 1 ) ) P ( α ( n + 1 ) ) \mathcal{P}(\alpha^{(n+1)}) P ( α ( n + 1 ) ) P ( α ( n ) ) \mathcal{P}(\alpha^{(n)}) P ( α ( n ) ) n ≥ 0 n \ge0 n ≥ 0 α ′ ∈ ( 0 , 1 / 2 ) \alpha' \in(0,1/2) α ′ ∈ ( 0 , 1/2 ) α ′ ≤ α ( 0 ) \alpha' \le\alpha^{(0)} α ′ ≤ α ( 0 ) P ( α ′ ) \mathcal{P}(\alpha') P ( α ′ ) α ( n ) → 1 / 2 \alpha^{(n)} \to1/2 α ( n ) → 1/2 n ≥ 0 n \ge0 n ≥ 0 α ( n ) < α ′ ≤ α ( n + 1 ) \alpha^{(n)} < \alpha' \le\alpha^{(n+1)} α ( n ) < α ′ ≤ α ( n + 1 ) ( α , α ′ ) = ( α ( n ) , α ′ ) (\alpha,\alpha') = (\alpha^{(n)},\alpha') ( α , α ′ ) = ( α ( n ) , α ′ ) P ( α ′ ) \mathcal{P}(\alpha') P ( α ′ )
Proof of Theorem 1.3. Fix α ∈ ( 0 , 1 / 2 ) \alpha\in(0,1/2) α ∈ ( 0 , 1/2 ) φ \varphi φ ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) f : [ 0 , ∞ ) → R f : [0,\infty) \to\mathbb{R} f : [ 0 , ∞ ) → R N 2 α − 1 = M α − 1 N^{2\alpha-1} = M_{\alpha}^{-1} N 2 α − 1 = M α − 1 P ( α ) \mathcal{P}(\alpha) P ( α ) f f f R ≥ 1 R \ge1 R ≥ 1 supp f ⊂ [ 0 , R ] \operatorname{supp} f \subset[0,R] supp f ⊂ [ 0 , R ]
Suppose first that φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C_{\mathrm{c}}^\infty((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) X α ( H ; φ , f ) \mathcal{X}_{\alpha}(H;\varphi,f) X α ( H ; φ , f ) j = i + 1 j=i+1 j = i + 1 j ≥ i + 2 j \ge i+2 j ≥ i + 2
X α ( H ; φ , f ) = ∑ i = 1 N − 1 φ ( E i ) f ( g i ℓ α ) + ∑ 1 ≤ i , j ≤ N j ≥ i + 2 φ ( λ i + λ j 2 ) f ( λ j − λ i ℓ α ) . (232) \mathcal{X}_{\alpha}(H;\varphi,f)=\sum_{i=1}^{N-1}\varphi(E_i)f\left(\frac{g_i}{\ell_\alpha}\right)+\sum_{\substack{1\le i,j\le N\\j\ge i+2}}\varphi\left(\frac{\lambda_i+\lambda_j}{2}\right)f\left(\frac{\lambda_j-\lambda_i}{\ell_\alpha}\right). \tag*{(232)} X α ( H ; φ , f ) = i = 1 ∑ N − 1 φ ( E i ) f ( ℓ α g i ) + 1 ≤ i , j ≤ N j ≥ i + 2 ∑ φ ( 2 λ i + λ j ) f ( ℓ α λ j − λ i ) . (232) To show that the second sum is negligible, we count its nonzero terms. A term in this sum vanishes unless λ j − λ i ≤ R ℓ α \lambda_j-\lambda_i \le R\ell_\alpha λ j − λ i ≤ R ℓ α N ≥ N 0 N \ge N_0 N ≥ N 0 R ℓ α ≤ N − 1 R\ell_\alpha\le N^{-1} R ℓ α ≤ N − 1 H ∈ G α / 2 H \in\mathcal{G}_{\alpha/2} H ∈ G α /2 i i i N α / 2 N^{\alpha/2} N α /2 j j j λ j ∈ [ λ i , λ i + N − 1 ] \lambda_j \in[\lambda_i,\lambda_i+N^{-1}] λ j ∈ [ λ i , λ i + N − 1 ] j ≥ i + 2 j \ge i+2 j ≥ i + 2 λ j − λ i ≤ R ℓ α \lambda_j-\lambda_i \le R\ell_\alpha λ j − λ i ≤ R ℓ α λ i + 2 − λ i ≤ R ℓ α \lambda_{i+2}-\lambda_i \le R\ell_\alpha λ i + 2 − λ i ≤ R ℓ α
# { ( i , j ) : 1 ≤ i < j ≤ N , j ≥ i + 2 , λ j − λ i ≤ R ℓ α } ≤ N α / 2 N 3 ( R ℓ α ) . (233) \#\{(i,j):1\le i<j\le N,\ j\ge i+2,\ \lambda_j-\lambda_i\le R\ell_\alpha\}\le N^{\alpha/2}\mathcal{N}_3(R\ell_\alpha). \tag*{(233)} # {( i , j ) : 1 ≤ i < j ≤ N , j ≥ i + 2 , λ j − λ i ≤ R ℓ α } ≤ N α /2 N 3 ( R ℓ α ) . (233) Using Proposition 3.4 with r = R ℓ α ≥ N − 2 r=R\ell_\alpha\ge N^{-2} r = R ℓ α ≥ N − 2
1 M α E [ N α / 2 N 3 ( R ℓ α ) ] ≤ C N α / 2 M α N ( N R ℓ α ) 3 = C R 3 N − α / 2 → 0. \frac{1}{M_\alpha}\mathbb{E}\left[N^{\alpha/2}\mathcal{N}_3(R\ell_\alpha)\right]\le\frac{CN^{\alpha/2}}{M_\alpha}N(NR\ell_\alpha)^3=CR^3N^{-\alpha/2}\to0. M α 1 E [ N α /2 N 3 ( R ℓ α ) ] ≤ M α C N α /2 N ( N R ℓ α ) 3 = C R 3 N − α /2 → 0. The second sum in (6.49) is bounded by ∥ φ ∥ ∞ ∥ f ∥ ∞ \|\varphi\|_\infty\|f\|_\infty ∥ φ ∥ ∞ ∥ f ∥ ∞ o P ( M α ) o_{\mathbb{P}}(M_\alpha) o P ( M α ) φ \varphi φ
1 M α ∑ i = 1 N − 1 φ ( E i ) f ( g i ℓ α ) → ∫ − 2 2 φ ( E ) c ( E ) d E ∫ 0 ∞ y f ( y ) d y (234) \frac{1}{M_\alpha}\sum_{i=1}^{N-1}\varphi(E_i)f\left(\frac{g_i}{\ell_\alpha}\right)\to\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E\int_0^\infty yf(y)\,\mathrm{d}y \tag*{(234)} M α 1 i = 1 ∑ N − 1 φ ( E i ) f ( ℓ α g i ) → ∫ − 2 2 φ ( E ) c ( E ) d E ∫ 0 ∞ y f ( y ) d y (234) in probability.
Next, we treat a general continuous φ \varphi φ ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) φ \varphi φ ( − 2 , 2 ) (-2,2) ( − 2 , 2 ) φ \varphi φ φ ≥ 0 \varphi\ge0 φ ≥ 0
Let δ > 0 \delta>0 δ > 0 supp φ ⊂ [ − 2 + 2 δ , 2 − 2 δ ] \operatorname{supp}\varphi\subset[-2+2\delta,2-2\delta] supp φ ⊂ [ − 2 + 2 δ , 2 − 2 δ ] I = [ − 2 + δ , 2 − δ ] I=[-2+\delta,2-\delta] I = [ − 2 + δ , 2 − δ ] ϖ > 0 \varpi>0 ϖ > 0 φ \varphi φ [ − δ , δ ] [-\delta,\delta] [ − δ , δ ] φ ~ ∈ C c ∞ ( ( − 2 , 2 ) ) \widetilde{\varphi} \in C^\infty_c((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) supp φ ~ ⊂ I \operatorname{supp}\widetilde{\varphi} \subset I supp φ ⊂ I ∥ φ − φ ~ ∥ ∞ ≤ ϖ \|\varphi-\widetilde{\varphi}\|_\infty\le\varpi ∥ φ − φ ∥ ∞ ≤ ϖ
C f = sup y ≥ 0 ( 1 + y 2 ) p ∗ ∣ f ( y ) ∣ ≤ ( 1 + R 2 ) p ∗ ∥ f ∥ ∞ . C_f=\sup_{y\ge0}(1+y^2)^{p_*}|f(y)|\le(1+R^2)^{p_*}\|f\|_\infty. C f = y ≥ 0 sup ( 1 + y 2 ) p ∗ ∣ f ( y ) ∣ ≤ ( 1 + R 2 ) p ∗ ∥ f ∥ ∞ . The function φ − φ ~ \varphi-\widetilde{\varphi} φ − φ I I I
∣ ∑ i = 1 N − 1 ( φ − φ ~ ) ( E i ) f ( ξ i ℓ α ) ∣ ≤ ϖ C f ∑ i : E i ∈ I f p ∗ ( ξ i ℓ α ) ≤ ϖ C f ∑ 1 ≤ i < j ≤ N ( λ i + λ j ) / 2 ∈ I f p ∗ ( λ j − λ i ℓ α ) . \left|\sum_{i=1}^{N-1}(\varphi-\widetilde{\varphi})(E_i)f\left(\frac{\xi_i}{\ell_\alpha}\right)\right|
\le\varpi C_f\sum_{i:E_i\in I}f_{p_*}\left(\frac{\xi_i}{\ell_\alpha}\right)
\le\varpi C_f\sum_{\substack{1\le i<j\le N\\(\lambda_i+\lambda_j)/2\in I}}f_{p_*}\left(\frac{\lambda_j-\lambda_i}{\ell_\alpha}\right). i = 1 ∑ N − 1 ( φ − φ ) ( E i ) f ( ℓ α ξ i ) ≤ ϖ C f i : E i ∈ I ∑ f p ∗ ( ℓ α ξ i ) ≤ ϖ C f 1 ≤ i < j ≤ N ( λ i + λ j ) /2 ∈ I ∑ f p ∗ ( ℓ α λ j − λ i ) . By (203) , the right side is bounded by ϖ C f C I M α \varpi C_f C_I M_\alpha ϖ C f C I M α (234) for φ \varphi φ φ ~ \widetilde{\varphi} φ ϖ ∫ − 2 2 c ( E ) d E ∫ 0 ∞ y ∣ f ( y ) ∣ d y \varpi\int_{-2}^{2}c(E)\,\mathrm{d}E\int_0^\infty y|f(y)|\,\mathrm{d}y ϖ ∫ − 2 2 c ( E ) d E ∫ 0 ∞ y ∣ f ( y ) ∣ d y (234) to φ ~ \widetilde{\varphi} φ
∣ 1 M α ∑ i = 1 N − 1 φ ( E i ) f ( ξ i ℓ α ) − ∫ − 2 2 φ ( E ) c ( E ) d E ∫ 0 ∞ y f ( y ) d y ∣ ≤ ϖ ( 1 + C f C I + ∫ − 2 2 c ( E ) d E ∫ 0 ∞ y ∣ f ( y ) ∣ d y ) . \left|\frac{1}{M_\alpha}\sum_{i=1}^{N-1}\varphi(E_i)f\left(\frac{\xi_i}{\ell_\alpha}\right)-\int_{-2}^{2}\varphi(E)c(E)\,\mathrm{d}E\int_0^\infty yf(y)\,\mathrm{d}y\right|
\le\varpi\left(1+C_fC_I+\int_{-2}^{2}c(E)\,\mathrm{d}E\int_0^\infty y|f(y)|\,\mathrm{d}y\right). M α 1 i = 1 ∑ N − 1 φ ( E i ) f ( ℓ α ξ i ) − ∫ − 2 2 φ ( E ) c ( E ) d E ∫ 0 ∞ y f ( y ) d y ≤ ϖ ( 1 + C f C I + ∫ − 2 2 c ( E ) d E ∫ 0 ∞ y ∣ f ( y ) ∣ d y ) . The constant C I C_I C I ϖ \varpi ϖ ϖ > 0 \varpi>0 ϖ > 0 □ \square □
Proof of Proposition 4.1. Recall that α ∗ = 99 / 200 \alpha_* = 99/200 α ∗ = 99/200 ℓ = ℓ α ∗ \ell=\ell_{\alpha_*} ℓ = ℓ α ∗ M α ∗ = τ M_{\alpha_*}=\tau M α ∗ = τ P ( α ∗ ) \mathcal{P}(\alpha_*) P ( α ∗ ) (203) with α = α ∗ \alpha=\alpha_* α = α ∗
For the first assertion, let f ∈ C c ( [ 0 , ∞ ) × ( − 2 , 2 ) ) f\in C_c([0,\infty)\times(-2,2)) f ∈ C c ([ 0 , ∞ ) × ( − 2 , 2 )) R ≥ 1 R\ge1 R ≥ 1 δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) supp f ⊂ [ 0 , R ] × [ − 2 + 2 δ , 2 − 2 δ ] \operatorname{supp}f\subset[0,R]\times[-2+2\delta,2-2\delta] supp f ⊂ [ 0 , R ] × [ − 2 + 2 δ , 2 − 2 δ ] I = [ − 2 + δ , 2 − δ ] I=[-2+\delta,2-\delta] I = [ − 2 + δ , 2 − δ ] ϖ > 0 \varpi>0 ϖ > 0 f f f δ ′ ∈ ( 0 , δ ) \delta'\in(0,\delta) δ ′ ∈ ( 0 , δ ) ∣ f ( r , E ) − f ( r , E ′ ) ∣ ≤ ϖ |f(r,E)-f(r,E')|\le\varpi ∣ f ( r , E ) − f ( r , E ′ ) ∣ ≤ ϖ r ≥ 0 r\ge0 r ≥ 0 E , E ′ E,E' E , E ′ ∣ E − E ′ ∣ ≤ δ ′ |E-E'|\le\delta' ∣ E − E ′ ∣ ≤ δ ′
We approximate f f f χ 1 , … , χ n ∈ C c ∞ ( ( − 2 + δ , 2 − δ ) ) \chi_1,\ldots,\chi_n\in C^\infty_c((-2+\delta,2-\delta)) χ 1 , … , χ n ∈ C c ∞ (( − 2 + δ , 2 − δ )) supp χ k \operatorname{supp}\chi_k supp χ k δ ′ \delta' δ ′
∑ k = 1 n χ k ≤ 1 on R , ∑ k = 1 n χ k = 1 on [ − 2 + 2 δ , 2 − 2 δ ] . (235) \sum_{k=1}^{n}\chi_k\le1\quad\text{on }\mathbb{R},\qquad
\sum_{k=1}^{n}\chi_k=1\quad\text{on }[-2+2\delta,2-2\delta]. \tag*{(235)} k = 1 ∑ n χ k ≤ 1 on R , k = 1 ∑ n χ k = 1 on [ − 2 + 2 δ , 2 − 2 δ ] . (235) For each k k k E k ′ ∈ supp χ k E'_k\in\operatorname{supp}\chi_k E k ′ ∈ supp χ k
f ‾ ( r , E ) = ∑ k = 1 n χ k ( E ) f ( r , E k ′ ) . \overline{f}(r,E)=\sum_{k=1}^{n}\chi_k(E)f(r,E'_k). f ( r , E ) = k = 1 ∑ n χ k ( E ) f ( r , E k ′ ) . We claim that
∣ f ( r , E ) − f ‾ ( r , E ) ∣ ≤ ϖ ( 1 + R 2 ) p ∗ f p ∗ ( r ) 1 I ( E ) for all ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) . (236) |f(r,E)-\overline{f}(r,E)|\le\varpi(1+R^2)^{p_*}f_{p_*}(r)\mathbf{1}_I(E)\quad\text{for all }(r,E)\in[0,\infty)\times(-2,2). \tag*{(236)} ∣ f ( r , E ) − f ( r , E ) ∣ ≤ ϖ ( 1 + R 2 ) p ∗ f p ∗ ( r ) 1 I ( E ) for all ( r , E ) ∈ [ 0 , ∞ ) × ( − 2 , 2 ) . (236) When E ∈ [ − 2 + 2 δ , 2 − 2 δ ] E \in[-2+2\delta,2-2\delta] E ∈ [ − 2 + 2 δ , 2 − 2 δ ]
f ( r , E ) − f ˉ ( r , E ) = ∑ k = 1 n χ k ( E ) ( f ( r , E ) − f ( r , E k ′ ) ) . f(r,E)-\bar{f}(r,E)=\sum_{k=1}^{n}\chi_k(E)\bigl(f(r,E)-f(r,E'_k)\bigr). f ( r , E ) − f ˉ ( r , E ) = k = 1 ∑ n χ k ( E ) ( f ( r , E ) − f ( r , E k ′ ) ) . When E ∉ [ − 2 + 2 δ , 2 − 2 δ ] E \notin[-2+2\delta,2-2\delta] E ∈ / [ − 2 + 2 δ , 2 − 2 δ ] f ( r , E ) = 0 f(r,E)=0 f ( r , E ) = 0 χ k ( E ) ≠ 0 \chi_k(E)\ne0 χ k ( E ) = 0 ∣ E − E k ′ ∣ ≤ δ ′ |E-E'_k|\le\delta' ∣ E − E k ′ ∣ ≤ δ ′
∣ f ( r , E ) − f ˉ ( r , E ) ∣ ≤ ω ∑ k = 1 n χ k ( E ) ≤ ω . |f(r,E)-\bar{f}(r,E)|\le\omega\sum_{k=1}^{n}\chi_k(E)\le\omega. ∣ f ( r , E ) − f ˉ ( r , E ) ∣ ≤ ω k = 1 ∑ n χ k ( E ) ≤ ω . The function f − f ˉ f-\bar{f} f − f ˉ r ≤ R r\le R r ≤ R E ∈ I E\in I E ∈ I α = α ∗ \alpha=\alpha_* α = α ∗
1 τ ∣ ∑ 1 ≤ i < j ≤ N ( f − f ˉ ) ( λ j − λ i ℓ , λ i + λ j 2 ) ∣ ≤ ω ( 1 + R 2 ) p ∗ C I . \frac{1}{\tau}\left|\sum_{1\le i<j\le N}(f-\bar{f})\left(\frac{\lambda_j-\lambda_i}{\ell},\frac{\lambda_i+\lambda_j}{2}\right)\right|\le\omega(1+R^2)^{p_*}C_I. τ 1 1 ≤ i < j ≤ N ∑ ( f − f ˉ ) ( ℓ λ j − λ i , 2 λ i + λ j ) ≤ ω ( 1 + R 2 ) p ∗ C I . On the other hand, for each k k k r ↦ f ( r , E k ′ ) r\mapsto f(r,E'_k) r ↦ f ( r , E k ′ ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
∑ 1 ≤ i < j ≤ N f ˉ ( λ j − λ i ℓ , λ i + λ j 2 ) = ∑ k = 1 n ν α ∗ ( H ; χ k f ( ⋅ , E k ′ ) ) . \sum_{1\le i<j\le N}\bar{f}\left(\frac{\lambda_j-\lambda_i}{\ell},\frac{\lambda_i+\lambda_j}{2}\right)=\sum_{k=1}^{n}\nu_{\alpha_*}\bigl(H;\chi_k f(\cdot,E'_k)\bigr). 1 ≤ i < j ≤ N ∑ f ˉ ( ℓ λ j − λ i , 2 λ i + λ j ) = k = 1 ∑ n ν α ∗ ( H ; χ k f ( ⋅ , E k ′ ) ) . We apply Lemma 6.6 with α = α ∗ \alpha=\alpha_* α = α ∗ χ k \chi_k χ k
1 τ ∑ 1 ≤ i < j ≤ N f ˉ ( λ j − λ i ℓ , λ i + λ j 2 ) ⟶ ∑ k = 1 n ∫ − 2 2 χ k ( E ) c ( E ) d E ∫ 0 ∞ r f ( r , E k ′ ) d r = ∫ − 2 2 ∫ 0 ∞ c ( E ) r f ˉ ( r , E ) d r d E \begin{aligned}
\frac{1}{\tau}\sum_{1\le i<j\le N}\bar{f}\left(\frac{\lambda_j-\lambda_i}{\ell},\frac{\lambda_i+\lambda_j}{2}\right)\longrightarrow\sum_{k=1}^{n}\int_{-2}^{2}\chi_k(E)c(E)\,\mathrm{d}E\int_{0}^{\infty}r f(r,E'_k)\,\mathrm{d}r \\
&=\int_{-2}^{2}\int_{0}^{\infty}c(E)r\bar{f}(r,E)\,\mathrm{d}r\,\mathrm{d}E
\end{aligned} τ 1 1 ≤ i < j ≤ N ∑ f ˉ ( ℓ λ j − λ i , 2 λ i + λ j ) ⟶ k = 1 ∑ n ∫ − 2 2 χ k ( E ) c ( E ) d E ∫ 0 ∞ r f ( r , E k ′ ) d r = ∫ − 2 2 ∫ 0 ∞ c ( E ) r f ˉ ( r , E ) d r d E in probability. Finally, (6.53) gives
∣ ∫ − 2 2 ∫ 0 ∞ c ( E ) r ( f − f ˉ ) ( r , E ) d r d E ∣ ≤ ω R 2 ∫ − 2 2 c ( E ) d E . \left|\int_{-2}^{2}\int_{0}^{\infty}c(E)r(f-\bar{f})(r,E)\,\mathrm{d}r\,\mathrm{d}E\right|\le\omega R^2\int_{-2}^{2}c(E)\,\mathrm{d}E. ∫ − 2 2 ∫ 0 ∞ c ( E ) r ( f − f ˉ ) ( r , E ) d r d E ≤ ω R 2 ∫ − 2 2 c ( E ) d E . The constant C I C_I C I ω \omega ω ω > 0 \omega>0 ω > 0
Actual and linearized gaps
The goal of this section is to prove Proposition 4.5 , which compares the gaps of the resampled matrix H ~ \widetilde{H} H 2 d ∣ z i + S i ∣ 2d\lvert z_i+S_i\rvert 2 d ∣ z i + S i ∣ Lemma 4.3 that, when the selected positions are distinct, the linearized gap of the pair ( i , i + 1 ) (i,i+1) ( i , i + 1 ) H ~ \widetilde{H} H u i u_i u i u i + 1 u_{i+1} u i + 1 g i g_i g i λ i \lambda_i λ i λ i + 1 \lambda_{i+1} λ i + 1 H H H g ~ i \widetilde{g}_i g i H H H Δ \Delta Δ
The proof has two steps. In § 9.1 , we prove a deterministic statement, Corollary 7.2 , by applying Lemma 3.8 in the eigenbasis of H H H § 9.2 , we show that these hypotheses hold with probability tending to one.
Throughout this section, we set ε = 1 / 100 \varepsilon=1/100 ε = 1/100 d = N − 3 / 2 d=N^{-3/2} d = N − 3/2 τ = N 1 / 100 \tau=N^{1/100} τ = N 1/100 s = N − 7 / 5 s=N^{-7/5} s = N − 7/5 m = 2 ⌊ τ / 8 ⌋ ≤ τ / 4 m=2\lfloor\tau/8\rfloor\le\tau/4 m = 2 ⌊ τ /8 ⌋ ≤ τ /4
A deterministic comparison
In § 9.1 , H H H N × N N\times N N × N λ 1 ≤ ⋯ ≤ λ N \lambda_1\le\cdots\le\lambda_N λ 1 ≤ ⋯ ≤ λ N u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N R N \mathbb{R}^N R N H u j = λ j u j Hu_j=\lambda_j u_j H u j = λ j u j j ∈ [ N ] j\in[N] j ∈ [ N ] g i g_i g i E i E_i E i i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] L i L_i L i
Definition 7.1. Let R ( H ) \mathcal{R}(H) R ( H ) [ N ] [N] [ N ] j j j j + 1 j+1 j + 1 j ∈ [ N − 1 ] j\in[N-1] j ∈ [ N − 1 ] g j ≤ s g_j\le s g j ≤ s R ( H ) \mathcal{R}(H) R ( H ) clusters of H H H
Observe that every cluster is a set of consecutive integers, and that for every j ∈ [ N − 1 ] j\in[N-1] j ∈ [ N − 1 ] j j j j + 1 j+1 j + 1 g j ≤ s g_j\le s g j ≤ s R ( H ) \mathcal{R}(H) R ( H ) § 4.2 .
We let Λ = diag ( λ 1 , … , λ N ) \Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_N) Λ = diag ( λ 1 , … , λ N ) U = ( u 1 , … , u N ) U=(u_1,\ldots,u_N) U = ( u 1 , … , u N ) N × N N\times N N × N u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N H = U Λ U T H=U\Lambda U^{T} H = U Λ U T K ∈ R ( H ) K\in\mathcal{R}(H) K ∈ R ( H ) U K U_K U K N × ∣ K ∣ N\times\lvert K\rvert N × ∣ K ∣ u j u_j u j j ∈ K j\in K j ∈ K j j j Λ K = diag ( λ j : j ∈ K ) \Lambda_K=\operatorname{diag}(\lambda_j:j\in K) Λ K = diag ( λ j : j ∈ K ) N × N N\times N N × N A A A A K K = ( A j j ′ ) j , j ′ ∈ K A_{KK}=(A_{jj'})_{j,j'\in K} A K K = ( A j j ′ ) j , j ′ ∈ K § 4.2 .
Recall the quantities w N w_N w N L N \mathcal{L}_N L N δ N \delta_N δ N (90) . Since ε = 1 / 100 \varepsilon= 1/100 ε = 1/100
w N = τ N − 3 / 2 + 3 ε , L N = 2 s − 1 log N , δ N = 8 w N 2 L N . (237) w_N = \tau N^{-3/2+3\varepsilon}, \qquad\mathcal{L}_N = 2s^{-1}\log N, \qquad\delta_N = 8w_N^2\mathcal{L}_N. \tag*{(237)} w N = τ N − 3/2 + 3 ε , L N = 2 s − 1 log N , δ N = 8 w N 2 L N . (237) Inserting the values of τ \tau τ s s s d d d ε \varepsilon ε δ N / d = 8 ( w N / d ) ( w N L N ) \delta_N/d = 8(w_N/d)(w_N\mathcal{L}_N) δ N / d = 8 ( w N / d ) ( w N L N ) w N / d = τ N 3 ε w_N/d = \tau N^{3\varepsilon} w N / d = τ N 3 ε
w N s = N − 9 / 100 + 3 ε = N − 3 / 50 , w N L N = 2 N − 3 / 50 log N , δ N d = 16 N − 2 / 25 + 6 ε log N = 16 N − 1 / 50 log N . (238) \begin{aligned}
\frac{w_N}{s} &= N^{-9/100+3\varepsilon} = N^{-3/50}, \qquad w_N\mathcal{L}_N = 2N^{-3/50}\log N,\\
\frac{\delta_N}{d} &= 16N^{-2/25+6\varepsilon}\log N = 16N^{-1/50}\log N. \tag*{(238)}
\end{aligned} s w N d δ N = N − 9/100 + 3 ε = N − 3/50 , w N L N = 2 N − 3/50 log N , = 16 N − 2/25 + 6 ε log N = 16 N − 1/50 log N . (238) All three quantities in (238) tend to zero.
Corollary 7.2. There exists an absolute constant N 0 ≥ 1 N_0 \ge1 N 0 ≥ 1 N ≥ N 0 N \ge N_0 N ≥ N 0 H H H u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N
(i) ∥ H ∥ ≤ 3 \lVert H\rVert\le3 ∥ H ∥ ≤ 3
(ii) ∣ u j ( a ) ∣ ≤ N − 1 / 2 + ε |u_j(a)| \le N^{-1/2+\varepsilon} ∣ u j ( a ) ∣ ≤ N − 1/2 + ε a , j ∈ [ N ] a,j \in[N] a , j ∈ [ N ]
(iii) λ j + 2 − λ j > 4 s \lambda_{j+2}-\lambda_j > 4s λ j + 2 − λ j > 4 s j ∈ [ N − 2 ] j \in[N-2] j ∈ [ N − 2 ]
Let Δ \Delta Δ N × N N \times N N × N Δ a a = 0 \Delta_{aa}=0 Δ aa = 0 a ∈ [ N ] a \in[N] a ∈ [ N ] ∣ Δ a b ∣ ≤ 2 N − 1 / 2 + ε |\Delta_{ab}| \le2N^{-1/2+\varepsilon} ∣ Δ ab ∣ ≤ 2 N − 1/2 + ε a , b ∈ [ N ] a,b \in[N] a , b ∈ [ N ] Δ a b ≠ 0 \Delta_{ab} \ne0 Δ ab = 0 τ / 4 \tau/4 τ /4 ( a , b ) (a,b) ( a , b ) a < b a < b a < b H H H H H H { i , i + 1 } \{i,i+1\} { i , i + 1 } i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] g i ≤ s g_i \le s g i ≤ s
(a) Let K ∈ R ( H ) \mathcal{K} \in\mathcal{R}(H) K ∈ R ( H ) j ∈ K j \in\mathcal{K} j ∈ K λ min K − s / 4 ≤ λ j ( H + Δ ) ≤ λ max K + s / 4 \lambda_{\min\mathcal{K}} - s/4 \le\lambda_j(H+\Delta) \le\lambda_{\max\mathcal{K}} + s/4 λ m i n K − s /4 ≤ λ j ( H + Δ ) ≤ λ m a x K + s /4 λ j ( H + Δ ) \lambda_j(H+\Delta) λ j ( H + Δ ) j ∈ K j \in\mathcal{K} j ∈ K δ N / 2 \delta_N/2 δ N /2 Λ K + U K T Δ U K \Lambda_{\mathcal{K}} + U_{\mathcal{K}}^{T}\Delta U_{\mathcal{K}} Λ K + U K T Δ U K
(b) For all K ∈ R ( H ) \mathcal{K} \in\mathcal{R}(H) K ∈ R ( H )
∣ ∑ j ∈ K ( λ j ( H + Δ ) − λ j ) ∣ ≤ ∣ K ∣ ( w N + δ N / 2 ) . \left|\sum_{j\in\mathcal{K}}\left(\lambda_j(H+\Delta)-\lambda_j\right)\right| \le|\mathcal{K}|(w_N+\delta_N/2). j ∈ K ∑ ( λ j ( H + Δ ) − λ j ) ≤ ∣ K ∣ ( w N + δ N /2 ) . (c) For all j ∈ [ N − 1 ] j \in[N-1] j ∈ [ N − 1 ] g j > s g_j > s g j > s λ j + 1 ( H + Δ ) − λ j ( H + Δ ) > s / 2 \lambda_{j+1}(H+\Delta)-\lambda_j(H+\Delta) > s/2 λ j + 1 ( H + Δ ) − λ j ( H + Δ ) > s /2
(d) For all i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] g i ≤ s g_i \le s g i ≤ s
∣ λ i + 1 ( H + Δ ) − λ i ( H + Δ ) − 2 ∣ L i ( H + Δ ) ∣ ∣ ≤ δ N . \left|\lambda_{i+1}(H+\Delta)-\lambda_i(H+\Delta)-2|L_i(H+\Delta)|\right| \le\delta_N. ∣ λ i + 1 ( H + Δ ) − λ i ( H + Δ ) − 2∣ L i ( H + Δ ) ∣ ∣ ≤ δ N . Proof. We first describe the clusters. Suppose that some K ∈ R ( H ) \mathcal{K} \in\mathcal{R}(H) K ∈ R ( H ) K \mathcal{K} K j ∈ [ N − 2 ] j \in[N-2] j ∈ [ N − 2 ] j , j + 1 , j + 2 ∈ K j,j+1,j+2 \in\mathcal{K} j , j + 1 , j + 2 ∈ K g j ≤ s g_j \le s g j ≤ s g j + 1 ≤ s g_{j+1} \le s g j + 1 ≤ s λ j + 2 − λ j ≤ 2 s \lambda_{j+2}-\lambda_j \le2s λ j + 2 − λ j ≤ 2 s
If i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] g i ≤ s g_i \le s g i ≤ s i i i i + 1 i+1 i + 1 { i , i + 1 } \{i,i+1\} { i , i + 1 } { i , i + 1 } \{i,i+1\} { i , i + 1 } g i ≤ s g_i \le s g i ≤ s Δ \Delta Δ H H H
W = U T Δ U , so that W j j ′ = u j T Δ u j ′ W = U^{T}\Delta U,\qquad\text{so that}\qquad W_{jj'} = u_j^{T}\Delta u_{j'} W = U T Δ U , so that W j j ′ = u j T Δ u j ′ for j , j ′ ∈ [ N ] j,j' \in[N] j , j ′ ∈ [ N ]
Since U U U H + Δ = U ( Λ + W ) U T H+\Delta=U(\Lambda+W)U^{T} H + Δ = U ( Λ + W ) U T
λ j ( H + Δ ) = λ j ( Λ + W ) \lambda_j(H+\Delta)=\lambda_j(\Lambda+W) λ j ( H + Δ ) = λ j ( Λ + W ) for all j ∈ [ N ] j \in[N] j ∈ [ N ]
As Δ \Delta Δ
∣ W j j ′ ∣ = ∣ ∑ 1 ≤ a < b ≤ N Δ a b ( u j ( a ) u j ′ ( b ) + u j ( b ) u j ′ ( a ) ) ∣ ≤ τ 4 ⋅ 2 N − 1 / 2 + ε ⋅ 2 N − 1 + 2 ε = w N (239) \lvert W_{jj'}\rvert
=
\left\lvert
\sum_{1\leq a<b\leq N}
\Delta_{ab}\left(u_j(a)u_{j'}(b)+u_j(b)u_{j'}(a)\right)
\right\rvert
\leq\frac{\tau}{4}\cdot2N^{-1/2+\varepsilon}\cdot2N^{-1+2\varepsilon}
= w_N \tag*{(239)} ∣ W j j ′ ∣ = 1 ≤ a < b ≤ N ∑ Δ ab ( u j ( a ) u j ′ ( b ) + u j ( b ) u j ′ ( a ) ) ≤ 4 τ ⋅ 2 N − 1/2 + ε ⋅ 2 N − 1 + 2 ε = w N (239) for all j , j ′ ∈ [ N ] j,j' \in[N] j , j ′ ∈ [ N ] τ / 4 \tau/4 τ /4 Δ a b \Delta_{ab} Δ ab a < b a<b a < b 2 N − 1 / 2 + ε 2N^{-1/2+\varepsilon} 2 N − 1/2 + ε 2 N − 1 + 2 ε 2N^{-1+2\varepsilon} 2 N − 1 + 2 ε
We will apply Lemma 3.8 with D y = Λ D_y=\Lambda D y = Λ y j = λ j y_j=\lambda_j y j = λ j K ( H ) \mathcal{K}(H) K ( H )
k 0 = 2 , ϑ = s / 4 , w 0 = w N , L = L N . k_0=2,\qquad\vartheta=s/4,\qquad w_0=w_N,\qquad\mathcal{L}=\mathcal{L}_N. k 0 = 2 , ϑ = s /4 , w 0 = w N , L = L N . For K ∈ K ( H ) \mathcal{K}\in\mathcal{K}(H) K ∈ K ( H ) (59) is then
R K = { w ∈ C : λ min K − s / 4 ≤ Re w ≤ λ max K + s / 4 , ∣ Im w ∣ ≤ s / 4 } . \mathcal{R}_{\mathcal{K}}=
\left\{
w\in\mathbb{C}:
\lambda_{\min\mathcal{K}}-s/4\leq\operatorname{Re}w\leq\lambda_{\max\mathcal{K}}+s/4,\
\lvert\operatorname{Im}w\rvert\leq s/4
\right\}. R K = { w ∈ C : λ m i n K − s /4 ≤ Re w ≤ λ m a x K + s /4 , ∣ Im w ∣ ≤ s /4 } . We next check that these rectangles are pairwise disjoint. Let K , K ′ ∈ K ( H ) \mathcal{K},\mathcal{K}'\in\mathcal{K}(H) K , K ′ ∈ K ( H ) max K + 1 = min K ′ \max\mathcal{K}+1=\min\mathcal{K}' max K + 1 = min K ′ g max K > s g_{\max\mathcal{K}}>s g m a x K > s R K ∩ R \mathcal{R}_{\mathcal{K}}\cap\mathbb{R} R K ∩ R R K ′ ∩ R \mathcal{R}_{\mathcal{K}'}\cap\mathbb{R} R K ′ ∩ R g max K − s / 2 > s / 2 g_{\max\mathcal{K}}-s/2>s/2 g m a x K − s /2 > s /2
We now bound the harmonic sum in (60) . Fix K ∈ K ( H ) \mathcal{K}\in\mathcal{K}(H) K ∈ K ( H ) w ∈ R K w\in\mathcal{R}_{\mathcal{K}} w ∈ R K i − = min K i_-=\min\mathcal{K} i − = min K i + = max K i_+=\max\mathcal{K} i + = max K j > i + j>i_+ j > i + i + < N i_+<N i + < N i + i_+ i + i + + 1 i_++1 i + + 1 g i + > s g_{i_+}>s g i + > s
Let r ≥ 0 r\geq0 r ≥ 0 j ∈ { i + + 2 r + 1 , i + + 2 r + 2 } j\in\{i_++2r+1,i_++2r+2\} j ∈ { i + + 2 r + 1 , i + + 2 r + 2 } j ≤ N j\leq N j ≤ N
λ j − Re w ≥ λ i + + 2 r + 1 − λ i + − s / 4 ≥ g i + + 4 s r − s / 4 > 3 s / 4 + 4 s r . \lambda_j-\operatorname{Re}w
\geq\lambda_{i_++2r+1}-\lambda_{i_+}-s/4
\geq g_{i_+}+4sr-s/4
>3s/4+4sr. λ j − Re w ≥ λ i + + 2 r + 1 − λ i + − s /4 ≥ g i + + 4 sr − s /4 > 3 s /4 + 4 sr . The first inequality follows from λ j ≥ λ i + + 2 r + 1 \lambda_j\geq\lambda_{i_++2r+1} λ j ≥ λ i + + 2 r + 1 Re w ≤ λ i + + s / 4 \operatorname{Re}w\leq\lambda_{i_+}+s/4 Re w ≤ λ i + + s /4 r r r i + + 1 , i + + 3 , … , i + + 2 r − 1 i_++1,i_++3,\ldots,i_++2r-1 i + + 1 , i + + 3 , … , i + + 2 r − 1 λ j − Re w < 7 \lambda_j-\operatorname{Re}w<7 λ j − Re w < 7 r r r 4 s r < 7 4sr<7 4 sr < 7 r r r j j j ∣ λ j − w ∣ ≥ ∣ λ j − Re w ∣ \lvert\lambda_j-w\rvert\geq\lvert\lambda_j-\operatorname{Re}w\rvert ∣ λ j − w ∣ ≥ ∣ λ j − Re w ∣
∑ j = i + + 1 N 1 ∣ λ j − w ∣ ≤ ∑ 0 ≤ r < 7 / ( 4 s ) 2 3 s / 4 + 4 s r ≤ 8 3 s + 1 2 s ∑ 1 ≤ r < 7 / ( 4 s ) 1 r ≤ 1 s ( 19 6 + 1 2 log 7 4 s ) , \sum_{j=i_++1}^{N}\frac{1}{\lvert\lambda_j-w\rvert}
\leq
\sum_{0\leq r<7/(4s)}\frac{2}{3s/4+4sr}
\leq
\frac{8}{3s}+\frac{1}{2s}\sum_{1\leq r<7/(4s)}\frac{1}{r}
\leq
\frac{1}{s}\left(\frac{19}{6}+\frac{1}{2}\log\frac{7}{4s}\right), j = i + + 1 ∑ N ∣ λ j − w ∣ 1 ≤ 0 ≤ r < 7/ ( 4 s ) ∑ 3 s /4 + 4 sr 2 ≤ 3 s 8 + 2 s 1 1 ≤ r < 7/ ( 4 s ) ∑ r 1 ≤ s 1 ( 6 19 + 2 1 log 4 s 7 ) , where we compared the harmonic sum with an integral in the last step.
The same argument, with the roles of the two sides exchanged, bounds the sum over j < i − j < i_{-} j < i − s = N − 7 / 5 s = N^{-7/5} s = N − 7/5
∑ j ∉ K 1 ∣ λ j − w ∣ ≤ 1 s ( 19 3 + log 7 4 + 7 5 log N ) ≤ 2 log N s = L N (240) \sum_{j \notin\mathcal{K}} \frac{1}{\lvert\lambda_j-w\rvert} \leq\frac{1}{s}\left(\frac{19}{3}+\log\frac{7}{4}+\frac{7}{5}\log N\right) \leq\frac{2\log N}{s} = \mathcal{L}_N \tag*{(240)} j ∈ / K ∑ ∣ λ j − w ∣ 1 ≤ s 1 ( 3 19 + log 4 7 + 5 7 log N ) ≤ s 2 log N = L N (240) for N ≥ N 0 N \geq N_0 N ≥ N 0 K \mathcal{K} K w ∈ R K w \in\mathcal{R}_{\mathcal{K}} w ∈ R K Lemma 3.8 . By (239) , we have max j , j ′ ∣ W j j ′ ∣ ≤ w N \max_{j,j'}\lvert W_{jj'}\rvert\leq w_N max j , j ′ ∣ W j j ′ ∣ ≤ w N (238) , we see that w N L N ≤ 1 / 2 w_N\mathcal{L}_N \leq1/2 w N L N ≤ 1/2
1 2 ( k 0 w N + 2 k 0 w N 2 L N ) = 2 w N s + 4 w N s ⋅ w N L N = 2 N − 3 / 50 + 8 N − 3 / 25 log N < 1 4 \frac{1}{2}\left(k_0w_N+2k_0w_N^2\mathcal{L}_N\right)=\frac{2w_N}{s}+\frac{4w_N}{s}\cdot w_N\mathcal{L}_N=2N^{-3/50}+8N^{-3/25}\log N<\frac{1}{4} 2 1 ( k 0 w N + 2 k 0 w N 2 L N ) = s 2 w N + s 4 w N ⋅ w N L N = 2 N − 3/50 + 8 N − 3/25 log N < 4 1 for N ≥ N 0 N \geq N_0 N ≥ N 0 Lemma 3.8 hold.
Let K ∈ R ( H ) \mathcal{K} \in\mathcal{R}(H) K ∈ R ( H ) λ j ( Λ + W ) = λ j ( H + Δ ) \lambda_j(\Lambda+W)=\lambda_j(H+\Delta) λ j ( Λ + W ) = λ j ( H + Δ ) Lemma 3.8 that the numbers λ j ( H + Δ ) \lambda_j(H+\Delta) λ j ( H + Δ ) j ∈ K j\in\mathcal{K} j ∈ K k 0 w N + 2 k 0 w N 2 L N < s / 4 k_0w_N+2k_0w_N^2\mathcal{L}_N<s/4 k 0 w N + 2 k 0 w N 2 L N < s /4 [ λ min K , λ max K ] [\lambda_{\min}\mathcal{K},\lambda_{\max}\mathcal{K}] [ λ m i n K , λ m a x K ] 2 k 0 w N 2 L N = δ N / 2 2k_0w_N^2\mathcal{L}_N=\delta_N/2 2 k 0 w N 2 L N = δ N /2
( Λ + W ) K K = Λ K + U K T Δ U K . (\Lambda+W)_{\mathcal{K}\mathcal{K}}=\Lambda_{\mathcal{K}}+U_{\mathcal{K}}^{T}\Delta U_{\mathcal{K}}. ( Λ + W ) KK = Λ K + U K T Δ U K . This proves (a).
To prove (b), we use that the sum of the eigenvalues of a matrix is its trace. Then (a) implies that
∣ ∑ j ∈ K λ j ( H + Δ ) − Tr ( Λ K + U K T Δ U K ) ∣ ≤ ∣ K ∣ δ N 2 , Tr ( Λ K + U K T Δ U K ) = ∑ j ∈ K ( λ j + W j j ) . \left\lvert\sum_{j\in\mathcal{K}}\lambda_j(H+\Delta)-\operatorname{Tr}\left(\Lambda_{\mathcal{K}}+U_{\mathcal{K}}^{T}\Delta U_{\mathcal{K}}\right)\right\rvert\leq\frac{\lvert\mathcal{K}\rvert\delta_N}{2},\qquad\operatorname{Tr}\left(\Lambda_{\mathcal{K}}+U_{\mathcal{K}}^{T}\Delta U_{\mathcal{K}}\right)=\sum_{j\in\mathcal{K}}(\lambda_j+W_{jj}). j ∈ K ∑ λ j ( H + Δ ) − Tr ( Λ K + U K T Δ U K ) ≤ 2 ∣ K ∣ δ N , Tr ( Λ K + U K T Δ U K ) = j ∈ K ∑ ( λ j + W j j ) . Combining these relations with the bound ∣ W j j ∣ ≤ w N \lvert W_{jj}\rvert\leq w_N ∣ W j j ∣ ≤ w N (239) , we obtain (b).
Next, we prove (c). Let j ∈ [ N − 1 ] j\in[N-1] j ∈ [ N − 1 ] g j > s g_j>s g j > s j j j j + 1 j+1 j + 1 K \mathcal{K} K K ′ \mathcal{K}' K ′ j = max K j=\max\mathcal{K} j = max K j + 1 = min K ′ j+1=\min\mathcal{K}' j + 1 = min K ′ K \mathcal{K} K K ′ \mathcal{K}' K ′
λ j + 1 ( H + Δ ) − λ j ( H + Δ ) ≥ ( λ j + 1 − s / 4 ) − ( λ j + s / 4 ) = g j − s / 2 > s / 2. \lambda_{j+1}(H+\Delta)-\lambda_j(H+\Delta)\geq(\lambda_{j+1}-s/4)-(\lambda_j+s/4)=g_j-s/2>s/2. λ j + 1 ( H + Δ ) − λ j ( H + Δ ) ≥ ( λ j + 1 − s /4 ) − ( λ j + s /4 ) = g j − s /2 > s /2. It remains to prove (d). Let i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] g i ≤ s g_i\leq s g i ≤ s K = { i , i + 1 } \mathcal{K}=\{i,i+1\} K = { i , i + 1 } u i T H u i = λ i u_i^{T}Hu_i=\lambda_i u i T H u i = λ i u i + 1 T H u i + 1 = λ i + 1 u_{i+1}^{T}Hu_{i+1}=\lambda_{i+1} u i + 1 T H u i + 1 = λ i + 1 u i T H u i + 1 = 0 u_i^{T}Hu_{i+1}=0 u i T H u i + 1 = 0
Λ K + U K T Δ U K = ( u j T ( H + Δ ) u j ′ ) j , j ′ ∈ { i , i + 1 } . \Lambda_{\mathcal{K}}+U_{\mathcal{K}}^{T}\Delta U_{\mathcal{K}}=\left(u_j^{T}(H+\Delta)u_{j'}\right)_{j,j'\in\{i,i+1\}}. Λ K + U K T Δ U K = ( u j T ( H + Δ ) u j ′ ) j , j ′ ∈ { i , i + 1 } . By (4.10) and the definition of L i L_i L i 2 ∣ L i ( H + Δ ) ∣ 2\lvert L_i(H+\Delta)\rvert 2 ∣ L i ( H + Δ )∣ λ i ( H + Δ ) \lambda_i(H+\Delta) λ i ( H + Δ ) λ i + 1 ( H + Δ ) \lambda_{i+1}(H+\Delta) λ i + 1 ( H + Δ ) δ N / 2 \delta_N/2 δ N /2
Proof of Proposition 4.5
We now return to the random matrix H H H u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N Corollary 7.2 to their realizations. The clusters K ( H ) \mathcal{K}(H) K ( H ) Definition 7.1 for each realization of H H H
Proof of Proposition 4.5 . Let N 0 N_0 N 0 Corollary 7.2 . For N ≥ N 0 N\geq N_0 N ≥ N 0 A N \mathcal{A}_N A N
A N = { H ∈ G ε } ∩ { λ i + 2 − λ i > 4 s for all i ∈ [ N − 2 ] } ∩ { ∣ X a b ∣ ≤ N ε for all a , b ∈ [ N ] with a < b } , (241) \mathcal{A}_N=\{H\in\mathcal{G}_{\varepsilon}\}\cap\{\lambda_{i+2}-\lambda_i>4s\text{ for all }i\in[N-2]\}
\cap\{|X_{ab}|\leq N^{\varepsilon}\text{ for all }a,b\in[N]\text{ with }a<b\}, \tag*{(241)} A N = { H ∈ G ε } ∩ { λ i + 2 − λ i > 4 s for all i ∈ [ N − 2 ]} ∩ { ∣ X ab ∣ ≤ N ε for all a , b ∈ [ N ] with a < b } , (241) and for N < N 0 N<N_0 N < N 0 A N = ∅ \mathcal{A}_N=\varnothing A N = ∅ A N \mathcal{A}_N A N H H H H ∈ G ε H\in\mathcal{G}_{\varepsilon} H ∈ G ε H H H u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N B N \mathcal{B}_N B N
B N = { ( a l , b l ) ≠ ( a l ′ , b l ′ ) for all l , l ′ ∈ [ m ] with l ≠ l ′ } ∩ { ∣ Y l ∣ ≤ N ε for all l ∈ [ m ] } , (242) \mathcal{B}_N=\{(a_l,b_l)\neq(a_{l'},b_{l'})\text{ for all }l,l'\in[m]\text{ with }l\neq l'\}\cap\{|Y_l|\leq N^{\varepsilon}\text{ for all }l\in[m]\}, \tag*{(242)} B N = {( a l , b l ) = ( a l ′ , b l ′ ) for all l , l ′ ∈ [ m ] with l = l ′ } ∩ { ∣ Y l ∣ ≤ N ε for all l ∈ [ m ]} , (242) which is measurable with respect to the resampling variables. We will show that P ( A N ) → 1 \mathbb{P}(\mathcal{A}_N)\to1 P ( A N ) → 1 P ( B N ∣ H ) = 1 − o ( 1 ) \mathbb{P}(\mathcal{B}_N\mid H)=1-o(1) P ( B N ∣ H ) = 1 − o ( 1 ) H H H Proposition 4.5 hold on A N ∩ B N \mathcal{A}_N\cap\mathcal{B}_N A N ∩ B N N ≥ N 0 N\geq N_0 N ≥ N 0
We begin with A N \mathcal{A}_N A N Proposition 3.2 with t = 0 t=0 t = 0 H ∈ G ε H\in\mathcal{G}_{\varepsilon} H ∈ G ε C 1 = 1 C_1=1 C 1 = 1 4 s = 4 N − 7 / 5 ≥ N − 2 4s=4N^{-7/5}\geq N^{-2} 4 s = 4 N − 7/5 ≥ N − 2
P ( min i ∈ [ N − 2 ] ( λ i + 2 − λ i ) ≤ 4 s ) ≤ E N 3 ( 4 s ) ≤ C N ( 4 N s ) 3 = 64 C N − 1 / 5 . (243) \mathbb{P}\left(\min_{i\in[N-2]}(\lambda_{i+2}-\lambda_i)\leq4s\right)\leq\mathbb{E}\mathcal{N}_3(4s)\leq CN(4Ns)^3=64CN^{-1/5}. \tag*{(243)} P ( i ∈ [ N − 2 ] min ( λ i + 2 − λ i ) ≤ 4 s ) ≤ E N 3 ( 4 s ) ≤ C N ( 4 N s ) 3 = 64 C N − 1/5 . (243) Note that (243) covers all i ∈ [ N − 2 ] i\in[N-2] i ∈ [ N − 2 ] Lemma 3.8 is applied to the whole spectrum.
Finally, the standardized entries X a b X_{ab} X ab a < b a<b a < b X X X N 2 N^2 N 2
P ( ∣ X a b ∣ > N ε for some a , b ∈ [ N ] with a < b ) ≤ C 0 N 2 e − c 0 N 2 ε . \mathbb{P}\left(|X_{ab}|>N^{\varepsilon}\text{ for some }a,b\in[N]\text{ with }a<b\right)\leq C_0N^2e^{-c_0N^{2\varepsilon}}. P ( ∣ X ab ∣ > N ε for some a , b ∈ [ N ] with a < b ) ≤ C 0 N 2 e − c 0 N 2 ε . Combining these three bounds, we obtain
P ( A N c ) ≤ C N − 1 / 5 . \mathbb{P}(\mathcal{A}_N^c)\leq CN^{-1/5}. P ( A N c ) ≤ C N − 1/5 . Next, we bound the conditional probability of B N c \mathcal{B}_N^c B N c Y l Y_l Y l H H H P ( B N ∣ H ) = P ( B N ) \mathbb{P}(\mathcal{B}_N\mid H)=\mathbb{P}(\mathcal{B}_N) P ( B N ∣ H ) = P ( B N ) l , l ′ ∈ [ m ] l,l'\in[m] l , l ′ ∈ [ m ] ( a l , b l ) (a_l,b_l) ( a l , b l ) ( a l ′ , b l ′ ) (a_{l'},b_{l'}) ( a l ′ , b l ′ ) N ( N − 1 ) / 2 N(N-1)/2 N ( N − 1 ) /2 2 / ( N ( N − 1 ) ) 2/(N(N-1)) 2/ ( N ( N − 1 )) m 2 / 2 m^2/2 m 2 /2 { l , l ′ } \{l,l'\} { l , l ′ } l ∈ [ m ] l \in[m] l ∈ [ m ] m ≤ τ / 4 m \le\tau/4 m ≤ τ /4
P ( B N c ∣ H ) ≤ m 2 N ( N − 1 ) + m C 0 e − c 0 N 2 ε ≤ τ 2 8 N 2 + C 0 τ e − c 0 N 2 ε . (244) \mathbb{P}(\mathcal{B}_N^c \mid H) \le\frac{m^2}{N(N-1)} + mC_0e^{-c_0N^{2\varepsilon}} \le\frac{\tau^2}{8N^2} + C_0\tau e^{-c_0N^{2\varepsilon}}. \tag*{(244)} P ( B N c ∣ H ) ≤ N ( N − 1 ) m 2 + m C 0 e − c 0 N 2 ε ≤ 8 N 2 τ 2 + C 0 τ e − c 0 N 2 ε . (244) The right side of (244) is deterministic and tends to zero.
We now fix a realization in A N ∩ B N \mathcal{A}_N \cap\mathcal{B}_N A N ∩ B N Corollary 7.2 . As H ∈ G ε H \in\mathcal{G}_{\varepsilon} H ∈ G ε ∥ H ∥ ≤ 3 \lVert H\rVert\le3 ∥ H ∥ ≤ 3 H H H u j u_j u j ∥ u j ∥ ∞ ≤ N − 1 / 2 + ε \lVert u_j\rVert_{\infty} \le N^{-1/2+\varepsilon} ∥ u j ∥ ∞ ≤ N − 1/2 + ε A N \mathcal{A}_N A N B N \mathcal{B}_N B N Lemma 4.3 gives
Δ = ∑ l = 1 m N − 1 / 2 ( Y l − X a l b l ) V a l b l . \Delta= \sum_{l=1}^{m}N^{-1/2}(Y_l-X_{a_lb_l})V^{a_lb_l}. Δ = l = 1 ∑ m N − 1/2 ( Y l − X a l b l ) V a l b l . Hence Δ \Delta Δ Δ a b ≠ 0 \Delta_{ab} \ne0 Δ ab = 0 m ≤ τ / 4 m \le\tau/4 m ≤ τ /4 ( a , b ) (a,b) ( a , b ) a < b a<b a < b
∣ Δ a l b l ∣ ≤ N − 1 / 2 ( ∣ Y l ∣ + ∣ X a l b l ∣ ) ≤ 2 N − 1 / 2 + ε for all l ∈ [ m ] , \lvert\Delta_{a_lb_l}\rvert\le N^{-1/2}(\lvert Y_l\rvert+\lvert X_{a_lb_l}\rvert) \le2N^{-1/2+\varepsilon} \qquad\text{for all } l \in[m], ∣ Δ a l b l ∣ ≤ N − 1/2 (∣ Y l ∣ + ∣ X a l b l ∣) ≤ 2 N − 1/2 + ε for all l ∈ [ m ] , where we used the bounds on Y l Y_l Y l X a l b l X_{a_lb_l} X a l b l B N \mathcal{B}_N B N A N \mathcal{A}_N A N Corollary 7.2 applies to H H H Δ \Delta Δ H + Δ = H ~ H+\Delta=\widetilde{H} H + Δ = H
By Corollary 7.2 , every cluster of H H H { i , i + 1 } \{i,i+1\} { i , i + 1 } g i ≤ s g_i \le s g i ≤ s Corollary 7.2 gives g ~ j > s / 2 \widetilde{g}_j>s/2 g j > s /2 j ∈ [ N − 1 ] j \in[N-1] j ∈ [ N − 1 ] g j > s g_j>s g j > s
Now let i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] g i ≤ s g_i \le s g i ≤ s { i , i + 1 } \{i,i+1\} { i , i + 1 } Lemma 4.3 , we have L i ( H ~ ) = d ( z i + S i ) L_i(\widetilde{H})=d(z_i+S_i) L i ( H ) = d ( z i + S i ) Corollary 7.2 , statement (b) with K = { i , i + 1 } \mathcal{K}=\{i,i+1\} K = { i , i + 1 } (238) therefore give
∣ g ~ i d − 2 ∣ z i + S i ∣ ∣ ≤ δ N d = 16 N − 1 / 50 log N , ∣ E ~ i − E i ∣ ≤ w N + δ N 2 . (245) \left\lvert\frac{\widetilde{g}_i}{d}-2\lvert z_i+S_i\rvert\right\rvert\le\frac{\delta_N}{d}=16N^{-1/50}\log N,\qquad\lvert\widetilde{E}_i-E_i\rvert\le w_N+\frac{\delta_N}{2}. \tag*{(245)} d g i − 2 ∣ z i + S i ∣ ≤ d δ N = 16 N − 1/50 log N , ∣ E i − E i ∣ ≤ w N + 2 δ N . (245) Both bounds in (245) are deterministic and tend to zero by (238) . Together with P ( A N ) → 1 \mathbb{P}(\mathcal{A}_N)\to1 P ( A N ) → 1 (244) , this establishes all assertions of Proposition 4.5 .
A local limit theorem for the linearized increments
The goal of this section is to prove Proposition 4.4 , whose normalization is explained heuristically in § 3.2 . Recall from Section 4 that, for a k k k i \mathbf{i} i S i S_{\mathbf{i}} S i m = 2 ⌊ τ / 8 ⌋ m=2\lfloor\tau/8\rfloor m = 2 ⌊ τ /8 ⌋ H H H S i S_{\mathbf{i}} S i τ 1 / 2 \tau^{1/2} τ 1/2 Proposition 4.4 concerns the probability that S i + x S_{\mathbf{i}}+x S i + x [67 ] . Here the law of an increment depends on N N N H H H H H H
This is a genuine issue when X X X X X X 1 1 1 − 1 -1 − 1 Y − X a b ∈ { − 2 , 0 , 2 } Y-X_{ab}\in\{-2,0,2\} Y − X ab ∈ { − 2 , 0 , 2 } u i u_i u i u i + 1 u_{i+1} u i + 1 N − 1 / 2 N^{-1/2} N − 1/2 D a b ( i ) D_{ab}^{(i)} D ab ( i ) { ( ± 2 , 0 ) , ( 0 , ± 2 ) } \{(\pm2,0),(0,\pm2)\} {( ± 2 , 0 ) , ( 0 , ± 2 )} S i S_i S i 4 Z 2 4\mathbb{Z}^2 4 Z 2 Proposition 4.4 would then fail for test functions supported near ( 2 , 2 ) (2,2) ( 2 , 2 ) G ε \mathcal{G}_{\varepsilon} G ε
After computing the moments of one increment in § 10.1 , we show in § 10.2 that with overwhelming probability every unit eigenvector of H H H ℓ 2 \ell^2 ℓ 2 τ \tau τ § 10.3 , we show by a deterministic argument that this property prevents the phases in the characteristic function of one increment from concentrating near 2 π Z 2\pi\mathbb{Z} 2 π Z § 10.4 then bounds the squared modulus of this characteristic function away from 1 1 1 N N N § 10.5 , this bound reads 1 − c τ − 1 / 4 1-c\tau^{-1/4} 1 − c τ − 1/4 Lemma 8.13 , which we prove in § 10.5 by Fourier inversion. Finally, in § 10.6 we add a large deviation estimate for pairs with large gaps and complete the proof of Proposition 4.4 . Throughout this section, we fix an integer k ≥ 1 k\ge1 k ≥ 1
Notation and moments of one increment
Throughout Section 8, i = ( i 1 , … , i k ) \mathbf{i}=(i_1,\ldots,i_k) i = ( i 1 , … , i k ) k k k [ N − 1 ] [N-1] [ N − 1 ] i j + 1 ≥ i j + 2 i_{j+1}\ge i_j+2 i j + 1 ≥ i j + 2 j ∈ [ k − 1 ] j\in[k-1] j ∈ [ k − 1 ] w ∈ R 2 k w\in\mathbb{R}^{2k} w ∈ R 2 k w j = ( w j , 1 , w j , 2 ) ∈ R 2 w_j=(w_{j,1},w_{j,2})\in\mathbb{R}^2 w j = ( w j , 1 , w j , 2 ) ∈ R 2 j j j w = ( w 1 , … , w k ) w=(w_1,\ldots,w_k) w = ( w 1 , … , w k ) w ( η ) w(\eta) w ( η ) η \eta η η ∈ [ 2 k ] \eta\in[2k] η ∈ [ 2 k ] U i U_i U i N × 2 k N\times2k N × 2 k u i 1 , u i 1 + 1 , … , u i k , u i k + 1 u_{i_1},u_{i_1+1},\ldots,u_{i_k},u_{i_k+1} u i 1 , u i 1 + 1 , … , u i k , u i k + 1 2 k 2k 2 k U i T U i = Id 2 k U_i^{T}U_i=\operatorname{Id}_{2k} U i T U i = Id 2 k b ∈ [ N ] b\in[N] b ∈ [ N ] r b ∈ R 2 k r_b\in\mathbb{R}^{2k} r b ∈ R 2 k b b b N 1 / 2 U i N^{1/2}U_i N 1/2 U i r b , j = N 1 / 2 ( u i j ( b ) , u i j + 1 ( b ) ) r_{b,j}=N^{1/2}(u_{i_j}(b),u_{i_j+1}(b)) r b , j = N 1/2 ( u i j ( b ) , u i j + 1 ( b )) U i T U i = Id 2 k U_i^{T}U_i=\operatorname{Id}_{2k} U i T U i = Id 2 k
1 N ∑ b = 1 N r b T r b = Id 2 k . (246) \frac{1}{N}\sum_{b=1}^{N}r_b^{T}r_b=\operatorname{Id}_{2k}. \tag*{(246)} N 1 b = 1 ∑ N r b T r b = Id 2 k . (246) For ζ ∈ R 2 k \zeta\in\mathbb{R}^{2k} ζ ∈ R 2 k Γ ( ζ ) \Gamma(\zeta) Γ ( ζ ) 2 k × 2 k 2k\times2k 2 k × 2 k j j j
\begin{pmatrix}
\zeta_{j,1} & \zeta_{j,2}\\
\zeta_{j,2} & -\zeta_{j,1}
\end{pmatrix}. \tag*{(247)}\end{pmatrix}
For k = 1 k=1 k = 1 Γ ( ζ j ) \Gamma(\zeta_j) Γ ( ζ j ) ζ j ∈ R 2 \zeta_j\in\mathbb{R}^2 ζ j ∈ R 2 Γ ( ζ ) = diag ( Γ ( ζ 1 ) , … , Γ ( ζ k ) ) \Gamma(\zeta)=\operatorname{diag}(\Gamma(\zeta_1),\ldots,\Gamma(\zeta_k)) Γ ( ζ ) = diag ( Γ ( ζ 1 ) , … , Γ ( ζ k )) Γ ( ζ j ) \Gamma(\zeta_j) Γ ( ζ j ) ∣ ζ j ∣ |\zeta_j| ∣ ζ j ∣ − ∣ ζ j ∣ -|\zeta_j| − ∣ ζ j ∣ ∥ Γ ( ζ ) ∥ = max j ∣ ζ j ∣ ≤ ∣ ζ ∣ \|\Gamma(\zeta)\|=\max_j|\zeta_j|\le|\zeta| ∥Γ ( ζ ) ∥ = max j ∣ ζ j ∣ ≤ ∣ ζ ∣ Γ ( ζ j ) \Gamma(\zeta_j) Γ ( ζ j ) ∥ Γ ( ζ j ) − 1 ∥ = ∣ ζ j ∣ − 1 \|\Gamma(\zeta_j)^{-1}\|=|\zeta_j|^{-1} ∥Γ ( ζ j ) − 1 ∥ = ∣ ζ j ∣ − 1 ζ j ≠ 0 \zeta_j\ne0 ζ j = 0 ζ ↦ Γ ( ζ ) \zeta\mapsto\Gamma(\zeta) ζ ↦ Γ ( ζ ) Γ η \Gamma_\eta Γ η η \eta η R 2 k \mathbb{R}^{2k} R 2 k
Γ ( ζ ) = ∑ η = 1 2 k ζ ( η ) Γ η . \Gamma(\zeta)=\sum_{\eta=1}^{2k}\zeta(\eta)\Gamma_\eta. Γ ( ζ ) = η = 1 ∑ 2 k ζ ( η ) Γ η . Recall the vectors D a b ( i ) ∈ R 2 D_{ab}^{(i)}\in\mathbb{R}^2 D ab ( i ) ∈ R 2 D a b ( i ) D_{ab}^{(i)} D ab ( i ) a , b ∈ [ N ] a,b\in[N] a , b ∈ [ N ] D a b ( i ) = D b a ( i ) D_{ab}^{(i)}=D_{ba}^{(i)} D ab ( i ) = D ba ( i ) D a b ( i ) ∈ R 2 k D_{ab}^{(i)}\in\mathbb{R}^{2k} D ab ( i ) ∈ R 2 k D a b ( i 1 ) , … , D a b ( i k ) D_{ab}^{(i_1)},\ldots,D_{ab}^{(i_k)} D ab ( i 1 ) , … , D ab ( i k ) D a b ( i j ) D_{ab}^{(i_j)} D ab ( i j )
ζ ⋅ D a b ( i ) = r a T Γ ( ζ ) r b , a , b ∈ [ N ] , ζ ∈ R 2 k . (248) \zeta\cdot D_{ab}^{(i)}=r_a^{T}\Gamma(\zeta)r_b,\qquad a,b\in[N],\quad\zeta\in\mathbb{R}^{2k}. \tag*{(248)} ζ ⋅ D ab ( i ) = r a T Γ ( ζ ) r b , a , b ∈ [ N ] , ζ ∈ R 2 k . (248) Further, after identifying R 2 \mathbb{R}^2 R 2 C \mathbb{C} C
D a b ( i j ) = N ( u i j ( a ) + i u i j + 1 ( a ) ) ( u i j ( b ) + i u i j + 1 ( b ) ) , (249) D_{ab}^{(i_j)}=N\bigl(u_{i_j}(a)+\mathrm{i}u_{i_j+1}(a)\bigr)\bigl(u_{i_j}(b)+\mathrm{i}u_{i_j+1}(b)\bigr), \tag*{(249)} D ab ( i j ) = N ( u i j ( a ) + i u i j + 1 ( a ) ) ( u i j ( b ) + i u i j + 1 ( b ) ) , (249) and therefore
∣ D a b ( i j ) ∣ = ∣ r a , j ∣ ∣ r b , j ∣ , a , b ∈ [ N ] , j ∈ [ k ] . (250) \left|D_{ab}^{(i_j)}\right|=|r_{a,j}|\,|r_{b,j}|,\qquad a,b\in[N],\quad j\in[k]. \tag*{(250)} D ab ( i j ) = ∣ r a , j ∣ ∣ r b , j ∣ , a , b ∈ [ N ] , j ∈ [ k ] . (250) Definition 8.1. Let ( a , b ) (a,b) ( a , b ) { ( a ′ , b ′ ) ∈ [ N ] 2 : a ′ < b ′ } \{(a',b')\in[N]^2:a'<b'\} {( a ′ , b ′ ) ∈ [ N ] 2 : a ′ < b ′ } Y Y Y X X X H H H ( a , b ) (a,b) ( a , b ) Y Y Y increment
J = ( Y − X a b ) D a b ( i ) ∈ R 2 k . (251) J=(Y-X_{ab})D_{ab}^{(i)}\in\mathbb{R}^{2k}. \tag*{(251)} J = ( Y − X ab ) D ab ( i ) ∈ R 2 k . (251) We define the conditional mean J ˉ = E [ J ∣ H ] \bar{J}=\mathbb{E}[J\mid H] J ˉ = E [ J ∣ H ] Σ H = E [ ( J − J ˉ ) ( J − J ˉ ) T ∣ H ] \Sigma_H=\mathbb{E}[(J-\bar{J})(J-\bar{J})^{T}\mid H] Σ H = E [( J − J ˉ ) ( J − J ˉ ) T ∣ H ]
Φ H ( ζ ) = E [ e i ζ ⋅ J ∣ H ] , ζ ∈ R 2 k . (252) \Phi_H(\zeta)=\mathbb{E}\left[e^{\mathrm{i}\zeta\cdot J}\mid H\right],\qquad\zeta\in\mathbb{R}^{2k}. \tag*{(252)} Φ H ( ζ ) = E [ e i ζ ⋅ J ∣ H ] , ζ ∈ R 2 k . (252) For an integer n ≥ 1 n \ge1 n ≥ 1 S ( n ) S^{(n)} S ( n ) R 2 k \mathbb{R}^{2k} R 2 k H H H n n n J J J H H H Φ Y \Phi_Y Φ Y Y Y Y Φ Y ( t ′ ) = E e i t ′ Y \Phi_Y(t') = \mathbb{E}e^{it'Y} Φ Y ( t ′ ) = E e i t ′ Y t ′ ∈ R t' \in\mathbb{R} t ′ ∈ R
These objects depend on i i i S ( n ) S^{(n)} S ( n )
E [ e i ξ ⋅ S ( n ) ∣ H ] = Φ H ( ξ ) n , ξ ∈ R 2 k . (253) \mathbb{E}\left[e^{i\xi\cdot S^{(n)}} \mid H\right] = \Phi_H(\xi)^n,\qquad\xi\in\mathbb{R}^{2k}. \tag*{(253)} E [ e i ξ ⋅ S ( n ) ∣ H ] = Φ H ( ξ ) n , ξ ∈ R 2 k . (253) Recall from Section 4 that S i = ∑ l = 1 m ( Y l − X a l b l ) D a l b l ( i ) S_i = \sum_{l=1}^{m} (Y_l-X_{a_l b_l})D_{a_l b_l}^{(i)} S i = ∑ l = 1 m ( Y l − X a l b l ) D a l b l ( i ) X a l b l X_{a_l b_l} X a l b l H H H l l l H H H m m m J J J S i S_i S i H H H S ( m ) S^{(m)} S ( m )
We next introduce the event on which the estimates below take place. It involves the matrix X ˙ \dot{X} X ˙ J J J
Definition 8.2. Let X ˙ \dot{X} X ˙ N × N N \times N N × N X ˙ a b = X a b 2 − 1 \dot{X}_{ab}=X_{ab}^{2}-1 X ˙ ab = X ab 2 − 1 a ≠ b a\ne b a = b ε > 0 \varepsilon>0 ε > 0 A N r e g ( ε ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon) A N reg ( ε )
H ∈ G ε , max a ≤ b ∣ X a b ∣ ≤ N ε , ∥ X ˙ ∥ ≤ N 1 / 2 + ε . (254) H \in\mathcal{G}_{\varepsilon},\qquad\max_{a\le b}|X_{ab}| \le N^\varepsilon,\qquad\|\dot{X}\| \le N^{1/2+\varepsilon}. \tag*{(254)} H ∈ G ε , a ≤ b max ∣ X ab ∣ ≤ N ε , ∥ X ˙ ∥ ≤ N 1/2 + ε . (254) Recall from Definition 3.1 that H ∈ G ε H \in\mathcal{G}_{\varepsilon} H ∈ G ε ∥ H ∥ ≤ 3 \|H\| \le3 ∥ H ∥ ≤ 3 ∥ v ∥ ∞ ≤ N − 1 / 2 + ε \|v\|_{\infty} \le N^{-1/2+\varepsilon} ∥ v ∥ ∞ ≤ N − 1/2 + ε v v v H H H A N r e g ( ε ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon) A N reg ( ε ) k k k i \mathbf{i} i a , b ∈ [ N ] a,b \in[N] a , b ∈ [ N ]
∣ r b , j ∣ ≤ 2 1 / 2 N ε , ∣ r b ∣ ≤ ( 2 k ) 1 / 2 N ε , ∣ D a b ( i ) ∣ ≤ 2 k 1 / 2 N 2 ε , (255) |r_{b,j}| \le2^{1/2}N^\varepsilon,\qquad|r_b| \le(2k)^{1/2}N^\varepsilon,\qquad|D_{ab}^{(i)}| \le2k^{1/2}N^{2\varepsilon}, \tag*{(255)} ∣ r b , j ∣ ≤ 2 1/2 N ε , ∣ r b ∣ ≤ ( 2 k ) 1/2 N ε , ∣ D ab ( i ) ∣ ≤ 2 k 1/2 N 2 ε , (255) where the last bound follows from (250) .
Lemma 8.3. For every ε > 0 \varepsilon>0 ε > 0 A N r e g ( ε ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon) A N reg ( ε )
Proof. The three conditions in (8.9) become weaker as ε \varepsilon ε ε ≤ 1 / 4 \varepsilon\le1/4 ε ≤ 1/4 H ∈ G ε H \in\mathcal{G}_{\varepsilon} H ∈ G ε X X X Z Z Z N 2 N^2 N 2
P ( max a ≤ b ∣ X a b ∣ > N ε / 4 ) ≤ C 0 N 2 exp ( − c 0 N ε / 2 ) , (256) \mathbb{P}\left(\max_{a\le b}|X_{ab}| > N^{\varepsilon/4}\right) \le C_0N^2\exp\left(-c_0N^{\varepsilon/2}\right), \tag*{(256)} P ( a ≤ b max ∣ X ab ∣ > N ε /4 ) ≤ C 0 N 2 exp ( − c 0 N ε /2 ) , (256) which is smaller than every negative power of N N N N N N
To bound ∥ X ˙ ∥ \|\dot{X}\| ∥ X ˙ ∥ N ε / 4 N^{\varepsilon/4} N ε /4
c N = E [ ( X 2 − 1 ) 1 { ∣ X ∣ ≤ N ε / 4 } ] = − E [ ( X 2 − 1 ) 1 { ∣ X ∣ > N ε / 4 } ] , (257) c_N=\mathbb{E}\left[(X^2-1)\mathbf{1}_{\{|X|\le N^{\varepsilon/4}\}}\right]
=-\mathbb{E}\left[(X^2-1)\mathbf{1}_{\{|X|>N^{\varepsilon/4}\}}\right], \tag*{(257)} c N = E [ ( X 2 − 1 ) 1 { ∣ X ∣ ≤ N ε /4 } ] = − E [ ( X 2 − 1 ) 1 { ∣ X ∣ > N ε /4 } ] , (257) where we used E X 2 = 1 \mathbb{E}X^2=1 E X 2 = 1 X X X
∣ c N ∣ ≤ C exp ( − c N ε / 2 ) . (258) |c_N| \le C\exp(-cN^{\varepsilon/2}). \tag*{(258)} ∣ c N ∣ ≤ C exp ( − c N ε /2 ) . (258) Recall that V a b = e a e b T + e b e a T V^{ab}=e_ae_b^{T}+e_be_a^{T} V ab = e a e b T + e b e a T a < b a<b a < b
A a b = ( ( X a b 2 − 1 ) 1 { ∣ X a b ∣ ≤ N ε / 4 } − c N ) V a b , X ˙ ′ = ∑ a < b A a b . (259) A_{ab}=\left((X_{ab}^2-1)\mathbf{1}_{\{|X_{ab}|\le N^{\varepsilon/4}\}}-c_N\right)V^{ab},\qquad\dot{X}'=\sum_{a<b}A_{ab}. \tag*{(259)} A ab = ( ( X ab 2 − 1 ) 1 { ∣ X ab ∣ ≤ N ε /4 } − c N ) V ab , X ˙ ′ = a < b ∑ A ab . (259) The matrices A a b A_{ab} A ab ∥ V a b ∥ = 1 \|V^{ab}\|=1 ∥ V ab ∥ = 1 ∣ c N ∣ ≤ 1 |c_N|\le1 ∣ c N ∣ ≤ 1 N N N ∥ A a b ∥ ≤ 4 N ε / 2 \|A_{ab}\|\le4N^{\varepsilon/2} ∥ A ab ∥ ≤ 4 N ε /2 ( V a b ) 2 = e a e a T + e b e b T (V^{ab})^2=e_ae_a^{T}+e_be_b^{T} ( V ab ) 2 = e a e a T + e b e b T ∑ a < b E A a b 2 \sum_{a<b}\mathbb{E}A_{ab}^2 ∑ a < b E A ab 2
∥ ∑ a < b E A a b 2 ∥ ≤ ( N − 1 ) E [ ( X 2 − 1 ) 2 ] ≤ C N . (260) \left\|\sum_{a<b}\mathbb{E}A_{ab}^2\right\|\le(N-1)\mathbb{E}\left[(X^2-1)^2\right]\le CN. \tag*{(260)} a < b ∑ E A ab 2 ≤ ( N − 1 ) E [ ( X 2 − 1 ) 2 ] ≤ C N . (260) We now use the matrix Bernstein inequality [73 Theorem 1.4] , which we state in the form needed here. Let ( A a b ) a < b (A_{ab})_{a<b} ( A ab ) a < b N × N N\times N N × N λ N ( A a b ) ≤ R ′ \lambda_N(A_{ab})\le R' λ N ( A ab ) ≤ R ′ a < b a<b a < b t ′ ≥ 0 t'\ge0 t ′ ≥ 0
P ( λ N ( ∑ a < b A a b ) ≥ t ′ ) ≤ N exp ( − t ′ 2 / 2 ∥ ∑ a < b E A a b 2 ∥ + R ′ t ′ / 3 ) . (261) \mathbb{P}\left(\lambda_N\left(\sum_{a<b}A_{ab}\right)\ge t'\right)\le N\exp\left(-\frac{t'^2/2}{\left\|\sum_{a<b}\mathbb{E}A_{ab}^2\right\|+R't'/3}\right). \tag*{(261)} P ( λ N ( a < b ∑ A ab ) ≥ t ′ ) ≤ N exp ( − ∑ a < b E A ab 2 + R ′ t ′ /3 t ′2 /2 ) . (261) We apply it to the matrices A a b A_{ab} A ab R ′ = 4 N ε / 2 R'=4N^{\varepsilon/2} R ′ = 4 N ε /2 t ′ = N 1 / 2 + ε / 2 t'=N^{1/2+\varepsilon/2} t ′ = N 1/2 + ε /2 ε ≤ 1 / 4 \varepsilon\le1/4 ε ≤ 1/4 R ′ t ′ ≤ 2 N R't'\le2N R ′ t ′ ≤ 2 N
P ( ∥ X ˙ ′ ∥ ≥ N 1 / 2 + ε 2 ) ≤ 2 N exp ( − c N 2 ε ) . (262) \mathbb{P}\left(\|\dot{X}'\|\ge\frac{N^{1/2+\varepsilon}}{2}\right)\le2N\exp(-cN^{2\varepsilon}). \tag*{(262)} P ( ∥ X ˙ ′ ∥ ≥ 2 N 1/2 + ε ) ≤ 2 N exp ( − c N 2 ε ) . (262) When max a < b ∣ X a b ∣ ≤ N ε / 4 \max_{a<b}|X_{ab}|\le N^{\varepsilon/4} max a < b ∣ X ab ∣ ≤ N ε /4
X ˙ = X ˙ ′ + c N ∑ a < b V a b , \dot{X}=\dot{X}'+c_N\sum_{a<b}V^{ab}, X ˙ = X ˙ ′ + c N a < b ∑ V ab , and hence
∥ X ˙ ∥ ≤ ∥ X ˙ ′ ∥ + N 2 ∣ c N ∣ ≤ ∥ X ˙ ′ ∥ + C N 2 e − c N ε / 2 . (263) \|\dot{X}\|\le\|\dot{X}'\|+N^2|c_N|\le\|\dot{X}'\|+CN^2e^{-cN^{\varepsilon/2}}. \tag*{(263)} ∥ X ˙ ∥ ≤ ∥ X ˙ ′ ∥ + N 2 ∣ c N ∣ ≤ ∥ X ˙ ′ ∥ + C N 2 e − c N ε /2 . (263) Combining the last two displays with (256) , we conclude that ∥ X ˙ ∥ ≤ N 1 / 2 + ε \|\dot{X}\|\le N^{1/2+\varepsilon} ∥ X ˙ ∥ ≤ N 1/2 + ε
For v , v ′ ∈ R N v,v'\in\mathbb{R}^N v , v ′ ∈ R N v ⊙ v ′ ∈ R N v\odot v'\in\mathbb{R}^N v ⊙ v ′ ∈ R N ( v ⊙ v ′ ) ( a ) = v ( a ) v ′ ( a ) (v\odot v')(a)=v(a)v'(a) ( v ⊙ v ′ ) ( a ) = v ( a ) v ′ ( a )
Lemma 8.4. Let 0 < ε ≤ 1 / 8 0<\varepsilon\le1/8 0 < ε ≤ 1/8 C > 0 C>0 C > 0 A N r e g ( ε ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon) A N reg ( ε ) k k k b o l d s y m b o l i boldsymbol{i} b o l d sy mb o l i
∣ J ∣ ≤ C N − 1 / 2 , E [ ∣ J ∣ 3 ∣ H ] ≤ C N 9 ε , |\boldsymbol{J}|\le CN^{-1/2},\qquad\mathbb{E}[|\boldsymbol{J}|^3\mid H]\le CN^{9\varepsilon}, ∣ J ∣ ≤ C N − 1/2 , E [ ∣ J ∣ 3 ∣ H ] ≤ C N 9 ε , ∥ E [ J J T ∣ H ] − 4 I d 2 k ∥ + ∥ Σ H − 4 I d 2 k ∥ ≤ C N 3 ε − 1 / 2 . (264) \left\|\mathbb{E}[\boldsymbol{J}\boldsymbol{J}^{T}\mid H]-4I_{d_{2k}}\right\|+\|\Sigma_H-4I_{d_{2k}}\|\le CN^{3\varepsilon-1/2}. \tag*{(264)} E [ J J T ∣ H ] − 4 I d 2 k + ∥ Σ H − 4 I d 2 k ∥ ≤ C N 3 ε − 1/2 . (264) Moreover, on A N r e g ( ε ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon) A N reg ( ε ) ∣ J ∣ ≤ 2 k 1 / 2 N 2 ε ( ∣ Y ∣ + N ε ) |\boldsymbol{J}|\le2k^{1/2}N^{2\varepsilon}(|Y|+N^\varepsilon) ∣ J ∣ ≤ 2 k 1/2 N 2 ε ( ∣ Y ∣ + N ε )
Proof. The last claim follows from (8.6), (8.10), and ∣ X a b ∣ ≤ N ε |X_{ab}| \le N^{\varepsilon} ∣ X ab ∣ ≤ N ε E Y = 0 \mathbb{E}Y = 0 E Y = 0 Y Y Y H H H ( a , b ) (a,b) ( a , b )
J ‾ = − ( N 2 ) − 1 ∑ a < b X a b D a b ( i j ) . (265) \overline{J}=-\binom{N}{2}^{-1}\sum_{a<b}X_{ab}D_{ab}^{(ij)}. \tag*{(265)} J = − ( 2 N ) − 1 a < b ∑ X ab D ab ( ij ) . (265) The main point is that the sum in (265) can be computed exactly with the eigenvalue equation; a bound by absolute values would give only ∣ J ‾ ∣ ≤ C N 3 ε |\overline{J}| \le CN^{3\varepsilon} ∣ J ∣ ≤ C N 3 ε m J ‾ m\overline{J} m J j ∈ [ k ] j \in[k] j ∈ [ k ] u = u i j u=u_{i_j} u = u i j v = u i j + 1 v=u_{i_{j+1}} v = u i j + 1 X b a = X a b X_{ba}=X_{ab} X ba = X ab X a a = N 1 / 2 H a a X_{aa}=N^{1/2}H_{aa} X aa = N 1/2 H aa
∑ a , b = 1 N X a b u ( a ) u ( b ) = N 1 / 2 u T H u = N 1 / 2 λ i j . (266) \sum_{a,b=1}^{N}X_{ab}u(a)u(b)=N^{1/2}u^{T}Hu=N^{1/2}\lambda_{i_j}. \tag*{(266)} a , b = 1 ∑ N X ab u ( a ) u ( b ) = N 1/2 u T H u = N 1/2 λ i j . (266) Separating the diagonal terms, we obtain
∑ a < b X a b u ( a ) u ( b ) = 1 2 ( N 1 / 2 λ i j − ∑ a = 1 N X a a u ( a ) 2 ) , (267) \sum_{a<b}X_{ab}u(a)u(b)=\frac{1}{2}\left(N^{1/2}\lambda_{i_j}-\sum_{a=1}^{N}X_{aa}u(a)^2\right), \tag*{(267)} a < b ∑ X ab u ( a ) u ( b ) = 2 1 ( N 1/2 λ i j − a = 1 ∑ N X aa u ( a ) 2 ) , (267) and the same identity holds with v v v λ i j + 1 \lambda_{i_{j+1}} λ i j + 1 u u u λ i j \lambda_{i_j} λ i j H v = λ i j + 1 v Hv=\lambda_{i_{j+1}}v H v = λ i j + 1 v u T v = 0 u^{T}v=0 u T v = 0
∑ a < b X a b ( u ( a ) v ( b ) + v ( a ) u ( b ) ) = N 1 / 2 u T H v − ∑ a = 1 N X a a u ( a ) v ( a ) = − ∑ a = 1 N X a a u ( a ) v ( a ) . (268) \sum_{a<b}X_{ab}\bigl(u(a)v(b)+v(a)u(b)\bigr)=N^{1/2}u^{T}Hv-\sum_{a=1}^{N}X_{aa}u(a)v(a)=-\sum_{a=1}^{N}X_{aa}u(a)v(a). \tag*{(268)} a < b ∑ X ab ( u ( a ) v ( b ) + v ( a ) u ( b ) ) = N 1/2 u T H v − a = 1 ∑ N X aa u ( a ) v ( a ) = − a = 1 ∑ N X aa u ( a ) v ( a ) . (268) Inserting these identities into the definition of D a b ( i j ) D_{ab}^{(ij)} D ab ( ij )
∑ a < b X a b D a b ( i j ) = N ( N 1 / 2 ( λ i j − λ i j + 1 ) − ∑ a X a a ( u ( a ) 2 − v ( a ) 2 ) 2 , − ∑ a X a a u ( a ) v ( a ) ) . (269) \sum_{a<b}X_{ab}D_{ab}^{(ij)}=N\left(\frac{N^{1/2}(\lambda_{i_j}-\lambda_{i_{j+1}})-\sum_a X_{aa}\bigl(u(a)^2-v(a)^2\bigr)}{2},-\sum_aX_{aa}u(a)v(a)\right). \tag*{(269)} a < b ∑ X ab D ab ( ij ) = N ( 2 N 1/2 ( λ i j − λ i j + 1 ) − ∑ a X aa ( u ( a ) 2 − v ( a ) 2 ) , − a ∑ X aa u ( a ) v ( a ) ) . (269) We have ∣ X a a ∣ ≤ N ε |X_{aa}| \le N^{\varepsilon} ∣ X aa ∣ ≤ N ε A N r e g ( ε ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon) A N reg ( ε )
∣ λ i j − λ i j + 1 ∣ ≤ 2 ∥ H ∥ ≤ 6. (270) |\lambda_{i_j}-\lambda_{i_{j+1}}|\le2\lVert H\rVert\le6. \tag*{(270)} ∣ λ i j − λ i j + 1 ∣ ≤ 2 ∥ H ∥ ≤ 6. (270) Hence the norm of the right side of (269) is bounded by
N ( 3 N 1 / 2 + 2 N ε ) ≤ 5 N 3 / 2 . (271) N(3N^{1/2}+2N^{\varepsilon})\le5N^{3/2}. \tag*{(271)} N ( 3 N 1/2 + 2 N ε ) ≤ 5 N 3/2 . (271) Inserting this into (265) gives ∣ J ‾ ∣ ≤ 20 k 1 / 2 N − 1 / 2 |\overline{J}|\le20k^{1/2}N^{-1/2} ∣ J ∣ ≤ 20 k 1/2 N − 1/2 E ( Y − y ) 2 = 1 + y 2 \mathbb{E}(Y-y)^2=1+y^2 E ( Y − y ) 2 = 1 + y 2 y ∈ R y\in\mathbb{R} y ∈ R
E [ J J T ∣ H ] = ( N 2 ) − 1 ∑ a < b ( 1 + X a b 2 ) D a b ( i ) ( D a b ( i ) ) T = ( N 2 ) − 1 ∑ a < b ( 2 + X ˚ a b ) D a b ( i ) ( D a b ( i ) ) T . (272) \mathbb{E}[JJ^{T}\mid H]=\binom{N}{2}^{-1}\sum_{a<b}(1+X_{ab}^{2})D_{ab}^{(i)}(D_{ab}^{(i)})^{T}=\binom{N}{2}^{-1}\sum_{a<b}(2+\mathring{X}_{ab})D_{ab}^{(i)}(D_{ab}^{(i)})^{T}. \tag*{(272)} E [ J J T ∣ H ] = ( 2 N ) − 1 a < b ∑ ( 1 + X ab 2 ) D ab ( i ) ( D ab ( i ) ) T = ( 2 N ) − 1 a < b ∑ ( 2 + X ˚ ab ) D ab ( i ) ( D ab ( i ) ) T . (272) To evaluate the first term, observe that D a b ( i ) ( η ) = r a T Γ η r b D_{ab}^{(i)}(\eta)=r_a^{T}\Gamma_{\eta}r_b D ab ( i ) ( η ) = r a T Γ η r b (248) . Using (246) twice and the symmetry of Γ η ′ \Gamma_{\eta'} Γ η ′ η , η ′ ∈ [ 2 k ] \eta,\eta'\in[2k] η , η ′ ∈ [ 2 k ]
∑ a , b = 1 N D a b ( i ) ( η ) D a b ( i ) ( η ′ ) = ∑ a = 1 N r a T Γ η ( ∑ b = 1 N r b r b T ) Γ η ′ r a = N Tr ( Γ η Γ η ′ ∑ a = 1 N r a r a T ) = N 2 Tr ( Γ η Γ η ′ ) = 2 N 2 δ η η ′ . (273) \begin{aligned}
\sum_{a,b=1}^{N}D_{ab}^{(i)}(\eta)D_{ab}^{(i)}(\eta')=\sum_{a=1}^{N}r_a^{T}\Gamma_{\eta}\left(\sum_{b=1}^{N}r_br_b^{T}\right)\Gamma_{\eta'}r_a=N\operatorname{Tr}\left(\Gamma_{\eta}\Gamma_{\eta'}\sum_{a=1}^{N}r_ar_a^{T}\right) \\
&=N^{2}\operatorname{Tr}(\Gamma_{\eta}\Gamma_{\eta'})=2N^{2}\delta_{\eta\eta'}. \tag*{(273)}
\end{aligned} a , b = 1 ∑ N D ab ( i ) ( η ) D ab ( i ) ( η ′ ) = a = 1 ∑ N r a T Γ η ( b = 1 ∑ N r b r b T ) Γ η ′ r a = N Tr ( Γ η Γ η ′ a = 1 ∑ N r a r a T ) = N 2 Tr ( Γ η Γ η ′ ) = 2 N 2 δ η η ′ . (273) In the last equality, we used that Γ η Γ η ′ = 0 \Gamma_{\eta}\Gamma_{\eta'}=0 Γ η Γ η ′ = 0 η \eta η η ′ \eta' η ′ Γ η 2 \Gamma_{\eta}^{2} Γ η 2 η \eta η Γ 2 j − 1 Γ 2 j \Gamma_{2j-1}\Gamma_{2j} Γ 2 j − 1 Γ 2 j (247) .
Since D a b ( i ) = D b a ( i ) D_{ab}^{(i)}=D_{ba}^{(i)} D ab ( i ) = D ba ( i ) ( N 2 ) = N ( N − 1 ) / 2 \binom{N}{2}=N(N-1)/2 ( 2 N ) = N ( N − 1 ) /2 (273) gives
2 ( N 2 ) − 1 ∑ a < b D a b ( i ) ( D a b ( i ) ) T = 2 N ( N − 1 ) ( 2 N 2 I d 2 k − ∑ a = 1 N D a a ( i ) ( D a a ( i ) ) T ) . (274) 2\binom{N}{2}^{-1}\sum_{a<b}D_{ab}^{(i)}(D_{ab}^{(i)})^{T}=\frac{2}{N(N-1)}\left(2N^{2}I_{d_{2k}}-\sum_{a=1}^{N}D_{aa}^{(i)}(D_{aa}^{(i)})^{T}\right). \tag*{(274)} 2 ( 2 N ) − 1 a < b ∑ D ab ( i ) ( D ab ( i ) ) T = N ( N − 1 ) 2 ( 2 N 2 I d 2 k − a = 1 ∑ N D aa ( i ) ( D aa ( i ) ) T ) . (274) Using ∣ D a a ( i ) ∣ 2 ≤ 4 k N 4 ε \lvert D_{aa}^{(i)}\rvert^{2}\leq4kN^{4\varepsilon} ∣ D aa ( i ) ∣ 2 ≤ 4 k N 4 ε (255) , we conclude that
∥ 2 ( N 2 ) − 1 ∑ a < b D a b ( i ) ( D a b ( i ) ) T − 4 I d 2 k ∥ ≤ 4 N − 1 + 8 k N 4 ε N − 1 ≤ C N 4 ε − 1 . (275) \left\|2\binom{N}{2}^{-1}\sum_{a<b}D_{ab}^{(i)}(D_{ab}^{(i)})^{T}-4I_{d_{2k}}\right\|\leq\frac{4}{N-1}+\frac{8kN^{4\varepsilon}}{N-1}\leq CN^{4\varepsilon-1}. \tag*{(275)} 2 ( 2 N ) − 1 a < b ∑ D ab ( i ) ( D ab ( i ) ) T − 4 I d 2 k ≤ N − 1 4 + N − 1 8 k N 4 ε ≤ C N 4 ε − 1 . (275) Next, we bound the contribution of X ~ \widetilde{X} X (272) . Since X ~ \widetilde{X} X ( η , η ′ ) (\eta,\eta') ( η , η ′ )
( N 2 ) − 1 ∑ a < b X ~ a b D a b ( i ) ( η ) D a b ( i ) ( η ′ ) = 1 N ( N − 1 ) ∑ a , b = 1 N X ~ a b D a b ( i ) ( η ) D a b ( i ) ( η ′ ) . (276) \binom{N}{2}^{-1}\sum_{a<b}\widetilde{X}_{ab}D_{ab}^{(i)}(\eta)D_{ab}^{(i)}(\eta')=\frac{1}{N(N-1)}\sum_{a,b=1}^{N}\widetilde{X}_{ab}D_{ab}^{(i)}(\eta)D_{ab}^{(i)}(\eta'). \tag*{(276)} ( 2 N ) − 1 a < b ∑ X ab D ab ( i ) ( η ) D ab ( i ) ( η ′ ) = N ( N − 1 ) 1 a , b = 1 ∑ N X ab D ab ( i ) ( η ) D ab ( i ) ( η ′ ) . (276) By the definition of D a b ( i ) D_{ab}^{(i)} D ab ( i ) D a b ( i ) ( η ) D_{ab}^{(i)}(\eta) D ab ( i ) ( η ) N N N ± v ( a ) v ′ ( b ) \pm v^{(a)}v'^{(b)} ± v ( a ) v ′ ( b ) v v v v ′ v' v ′ U i U_i U i (276) is N 2 N^{2} N 2 ± ( v 1 ⊙ v 2 ) T X ~ ( v 3 ⊙ v 4 ) \pm(v_1\odot v_2)^{T}\widetilde{X}(v_3\odot v_4) ± ( v 1 ⊙ v 2 ) T X ( v 3 ⊙ v 4 ) v 1 , … , v 4 v_1,\ldots,v_4 v 1 , … , v 4 U i U_i U i v , v ′ v,v' v , v ′ U i U_i U i A N r e g ( ε ) \mathcal{A}_{N}^{\mathrm{reg}}(\varepsilon) A N reg ( ε )
∥ v ⊙ v ′ ∥ 2 ≤ ∥ v ∥ ∞ ∥ v ′ ∥ 2 ≤ N ε − 1 / 2 , (277) \|v\odot v'\|_{2}\leq\|v\|_{\infty}\|v'\|_{2}\leq N^{\varepsilon-1/2}, \tag*{(277)} ∥ v ⊙ v ′ ∥ 2 ≤ ∥ v ∥ ∞ ∥ v ′ ∥ 2 ≤ N ε − 1/2 , (277) and therefore
∣ ( v 1 ⊙ v 2 ) T X ~ ( v 3 ⊙ v 4 ) ∣ ≤ ∥ X ~ ∥ N 2 ε − 1 ≤ N 3 ε − 1 / 2 . (278) \left|(v_1\odot v_2)^{T}\widetilde{X}(v_3\odot v_4)\right|\leq\|\widetilde{X}\|N^{2\varepsilon-1}\leq N^{3\varepsilon-1/2}. \tag*{(278)} ( v 1 ⊙ v 2 ) T X ( v 3 ⊙ v 4 ) ≤ ∥ X ∥ N 2 ε − 1 ≤ N 3 ε − 1/2 . (278) It follows that the absolute value of (276) is bounded by
4 N 2 N 3 ε − 1 / 2 N ( N − 1 ) ≤ 8 N 3 ε − 1 / 2 . (279) \frac{4N^{2}N^{3\varepsilon-1/2}}{N(N-1)}\leq8N^{3\varepsilon-1/2}. \tag*{(279)} N ( N − 1 ) 4 N 2 N 3 ε − 1/2 ≤ 8 N 3 ε − 1/2 . (279) The operator norm of a 2 k × 2 k 2k\times2k 2 k × 2 k 2 k 2k 2 k (272) and (275) give
∥ E [ J J T ∣ H ] − 4 I d 2 k ∥ ≤ C N 4 ε − 1 + 16 k N 3 ε − 1 / 2 ≤ C N 3 ε − 1 / 2 . (280) \left\|\mathbb{E}[JJ^{T}\mid H]-4I_{d_{2k}}\right\|\leq CN^{4\varepsilon-1}+16kN^{3\varepsilon-1/2}\leq CN^{3\varepsilon-1/2}. \tag*{(280)} E [ J J T ∣ H ] − 4 I d 2 k ≤ C N 4 ε − 1 + 16 k N 3 ε − 1/2 ≤ C N 3 ε − 1/2 . (280) where we used ε ≤ 1 / 8 \varepsilon\le1/8 ε ≤ 1/8 Σ ‾ H = E [ J J T ∣ H ] − J ‾ J ‾ T \overline{\Sigma}_H = \mathbb{E}[JJ^{\mathsf{T}} \mid H] - \overline{J}\overline{J}^{\mathsf{T}} Σ H = E [ J J T ∣ H ] − J J T ∣ J ‾ ∣ 2 ≤ C N − 1 |\overline{J}|^2 \le CN^{-1} ∣ J ∣ 2 ≤ C N − 1 Σ H \Sigma_H Σ H
Finally, the bound on ∣ J ∣ |J| ∣ J ∣ ( y 1 + y 2 ) 3 ≤ 4 ( y 1 3 + y 2 3 ) (y_1+y_2)^3 \le4(y_1^3+y_2^3) ( y 1 + y 2 ) 3 ≤ 4 ( y 1 3 + y 2 3 ) y 1 , y 2 ≥ 0 y_1,y_2 \ge0 y 1 , y 2 ≥ 0
E [ ∣ J ∣ 3 ∣ H ] ≤ 8 k 3 / 2 N 6 ε E [ ( ∣ Y ∣ + N ε ) 3 ] ≤ 32 k 3 / 2 N 6 ε ( E ∣ Y ∣ 3 + N 3 ε ) ≤ C N 9 ε . (281) \mathbb{E}[|J|^3 \mid H] \le8k^{3/2}N^{6\varepsilon}\mathbb{E}[(|Y|+N^\varepsilon)^3] \le32k^{3/2}N^{6\varepsilon}\left(\mathbb{E}|Y|^3+N^{3\varepsilon}\right) \le CN^{9\varepsilon}. \tag*{(281)} E [ ∣ J ∣ 3 ∣ H ] ≤ 8 k 3/2 N 6 ε E [( ∣ Y ∣ + N ε ) 3 ] ≤ 32 k 3/2 N 6 ε ( E ∣ Y ∣ 3 + N 3 ε ) ≤ C N 9 ε . (281) This completes the proof.
Eigenvectors with few coordinate values
For an integer n ≥ 1 n \ge1 n ≥ 1 V n V_n V n v ∈ R N v \in\mathbb{R}^N v ∈ R N v ( 1 ) , … , v ( N ) v(1),\ldots,v(N) v ( 1 ) , … , v ( N ) n n n v ∈ R N v \in\mathbb{R}^N v ∈ R N
∥ v ∥ 2 , N = ( 1 N ∑ b = 1 N v ( b ) 2 ) 1 / 2 = N − 1 / 2 ∥ v ∥ 2 . (282) \|v\|_{2,N}=\left(\frac{1}{N}\sum_{b=1}^{N}v(b)^2\right)^{1/2}=N^{-1/2}\|v\|_2. \tag*{(282)} ∥ v ∥ 2 , N = ( N 1 b = 1 ∑ N v ( b ) 2 ) 1/2 = N − 1/2 ∥ v ∥ 2 . (282) Observe that V n V_n V n [ N ] [N] [ N ] n n n V n V_n V n ∥ w − v ∥ 2 , N \|w-v\|_{2,N} ∥ w − v ∥ 2 , N v ∈ V n v \in V_n v ∈ V n w ∈ R N w \in\mathbb{R}^N w ∈ R N
The following lemma states that, with overwhelming probability, no delocalized unit eigenvector of H H H V n V_n V n n n n [60 , 74 ] .
Lemma 8.5. Let p > 0 p>0 p > 0 a > 4 p a>4p a > 4 p ( Ω N ) (\Omega_N) ( Ω N ) ( K N ) (K_N) ( K N ) Ω N → ∞ \Omega_N \to\infty Ω N → ∞ Ω N p = o ( N ) \Omega_N^p=o(N) Ω N p = o ( N ) Ω N a + 1 = o ( N 1 / 2 ) \Omega_N^{a+1}=o(N^{1/2}) Ω N a + 1 = o ( N 1/2 ) 1 ≤ K N ≤ Ω N 1 \le K_N \le\Omega_N 1 ≤ K N ≤ Ω N N N N Ω = Ω N \Omega=\Omega_N Ω = Ω N K = K N K=K_N K = K N c > 0 c>0 c > 0 N N N ∥ H ∥ ≤ 3 \|H\|\le3 ∥ H ∥ ≤ 3 w w w H H H
N 1 / 2 ∥ w ∥ ∞ ≤ K , inf v ∈ V ⌊ Ω p ⌋ ∥ N 1 / 2 w − v ∥ 2 , N ≤ Ω − a (283) N^{1/2}\|w\|_\infty\le K,\qquad\inf_{v\in V_{\lfloor\Omega^p\rfloor}}\|N^{1/2}w-v\|_{2,N}\le\Omega^{-a} \tag*{(283)} N 1/2 ∥ w ∥ ∞ ≤ K , v ∈ V ⌊ Ω p ⌋ inf ∥ N 1/2 w − v ∥ 2 , N ≤ Ω − a (283) has probability at most exp ( − c N log Ω ) \exp(-cN\log\Omega) exp ( − c N log Ω )
Proof. Throughout the proof, we write n = ⌊ Ω p ⌋ n=\lfloor\Omega^p\rfloor n = ⌊ Ω p ⌋ N N N 1 ≤ K ≤ Ω 1\le K\le\Omega 1 ≤ K ≤ Ω 4 Ω − a ≤ 1 4\Omega^{-a}\le1 4 Ω − a ≤ 1 w w w E ∈ [ − 3 , 3 ] E\in[-3,3] E ∈ [ − 3 , 3 ] w w w v ∈ V n v\in V_n v ∈ V n ∥ N 1 / 2 w − v ∥ 2 , N ≤ Ω − a \|N^{1/2}w-v\|_{2,N}\le\Omega^{-a} ∥ N 1/2 w − v ∥ 2 , N ≤ Ω − a v ^ ∈ R N \widehat{v}\in\mathbb{R}^N v ∈ R N v v v v ( b ) v(b) v ( b ) [ − K , K ] [-K,K] [ − K , K ] ∣ N 1 / 2 w ( b ) ∣ ≤ K |N^{1/2}w(b)|\le K ∣ N 1/2 w ( b ) ∣ ≤ K ∣ N 1 / 2 w ( b ) − v ( b ) ∣ |N^{1/2}w(b)-v(b)| ∣ N 1/2 w ( b ) − v ( b ) ∣ Ω − a Z ∩ [ − K , K ] \Omega^{-a}\mathbb{Z}\cap[-K,K] Ω − a Z ∩ [ − K , K ] Ω − a \Omega^{-a} Ω − a
v ^ ∈ V n , ∥ v ^ ∥ ∞ ≤ K , ∥ N 1 / 2 w − v ^ ∥ 2 , N ≤ 2 Ω − α . (284) \widehat{v} \in\mathcal{V}_n,\qquad\|\widehat{v}\|_{\infty} \le K,\qquad\|N^{1/2}w-\widehat{v}\|_{2,N} \le2\Omega^{-\alpha}. \tag*{(284)} v ∈ V n , ∥ v ∥ ∞ ≤ K , ∥ N 1/2 w − v ∥ 2 , N ≤ 2 Ω − α . (284) Set y = N − 1 / 2 v ^ y=N^{-1/2}\widehat{v} y = N − 1/2 v E ′ E' E ′ E E E Ω − α Z ∩ [ − 3 , 3 ] \Omega^{-\alpha}\mathbb{Z}\cap[-3,3] Ω − α Z ∩ [ − 3 , 3 ]
∥ y − w ∥ 2 ≤ 2 Ω − α , 1 2 ≤ ∥ y ∥ 2 ≤ 2 , ∥ y ∥ ∞ ≤ K N − 1 / 2 . (285) \|y-w\|_2\le2\Omega^{-\alpha},\qquad\frac{1}{2}\le\|y\|_2\le2,\qquad\|y\|_\infty\le KN^{-1/2}. \tag*{(285)} ∥ y − w ∥ 2 ≤ 2 Ω − α , 2 1 ≤ ∥ y ∥ 2 ≤ 2 , ∥ y ∥ ∞ ≤ K N − 1/2 . (285) Using ( H − E ) w = 0 (H-E)w=0 ( H − E ) w = 0 ∥ H − E ∥ ≤ 6 \|H-E\|\le6 ∥ H − E ∥ ≤ 6 ∣ E − E ′ ∣ ≤ Ω − α |E-E'|\le\Omega^{-\alpha} ∣ E − E ′ ∣ ≤ Ω − α
∥ ( H − E ′ ) y ∥ 2 ≤ ∥ ( H − E ) ( y − w ) ∥ 2 + ∣ E − E ′ ∣ ∥ y ∥ 2 ≤ 14 Ω − α . (286) \|(H-E')y\|_2\le\|(H-E)(y-w)\|_2+|E-E'|\|y\|_2\le14\Omega^{-\alpha}. \tag*{(286)} ∥ ( H − E ′ ) y ∥ 2 ≤ ∥ ( H − E ) ( y − w ) ∥ 2 + ∣ E − E ′ ∣∥ y ∥ 2 ≤ 14 Ω − α . (286) Let Y \mathcal{Y} Y y = N − 1 / 2 v ^ y=N^{-1/2}\widehat{v} y = N − 1/2 v v ^ ∈ V n \widehat{v}\in\mathcal{V}_n v ∈ V n Ω − α Z ∩ [ − K , K ] \Omega^{-\alpha}\mathbb{Z}\cap[-K,K] Ω − α Z ∩ [ − K , K ] 1 / 2 ≤ ∥ y ∥ 2 ≤ 2 1/2\le\|y\|_2\le2 1/2 ≤ ∥ y ∥ 2 ≤ 2 Ω − α Z ∩ [ − K , K ] \Omega^{-\alpha}\mathbb{Z}\cap[-K,K] Ω − α Z ∩ [ − K , K ] 2 K Ω α + 1 ≤ 3 Ω α + 1 2K\Omega^\alpha+1\le3\Omega^{\alpha+1} 2 K Ω α + 1 ≤ 3 Ω α + 1 n n n
∣ Y ∣ ≤ ( 3 Ω α + 1 + 1 ) n n N ≤ ( 4 Ω α + 1 ) n n N . (287) |\mathcal{Y}|\le(3\Omega^{\alpha+1}+1)^n n^N\le(4\Omega^{\alpha+1})^n n^N. \tag*{(287)} ∣ Y ∣ ≤ ( 3 Ω α + 1 + 1 ) n n N ≤ ( 4 Ω α + 1 ) n n N . (287) The grid Ω − α Z ∩ [ − 3 , 3 ] \Omega^{-\alpha}\mathbb{Z}\cap[-3,3] Ω − α Z ∩ [ − 3 , 3 ] 7 Ω α 7\Omega^\alpha 7 Ω α
⋃ y ∈ Y ⋃ E ′ ∈ Ω − α Z ∩ [ − 3 , 3 ] { ∥ ( H − E ′ ) y ∥ 2 ≤ 14 Ω − α } . (288) \bigcup_{y\in\mathcal{Y}}\ \bigcup_{E'\in\Omega^{-\alpha}\mathbb{Z}\cap[-3,3]}
\left\{\|(H-E')y\|_2\le14\Omega^{-\alpha}\right\}. \tag*{(288)} y ∈ Y ⋃ E ′ ∈ Ω − α Z ∩ [ − 3 , 3 ] ⋃ { ∥ ( H − E ′ ) y ∥ 2 ≤ 14 Ω − α } . (288) These vectors and energies are deterministic.
We now fix y ∈ Y y\in\mathcal{Y} y ∈ Y E ′ ∈ Ω − α Z ∩ [ − 3 , 3 ] E'\in\Omega^{-\alpha}\mathbb{Z}\cap[-3,3] E ′ ∈ Ω − α Z ∩ [ − 3 , 3 ] ∥ ( H − E ′ ) y ∥ 2 ≤ 14 Ω − α \|(H-E')y\|_2\le14\Omega^{-\alpha} ∥ ( H − E ′ ) y ∥ 2 ≤ 14 Ω − α T ⊂ [ N ] \mathcal{T}\subset[N] T ⊂ [ N ] ⌊ N / 2 ⌋ \lfloor N/2\rfloor ⌊ N /2 ⌋ b b b ∣ y ( b ) ∣ |y(b)| ∣ y ( b ) ∣ T c = [ N ] ∖ T \mathcal{T}^{c}=[N]\setminus\mathcal{T} T c = [ N ] ∖ T y T ∈ R N y_{\mathcal{T}}\in\mathbb{R}^N y T ∈ R N y T ( b ) = y ( b ) y_{\mathcal{T}}(b)=y(b) y T ( b ) = y ( b ) b ∈ T b\in\mathcal{T} b ∈ T y T ( b ) = 0 y_{\mathcal{T}}(b)=0 y T ( b ) = 0 y ( b ) 2 y(b)^2 y ( b ) 2 b ∈ T b\in\mathcal{T} b ∈ T [ N ] [N] [ N ]
∥ y T ∥ 2 2 ≥ ⌊ N / 2 ⌋ N ∥ y ∥ 2 2 ≥ 1 12 . (289) \|y_{\mathcal{T}}\|_2^2\ge\frac{\lfloor N/2\rfloor}{N}\|y\|_2^2\ge\frac{1}{12}. \tag*{(289)} ∥ y T ∥ 2 2 ≥ N ⌊ N /2 ⌋ ∥ y ∥ 2 2 ≥ 12 1 . (289) Suppose that ∥ ( H − E ′ ) y ∥ 2 ≤ 14 Ω − α \|(H-E')y\|_2\le14\Omega^{-\alpha} ∥ ( H − E ′ ) y ∥ 2 ≤ 14 Ω − α N / 4 N/4 N /4 a ∈ [ N ] a\in[N] a ∈ [ N ] ∣ ( ( H − E ′ ) y ) ( a ) ∣ > 28 Ω − α N − 1 / 2 |((H-E')y)(a)|>28\Omega^{-\alpha}N^{-1/2} ∣ (( H − E ′ ) y ) ( a ) ∣ > 28 Ω − α N − 1/2 ∣ T c ∣ ≥ N / 2 |\mathcal{T}^{c}|\ge N/2 ∣ T c ∣ ≥ N /2 N / 4 N/4 N /4 a ∈ T c a\in\mathcal{T}^{c} a ∈ T c
∣ N 1 / 2 ( ( H − E ′ ) y ) ( a ) ∣ ≤ 28 Ω − α . (290) \left|N^{1/2}((H-E')y)(a)\right|\le28\Omega^{-\alpha}. \tag*{(290)} N 1/2 (( H − E ′ ) y ) ( a ) ≤ 28 Ω − α . (290) The rows indexed by T c \mathcal{T}^{c} T c T c × T \mathcal{T}^{c}\times\mathcal{T} T c × T a ∈ T c a\in\mathcal{T}^{c} a ∈ T c
N 1 / 2 ( ( H − E ′ ) y ) ( a ) = ∑ b ∈ T X a b y ( b ) + χ a , χ a = ∑ b ∈ T c X a b y ( b ) − N 1 / 2 E ′ y ( a ) . (291) N^{1/2}((H-E')y)(a)=\sum_{b\in\mathcal{T}}X_{ab}y(b)+\chi_a,\qquad
\chi_a=\sum_{b\in\mathcal{T}^{c}}X_{ab}y(b)-N^{1/2}E'y(a). \tag*{(291)} N 1/2 (( H − E ′ ) y ) ( a ) = b ∈ T ∑ X ab y ( b ) + χ a , χ a = b ∈ T c ∑ X ab y ( b ) − N 1/2 E ′ y ( a ) . (291) Let F T \mathcal{F}_{T} F T σ \sigma σ X a b X_{ab} X ab a ≤ b a \le b a ≤ b a a a b b b { T , T c } \{T,T^c\} { T , T c } χ a \chi_a χ a F T \mathcal{F}_{T} F T X a b X_{ab} X ab a ∈ T c a \in T^c a ∈ T c b ∈ T b \in T b ∈ T ( a , b ) (a,b) ( a , b ) X X X F T \mathcal{F}_{T} F T F T \mathcal{F}_{T} F T ∑ b ∈ T X a b y ( b ) + χ a \sum_{b\in T}X_{ab}y(b)+\chi_a ∑ b ∈ T X ab y ( b ) + χ a a ∈ T c a \in T^c a ∈ T c
We bound the conditional probability of (8.45) for one row with the Berry–Esseen inequality for sums of independent, not necessarily identically distributed, random variables [58 Chapter V] . Since E X = 0 \mathbb{E}X=0 E X = 0 E X 2 = 1 \mathbb{E}X^2=1 E X 2 = 1 ∥ y T ∥ 2 − 1 ∑ b ∈ T X a b y ( b ) \lVert y_T\rVert_2^{-1}\sum_{b\in T}X_{ab}y(b) ∥ y T ∥ 2 − 1 ∑ b ∈ T X ab y ( b ) C B E ∥ y T ∥ 2 − 3 ∑ b ∈ T E ∣ X ∣ 3 ∣ y ( b ) ∣ 3 C_{\mathrm{BE}}\lVert y_T\rVert_2^{-3}\sum_{b\in T}\mathbb{E}|X|^3|y(b)|^3 C BE ∥ y T ∥ 2 − 3 ∑ b ∈ T E ∣ X ∣ 3 ∣ y ( b ) ∣ 3 C B E C_{\mathrm{BE}} C BE ( 2 π ) − 1 / 2 (2\pi)^{-1/2} ( 2 π ) − 1/2 χ ∈ R \chi\in\mathbb{R} χ ∈ R
P ( ∣ ∑ b ∈ T X a b y ( b ) + χ ∣ ≤ 28 Ω − a ) ≤ 56 Ω − a ( 2 π ) 1 / 2 ∥ y T ∥ 2 + 2 C B E E ∣ X ∣ 3 ∥ y ∥ ∞ ∥ y T ∥ 2 2 ∥ y T ∥ 2 3 ≤ C ( Ω − a + K N − 1 / 2 ) . (292) \mathbb{P}\left(\left|\sum_{b\in T}X_{ab}y(b)+\chi\right|\le28\Omega^{-a}\right)
\le\frac{56\Omega^{-a}}{(2\pi)^{1/2}\lVert y_T\rVert_2}+\frac{2C_{\mathrm{BE}}\mathbb{E}|X|^3\lVert y\rVert_\infty\lVert y_T\rVert_2^2}{\lVert y_T\rVert_2^3}
\le C\left(\Omega^{-a}+KN^{-1/2}\right). \tag*{(292)} P ( b ∈ T ∑ X ab y ( b ) + χ ≤ 28 Ω − a ) ≤ ( 2 π ) 1/2 ∥ y T ∥ 2 56 Ω − a + ∥ y T ∥ 2 3 2 C BE E ∣ X ∣ 3 ∥ y ∥ ∞ ∥ y T ∥ 2 2 ≤ C ( Ω − a + K N − 1/2 ) . (292) Recall that K ≤ Ω K\le\Omega K ≤ Ω Ω a + 1 N − 1 / 2 → 0 \Omega^{a+1}N^{-1/2}\to0 Ω a + 1 N − 1/2 → 0 C 2 Ω − a C_2\Omega^{-a} C 2 Ω − a C 2 > 0 C_2>0 C 2 > 0 N N N N N N C 2 Ω − a ≤ 1 C_2\Omega^{-a}\le1 C 2 Ω − a ≤ 1
We apply this bound with χ = χ a \chi=\chi_a χ = χ a 2 N 2^N 2 N ⌈ N / 4 ⌉ \lceil N/4\rceil ⌈ N /4 ⌉ T c T^c T c
P ( ∥ ( H − E ′ ) y ∥ 2 ≤ 14 Ω − a | F T ) ≤ 2 N ( C 2 Ω − a ) N / 4 . (293) \mathbb{P}\left(\lVert(H-E')y\rVert_2\le14\Omega^{-a}\,\middle|\,\mathcal{F}_T\right)\le2^N(C_2\Omega^{-a})^{N/4}. \tag*{(293)} P ( ∥( H − E ′ ) y ∥ 2 ≤ 14 Ω − a F T ) ≤ 2 N ( C 2 Ω − a ) N /4 . (293) The right side is deterministic, so the same bound holds for the unconditional probability. Combining the last display with (8.42) and the bound on the number of energies, we find that the logarithm of the probability of the event in the statement is bounded by
n log ( 4 Ω a + 1 ) + N log n + log ( 7 Ω a ) + N log 2 + N 4 log C 2 − a 4 N log Ω . (294) n\log(4\Omega^{a+1})+N\log n+\log(7\Omega^a)+N\log2+\frac{N}{4}\log C_2-\frac{a}{4}N\log\Omega. \tag*{(294)} n log ( 4 Ω a + 1 ) + N log n + log ( 7 Ω a ) + N log 2 + 4 N log C 2 − 4 a N log Ω. (294) We have N log n ≤ p N log Ω N\log n\le pN\log\Omega N log n ≤ pN log Ω n ≤ Ω p = o ( N ) n\le\Omega^p=o(N) n ≤ Ω p = o ( N ) log Ω → ∞ \log\Omega\to\infty log Ω → ∞ o ( N log Ω ) o(N\log\Omega) o ( N log Ω ) − ( a / 4 − p − o ( 1 ) ) N log Ω -(a/4-p-o(1))N\log\Omega − ( a /4 − p − o ( 1 )) N log Ω c = ( a − 4 p ) / 8 c=(a-4p)/8 c = ( a − 4 p ) /8
Remark 8.6. The vectors y ∈ Y y\in\mathcal{Y} y ∈ Y E ′ E' E ′ w w w ( y , E ′ ) (y,E') ( y , E ′ ) H H H H H H H H H
A deterministic nonlattice estimate
In Section 8.3, U U U N × 2 k N \times2k N × 2 k U T U = Id 2 k U^{T}U = \operatorname{Id}_{2k} U T U = Id 2 k U U U r b r_b r b b b b N 1 / 2 U N^{1/2}U N 1/2 U (246) holds. For ξ ∈ R 2 k \xi\in\mathbb{R}^{2k} ξ ∈ R 2 k
D U ( ξ ) = 1 N 2 ∑ a , b = 1 N [ 1 − cos ( r a T Γ ( ξ ) r b ) ] . (295) D_U(\xi) = \frac{1}{N^2}\sum_{a,b=1}^{N}\left[1-\cos\left(r_a^T\Gamma(\xi)r_b\right)\right]. \tag*{(295)} D U ( ξ ) = N 2 1 a , b = 1 ∑ N [ 1 − cos ( r a T Γ ( ξ ) r b ) ] . (295) For U = U i U = U_i U = U i (248) shows that D U ( ξ ) D_U(\xi) D U ( ξ ) 1 − cos ( ξ ⋅ D a b ( i ) ) 1-\cos\left(\xi\cdot D_{ab}^{(i)}\right) 1 − cos ( ξ ⋅ D ab ( i ) ) ( a , b ) ∈ [ N ] 2 (a,b) \in[N]^2 ( a , b ) ∈ [ N ] 2 ξ ⋅ D a b ( i ) \xi\cdot D_{ab}^{(i)} ξ ⋅ D ab ( i ) 2 π Z 2\pi\mathbb{Z} 2 π Z
p k = 4 k + 4 , a k = 100 ( k + 1 ) , q k = 2 a k + 2 k + 10 , (296) p_k = 4k+4,\qquad a_k = 100(k+1),\qquad q_k = 2a_k+2k+10, \tag*{(296)} p k = 4 k + 4 , a k = 100 ( k + 1 ) , q k = 2 a k + 2 k + 10 , (296) and we note that a k > 4 p k a_k > 4p_k a k > 4 p k
Lemma 8.7. There exists Ω 0 ( k ) ≥ 3 \Omega_0(k) \ge3 Ω 0 ( k ) ≥ 3 K ≥ 1 K \ge1 K ≥ 1 Ω ≥ max { Ω 0 ( k ) , K 2 } \Omega\ge\max\{\Omega_0(k),K^2\} Ω ≥ max { Ω 0 ( k ) , K 2 } max b ∣ r b ∣ ≤ K \max_b |r_b| \le K max b ∣ r b ∣ ≤ K u u u U U U
inf v ∈ V ⌊ Ω p k ⌋ ∥ N 1 / 2 u − v ∥ 2 , N > Ω − a k . (297) \inf_{v\in V_{\lfloor\Omega^{p_k}\rfloor}}\lVert N^{1/2}u-v\rVert_{2,N}>\Omega^{-a_k}. \tag*{(297)} v ∈ V ⌊ Ω p k ⌋ inf ∥ N 1/2 u − v ∥ 2 , N > Ω − a k . (297) Then D U ( ξ ) ≥ Ω − q k D_U(\xi) \ge\Omega^{-q_k} D U ( ξ ) ≥ Ω − q k ξ ∈ R 2 k \xi\in\mathbb{R}^{2k} ξ ∈ R 2 k Ω − 1 ≤ ∣ ξ ∣ ≤ Ω K − 2 \Omega^{-1} \le|\xi| \le\Omega K^{-2} Ω − 1 ≤ ∣ ξ ∣ ≤ Ω K − 2
Proof. We argue by contradiction. Suppose that there exists ξ \xi ξ Ω − 1 ≤ ∣ ξ ∣ ≤ Ω K − 2 \Omega^{-1} \le|\xi| \le\Omega K^{-2} Ω − 1 ≤ ∣ ξ ∣ ≤ Ω K − 2 D U ( ξ ) < Ω − q k D_U(\xi) < \Omega^{-q_k} D U ( ξ ) < Ω − q k D = D U ( ξ ) D = D_U(\xi) D = D U ( ξ ) a , b ∈ [ N ] a,b \in[N] a , b ∈ [ N ]
Θ a b = dist ( r a T Γ ( ξ ) r b , 2 π Z ) ∈ [ 0 , π ] . (298) \Theta_{ab}=\operatorname{dist}\left(r_a^T\Gamma(\xi)r_b,2\pi\mathbb{Z}\right)\in[0,\pi]. \tag*{(298)} Θ ab = dist ( r a T Γ ( ξ ) r b , 2 π Z ) ∈ [ 0 , π ] . (298) For real t ′ t' t ′ dist ( t ′ , 2 π Z ) = Θ \operatorname{dist}(t',2\pi\mathbb{Z})=\Theta dist ( t ′ , 2 π Z ) = Θ
1 − cos t ′ = 2 sin 2 ( Θ / 2 ) ≥ 2 Θ 2 π 2 , (299) 1-\cos t'=2\sin^2(\Theta/2)\ge\frac{2\Theta^2}{\pi^2}, \tag*{(299)} 1 − cos t ′ = 2 sin 2 ( Θ/2 ) ≥ π 2 2 Θ 2 , (299) where we used sin y ′ ≥ 2 y ′ / π \sin y' \ge2y'/\pi sin y ′ ≥ 2 y ′ / π y ′ ∈ [ 0 , π / 2 ] y' \in[0,\pi/2] y ′ ∈ [ 0 , π /2 ] a a a b b b
1 N 2 ∑ a , b = 1 N Θ a b 2 ≤ π 2 2 D . (300) \frac{1}{N^2}\sum_{a,b=1}^{N}\Theta_{ab}^2\le\frac{\pi^2}{2}D. \tag*{(300)} N 2 1 a , b = 1 ∑ N Θ ab 2 ≤ 2 π 2 D . (300) We first discard the rows with large errors. For a ∈ [ N ] a \in[N] a ∈ [ N ]
Θ ‾ a = 1 N ∑ b = 1 N Θ a b 2 , O = { a ∈ [ N ] : Θ ‾ a ≤ 2 π 2 K 2 D } . (301) \overline{\Theta}_a=\frac{1}{N}\sum_{b=1}^{N}\Theta_{ab}^2,\qquad\mathcal{O}=\left\{a\in[N]:\overline{\Theta}_a\le2\pi^2K^2D\right\}. \tag*{(301)} Θ a = N 1 b = 1 ∑ N Θ ab 2 , O = { a ∈ [ N ] : Θ a ≤ 2 π 2 K 2 D } . (301) When D > 0 \mathcal{D}>0 D > 0
∣ [ N ] ∖ O ∣ ≤ N 4 K 2 , (302) |[N]\setminus\mathcal{O}|\le\frac{N}{4K^2}, \tag*{(302)} ∣ [ N ] ∖ O ∣ ≤ 4 K 2 N , (302) and when D = 0 \mathcal{D}=0 D = 0 O = [ N ] \mathcal{O}=[N] O = [ N ] ∣ r a ∣ ≤ K |r_a|\le K ∣ r a ∣ ≤ K
1 N ∑ a ∈ O r a r a T = I d 2 k − 1 N ∑ a ∉ O r a r a T ≥ ( 1 − K 2 ∣ [ N ] ∖ O ∣ N ) I d 2 k ≥ 3 4 I d 2 k . (303) \frac{1}{N}\sum_{a\in\mathcal{O}}r_a r_a^{T}=\mathrm{Id}_{2k}-\frac{1}{N}\sum_{a\notin\mathcal{O}}r_a r_a^{T}\ge\left(1-\frac{K^2|[N]\setminus\mathcal{O}|}{N}\right)\mathrm{Id}_{2k}\ge\frac{3}{4}\mathrm{Id}_{2k}. \tag*{(303)} N 1 a ∈ O ∑ r a r a T = Id 2 k − N 1 a ∈ / O ∑ r a r a T ≥ ( 1 − N K 2 ∣ [ N ] ∖ O ∣ ) Id 2 k ≥ 4 3 Id 2 k . (303) Next, we select 2 k 2k 2 k O 0 ⊂ O \mathcal{O}_0\subset\mathcal{O} O 0 ⊂ O 2 k 2k 2 k Q ( O 0 ) Q(\mathcal{O}_0) Q ( O 0 ) 2 k × 2 k 2k\times2k 2 k × 2 k r a T r_a^T r a T a ∈ O 0 a\in\mathcal{O}_0 a ∈ O 0 a a a r a T r_a^T r a T a ∈ O a\in\mathcal{O} a ∈ O
det ( ∑ a ∈ O r a r a T ) = ∑ O 0 ⊂ O , ∣ O 0 ∣ = 2 k det ( Q ( O 0 ) ) 2 . (304) \det\left(\sum_{a\in\mathcal{O}}r_a r_a^T\right)=\sum_{\mathcal{O}_0\subset\mathcal{O},\,|\mathcal{O}_0|=2k}\det\left(Q(\mathcal{O}_0)\right)^2. \tag*{(304)} det ( a ∈ O ∑ r a r a T ) = O 0 ⊂ O , ∣ O 0 ∣ = 2 k ∑ det ( Q ( O 0 ) ) 2 . (304) By (8.58), the left side is at least ( 3 N / 4 ) 2 k (3N/4)^{2k} ( 3 N /4 ) 2 k ∣ O ∣ ≥ 2 k |\mathcal{O}|\ge2k ∣ O ∣ ≥ 2 k ( N 2 k ) ≤ N 2 k / ( 2 k ) ! \binom{N}{2k}\le N^{2k}/(2k)! ( 2 k N ) ≤ N 2 k / ( 2 k )! O 0 \mathcal{O}_0 O 0 Q = Q ( O 0 ) Q=Q(\mathcal{O}_0) Q = Q ( O 0 )
∣ det Q ∣ ≥ ( 3 / 4 ) k ( ( 2 k ) ! ) 1 / 2 . (305) |\det Q|\ge(3/4)^k((2k)!)^{1/2}. \tag*{(305)} ∣ det Q ∣ ≥ ( 3/4 ) k (( 2 k )! ) 1/2 . (305) We now bound ∥ Q − 1 ∥ \|Q^{-1}\| ∥ Q − 1 ∥ Q Q Q K K K ∥ Q ∥ ≤ ( 2 k ) 1 / 2 K \|Q\|\le(2k)^{1/2}K ∥ Q ∥ ≤ ( 2 k ) 1/2 K 2 k 2k 2 k Q Q Q Q Q Q ∣ det Q ∣ ∥ Q ∥ 1 − 2 k |\det Q|\|Q\|^{1-2k} ∣ det Q ∣∥ Q ∥ 1 − 2 k
∥ Q − 1 ∥ ≤ ∥ Q ∥ 2 k − 1 ∣ det Q ∣ ≤ C k K 2 k − 1 . (306) \|Q^{-1}\|\le\frac{\|Q\|^{2k-1}}{|\det Q|}\le C_kK^{2k-1}. \tag*{(306)} ∥ Q − 1 ∥ ≤ ∣ det Q ∣ ∥ Q ∥ 2 k − 1 ≤ C k K 2 k − 1 . (306) Next, we round the phases against the rows of Q Q Q a ∈ O 0 a\in\mathcal{O}_0 a ∈ O 0 b ∈ [ N ] b\in[N] b ∈ [ N ] n b ( a ) ∈ Z n_b(a)\in\mathbb{Z} n b ( a ) ∈ Z
∣ r a T Γ ( ξ ) r b − 2 π n b ( a ) ∣ = Θ a b . (307) \left|r_a^T\Gamma(\xi)r_b-2\pi n_b(a)\right|=\Theta_{ab}. \tag*{(307)} r a T Γ ( ξ ) r b − 2 π n b ( a ) = Θ ab . (307) We let n b ∈ Z 2 k \mathbf{n}_b\in\mathbb{Z}^{2k} n b ∈ Z 2 k n b ( a ) n_b(a) n b ( a ) a ∈ O 0 a\in\mathcal{O}_0 a ∈ O 0 a a a Q Γ ( ξ ) r b Q\Gamma(\xi)r_b Q Γ ( ξ ) r b r a T Γ ( ξ ) r b r_a^T\Gamma(\xi)r_b r a T Γ ( ξ ) r b a ∈ O 0 a\in\mathcal{O}_0 a ∈ O 0 O 0 ⊂ O \mathcal{O}_0\subset\mathcal{O} O 0 ⊂ O
1 N ∑ b = 1 N ∣ Q Γ ( ξ ) r b − 2 π n b ∣ 2 = ∑ a ∈ O 0 Θ ‾ a ≤ 4 k π 2 K 2 D . (308) \frac{1}{N}\sum_{b=1}^{N}|Q\Gamma(\xi)r_b-2\pi\mathbf{n}_b|^2=\sum_{a\in\mathcal{O}_0}\overline{\Theta}_a\le4k\pi^2K^2\mathcal{D}. \tag*{(308)} N 1 b = 1 ∑ N ∣ Q Γ ( ξ ) r b − 2 π n b ∣ 2 = a ∈ O 0 ∑ Θ a ≤ 4 k π 2 K 2 D . (308) Further, the bound ∣ ξ ∣ ≤ Ω K − 2 |\xi|\le\Omega K^{-2} ∣ ξ ∣ ≤ Ω K − 2
∣ r a T Γ ( ξ ) r b ∣ ≤ ∥ Γ ( ξ ) ∥ K 2 ≤ ∣ ξ ∣ K 2 ≤ Ω , so ∣ n b ( a ) ∣ ≤ Ω 2 π + 1 2 ≤ Ω . (309) |r_a^T\Gamma(\xi)r_b|\le\|\Gamma(\xi)\|K^2\le|\xi|K^2\le\Omega,
\qquad\text{so}\qquad
|n_b(a)|\le\frac{\Omega}{2\pi}+\frac{1}{2}\le\Omega. \tag*{(309)} ∣ r a T Γ ( ξ ) r b ∣ ≤ ∥Γ ( ξ ) ∥ K 2 ≤ ∣ ξ ∣ K 2 ≤ Ω , so ∣ n b ( a ) ∣ ≤ 2 π Ω + 2 1 ≤ Ω. (309) Hence, as b b b [ N ] [N] [ N ] n b \mathbf{n}_b n b ( 2 Ω + 1 ) 2 k ≤ ( 3 Ω ) 2 k (2\Omega+1)^{2k}\le(3\Omega)^{2k} ( 2Ω + 1 ) 2 k ≤ ( 3Ω ) 2 k
We now reconstruct one block of the rows r b r_b r b n b n_b n b j ∈ [ k ] j \in[k] j ∈ [ k ] ∣ ζ j ∣ = max j ′ ∣ ζ j ′ ∣ |\zeta_j| = \max_{j'} |\zeta_{j'}| ∣ ζ j ∣ = max j ′ ∣ ζ j ′ ∣
∣ ζ j ∣ ≥ k − 1 / 2 ∣ ζ ∣ ≥ k − 1 / 2 Ω − 1 . (310) |\zeta_j| \ge k^{-1/2}|\zeta| \ge k^{-1/2}\Omega^{-1}. \tag*{(310)} ∣ ζ j ∣ ≥ k − 1/2 ∣ ζ ∣ ≥ k − 1/2 Ω − 1 . (310) Since Γ ( ζ ) \Gamma(\zeta) Γ ( ζ ) j j j Γ ( ζ ) r b \Gamma(\zeta)r_b Γ ( ζ ) r b Γ ( ζ j ) r b , j \Gamma(\zeta_j)r_{b,j} Γ ( ζ j ) r b , j b ∈ [ N ] b \in[N] b ∈ [ N ]
r ^ b = Γ ( ζ j ) − 1 ( Q − 1 2 π n b ) j ∈ R 2 . (311) \hat r_b = \Gamma(\zeta_j)^{-1}\left(Q^{-1}2\pi n_b\right)_j \in\mathbb{R}^2. \tag*{(311)} r ^ b = Γ ( ζ j ) − 1 ( Q − 1 2 π n b ) j ∈ R 2 . (311) Then
r b , j − r ^ b = Γ ( ζ j ) − 1 ( Q − 1 ( Q Γ ( ζ ) r b − 2 π n b ) ) j . (312) r_{b,j}-\hat r_b=\Gamma(\zeta_j)^{-1}\left(Q^{-1}\left(Q\Gamma(\zeta)r_b-2\pi n_b\right)\right)_j. \tag*{(312)} r b , j − r ^ b = Γ ( ζ j ) − 1 ( Q − 1 ( Q Γ ( ζ ) r b − 2 π n b ) ) j . (312) Using ∥ Γ ( ζ j ) − 1 ∥ = ∣ ζ j ∣ − 1 \|\Gamma(\zeta_j)^{-1}\|=|\zeta_j|^{-1} ∥Γ ( ζ j ) − 1 ∥ = ∣ ζ j ∣ − 1 (306) , and (308) , we obtain
( 1 N ∑ b = 1 N ∣ r b , j − r ^ b ∣ 2 ) 1 / 2 ≤ ∥ Q − 1 ∥ ∣ ζ j ∣ ( 4 k π 2 K 2 D ) 1 / 2 ≤ C k Ω K 2 k D 1 / 2 ≤ C k Ω k + 1 − q k / 2 = C k Ω − α k − 4 . (313) \begin{aligned}
\left(\frac{1}{N}\sum_{b=1}^{N}|r_{b,j}-\hat r_b|^2\right)^{1/2}
&\le\frac{\|Q^{-1}\|}{|\zeta_j|}(4k\pi^2K^2\mathcal{D})^{1/2}
\le C_k\Omega K^{2k}\mathcal{D}^{1/2}\\
&\le C_k\Omega^{k+1-q_k/2}=C_k\Omega^{-\alpha_k-4}. \tag*{(313)}
\end{aligned} ( N 1 b = 1 ∑ N ∣ r b , j − r ^ b ∣ 2 ) 1/2 ≤ ∣ ζ j ∣ ∥ Q − 1 ∥ ( 4 k π 2 K 2 D ) 1/2 ≤ C k Ω K 2 k D 1/2 ≤ C k Ω k + 1 − q k /2 = C k Ω − α k − 4 . (313) In the third inequality, we used K 2 ≤ Ω K^2\le\Omega K 2 ≤ Ω D < Ω − q k \mathcal{D}<\Omega^{-q_k} D < Ω − q k q k q_k q k (296) .
It remains to derive the contradiction. Let u u u ( 2 j − 1 ) (2j-1) ( 2 j − 1 ) U U U N 1 / 2 u ( b ) N^{1/2}u(b) N 1/2 u ( b ) r b , j r_{b,j} r b , j v ∈ R N v\in\mathbb{R}^N v ∈ R N b b b r ^ b \hat r_b r ^ b r ^ b \hat r_b r ^ b b b b n b n_b n b v v v ( 3 Ω ) 2 k (3\Omega)^{2k} ( 3Ω ) 2 k Ω ≥ 3 \Omega\ge3 Ω ≥ 3 ( 3 Ω ) 2 k ≤ ⌊ Ω p k ⌋ (3\Omega)^{2k}\le\lfloor\Omega^{p_k}\rfloor ( 3Ω ) 2 k ≤ ⌊ Ω p k ⌋ v ∈ V ⌊ Ω p k ⌋ v\in V_{\lfloor\Omega^{p_k}\rfloor} v ∈ V ⌊ Ω p k ⌋ (313) , we have
∥ N 1 / 2 u − v ∥ 2 , N ≤ C k Ω − α k − 4 ≤ Ω − α k , (314) \|N^{1/2}u-v\|_{2,N}\le C_k\Omega^{-\alpha_k-4}\le\Omega^{-\alpha_k}, \tag*{(314)} ∥ N 1/2 u − v ∥ 2 , N ≤ C k Ω − α k − 4 ≤ Ω − α k , (314) where the second inequality holds when Ω 4 ≥ C k \Omega^4\ge C_k Ω 4 ≥ C k (297) , and the lemma follows with Ω 0 ( k ) = max { 3 , C k 1 / 4 } \Omega_0(k)=\max\{3,C_k^{1/4}\} Ω 0 ( k ) = max { 3 , C k 1/4 } C k C_k C k (313) . □ \square □
Remark 8.8. The resampling selects only pairs a < b a<b a < b Φ H \Phi_H Φ H r a T Γ ( ζ ) r b r_a^T\Gamma(\zeta)r_b r a T Γ ( ζ ) r b a < b a<b a < b
For ζ ∈ R 2 k \zeta\in\mathbb{R}^{2k} ζ ∈ R 2 k
D U o f f ( ζ ) = ( N 2 ) − 1 ∑ a < b [ 1 − cos ( r a T Γ ( ζ ) r b ) ] . (315) \mathcal{D}^{\mathrm{off}}_U(\zeta)=\binom{N}{2}^{-1}\sum_{a<b}\left[1-\cos\left(r_a^T\Gamma(\zeta)r_b\right)\right]. \tag*{(315)} D U off ( ζ ) = ( 2 N ) − 1 a < b ∑ [ 1 − cos ( r a T Γ ( ζ ) r b ) ] . (315) Corollary 8.9. Let K K K Ω \Omega Ω U U U Ω q k ≤ N / 4 \Omega^{q_k}\le N/4 Ω q k ≤ N /4 D U o f f ( ζ ) ≥ Ω − q k / 2 \mathcal{D}^{\mathrm{off}}_U(\zeta)\ge\Omega^{-q_k/2} D U off ( ζ ) ≥ Ω − q k /2 ζ ∈ R 2 k \zeta\in\mathbb{R}^{2k} ζ ∈ R 2 k Ω − 1 ≤ ∣ ζ ∣ ≤ Ω K − 2 \Omega^{-1}\le|\zeta|\le\Omega K^{-2} Ω − 1 ≤ ∣ ζ ∣ ≤ Ω K − 2
Proof. The summand in (295) is symmetric in ( a , b ) (a,b) ( a , b ) [ 0 , 2 ] [0,2] [ 0 , 2 ]
D U ( ξ ) = ( 1 − 1 N ) D U o f f ( ξ ) + 1 N 2 ∑ a = 1 N [ 1 − cos ( r a T Γ ( ξ ) r a ) ] ≤ D U o f f ( ξ ) + 2 N . (316) \mathcal{D}_U(\xi)=\left(1-\frac{1}{N}\right)\mathcal{D}^{\mathrm{off}}_U(\xi)+\frac{1}{N^2}\sum_{a=1}^{N}\left[1-\cos\left(r_a^{\mathsf T}\Gamma(\xi)r_a\right)\right]\leq\mathcal{D}^{\mathrm{off}}_U(\xi)+\frac{2}{N}. \tag*{(316)} D U ( ξ ) = ( 1 − N 1 ) D U off ( ξ ) + N 2 1 a = 1 ∑ N [ 1 − cos ( r a T Γ ( ξ ) r a ) ] ≤ D U off ( ξ ) + N 2 . (316) For Ω − 1 ≤ ∣ ξ ∣ ≤ Ω K − 2 \Omega^{-1}\leq|\xi|\leq\Omega K^{-2} Ω − 1 ≤ ∣ ξ ∣ ≤ Ω K − 2 Lemma 8.7 and the condition Ω q k ≤ N / 4 \Omega^{q_k}\leq N/4 Ω q k ≤ N /4 D U o f f ( ξ ) ≥ Ω − q k − 2 N − 1 ≥ Ω − q k / 2 \mathcal{D}^{\mathrm{off}}_U(\xi)\geq\Omega^{-q_k}-2N^{-1}\geq\Omega^{-q_k}/2 D U off ( ξ ) ≥ Ω − q k − 2 N − 1 ≥ Ω − q k /2
The characteristic function of one increment
We now convert Lemma 8.7 into a bound on ∣ Φ H ∣ |\Phi_H| ∣ Φ H ∣ Y ′ Y' Y ′ Y Y Y Y Y Y E ( Y − Y ′ ) 2 = 2 \mathbb{E}(Y-Y')^2=2 E ( Y − Y ′ ) 2 = 2 P ( Y ≠ Y ′ ) > 0 \mathbb{P}(Y\neq Y')>0 P ( Y = Y ′ ) > 0 0 < c − < c + 0<c_-<c_+ 0 < c − < c + X X X
p Y = P ( c − ≤ ∣ Y − Y ′ ∣ ≤ c + ) > 0. (317) p_Y=\mathbb{P}\left(c_-\leq|Y-Y'|\leq c_+\right)>0. \tag*{(317)} p Y = P ( c − ≤ ∣ Y − Y ′ ∣ ≤ c + ) > 0. (317) Lemma 8.10. Let K ≥ 1 K\geq1 K ≥ 1 Ω ≥ max { Ω 0 ( k ) , K 2 } \Omega\geq\max\{\Omega_0(k),K^2\} Ω ≥ max { Ω 0 ( k ) , K 2 } Ω q k ≤ N / 4 \Omega^{q_k}\leq N/4 Ω q k ≤ N /4 i i i k k k H H H U = U i U=U_i U = U i Lemma 8.7 . Then
1 − ∣ Φ H ( ξ ) ∣ 2 ≥ p Y 2 Ω − q k for all ξ ∈ R 2 k such that 1 c − Ω ≤ ∣ ξ ∣ ≤ Ω c + K 2 . (318) 1-|\Phi_H(\xi)|^2\geq\frac{p_Y}{2}\Omega^{-q_k}\qquad\text{for all }\xi\in\mathbb{R}^{2k}\text{ such that }\frac{1}{c_-\Omega}\leq|\xi|\leq\frac{\Omega}{c_+K^2}. \tag*{(318)} 1 − ∣ Φ H ( ξ ) ∣ 2 ≥ 2 p Y Ω − q k for all ξ ∈ R 2 k such that c − Ω 1 ≤ ∣ ξ ∣ ≤ c + K 2 Ω . (318) Proof. Conditioning on ( a , b ) (a,b) ( a , b ) (252) and using the independence of Y Y Y
Φ H ( ξ ) = ( N 2 ) − 1 ∑ a < b e − i X a b ξ ⋅ D a b ( i ) Φ Y ( ξ ⋅ D a b ( i ) ) . (319) \Phi_H(\xi)=\binom{N}{2}^{-1}\sum_{a<b}e^{-\mathrm{i}X_{ab}\xi\cdot D_{ab}^{(i)}}\Phi_Y\left(\xi\cdot D_{ab}^{(i)}\right). \tag*{(319)} Φ H ( ξ ) = ( 2 N ) − 1 a < b ∑ e − i X ab ξ ⋅ D ab ( i ) Φ Y ( ξ ⋅ D ab ( i ) ) . (319) We symmetrize with the Cauchy–Schwarz inequality, which removes the phases e − i X a b ξ ⋅ D a b ( i ) e^{-\mathrm{i}X_{ab}\xi\cdot D_{ab}^{(i)}} e − i X ab ξ ⋅ D ab ( i )
∣ Φ Y ( t ′ ) ∣ 2 = E e i t ′ ( Y − Y ′ ) = E cos ( t ′ ( Y − Y ′ ) ) , t ′ ∈ R , (320) |\Phi_Y(t')|^2=\mathbb{E}e^{\mathrm{i}t'(Y-Y')}=\mathbb{E}\cos\left(t'(Y-Y')\right),\qquad t'\in\mathbb{R}, \tag*{(320)} ∣ Φ Y ( t ′ ) ∣ 2 = E e i t ′ ( Y − Y ′ ) = E cos ( t ′ ( Y − Y ′ ) ) , t ′ ∈ R , (320) we obtain
∣ Φ H ( ξ ) ∣ 2 ≤ ( N 2 ) − 1 ∑ a < b ∣ Φ Y ( ξ ⋅ D a b ( i ) ) ∣ 2 = E [ ( N 2 ) − 1 ∑ a < b cos ( ( Y − Y ′ ) ξ ⋅ D a b ( i ) ) ] . (321) |\Phi_H(\xi)|^2\leq\binom{N}{2}^{-1}\sum_{a<b}\left|\Phi_Y\left(\xi\cdot D_{ab}^{(i)}\right)\right|^2=\mathbb{E}\left[\binom{N}{2}^{-1}\sum_{a<b}\cos\left((Y-Y')\xi\cdot D_{ab}^{(i)}\right)\right]. \tag*{(321)} ∣ Φ H ( ξ ) ∣ 2 ≤ ( 2 N ) − 1 a < b ∑ Φ Y ( ξ ⋅ D ab ( i ) ) 2 = E [ ( 2 N ) − 1 a < b ∑ cos ( ( Y − Y ′ ) ξ ⋅ D ab ( i ) ) ] . (321) By (248) and the linearity of Γ \Gamma Γ
( Y − Y ′ ) ξ ⋅ D a b ( i ) = r a T Γ ( ( Y − Y ′ ) ξ ) r b . (322) (Y-Y')\xi\cdot D_{ab}^{(i)}=r_a^{\mathsf T}\Gamma\left((Y-Y')\xi\right)r_b. \tag*{(322)} ( Y − Y ′ ) ξ ⋅ D ab ( i ) = r a T Γ ( ( Y − Y ′ ) ξ ) r b . (322) Hence (321) and (315) give
1 − ∣ Φ H ( ξ ) ∣ 2 ≥ E [ D U o f f ( ( Y − Y ′ ) ξ ) ] ≥ E [ D U o f f ( ( Y − Y ′ ) ξ ) 1 { c − ≤ ∣ Y − Y ′ ∣ ≤ c + } ] . (323) 1-|\Phi_H(\xi)|^2\geq\mathbb{E}\left[\mathcal{D}^{\mathrm{off}}_U\left((Y-Y')\xi\right)\right]\geq\mathbb{E}\left[\mathcal{D}^{\mathrm{off}}_U\left((Y-Y')\xi\right)\mathbf{1}\{c_-\leq|Y-Y'|\leq c_+\}\right]. \tag*{(323)} 1 − ∣ Φ H ( ξ ) ∣ 2 ≥ E [ D U off ( ( Y − Y ′ ) ξ ) ] ≥ E [ D U off ( ( Y − Y ′ ) ξ ) 1 { c − ≤ ∣ Y − Y ′ ∣ ≤ c + } ] . (323) where the second inequality holds because D U o f f ≥ 0 D^{\mathrm{off}}_{U} \ge0 D U off ≥ 0 ξ ~ \widetilde{\xi} ξ (318) and every realization with c − ≤ ∣ Y − Y ′ ∣ ≤ c + c_- \le|Y-Y'| \le c_+ c − ≤ ∣ Y − Y ′ ∣ ≤ c +
Ω − 1 ≤ c − ∣ ξ ~ ∣ ≤ ∣ ( Y − Y ′ ) ξ ~ ∣ ≤ c + ∣ ξ ~ ∣ ≤ Ω K − 2 . (324) \Omega^{-1} \le c_-|\widetilde{\xi}| \le|(Y-Y')\widetilde{\xi}| \le c_+|\widetilde{\xi}| \le\Omega K^{-2}. \tag*{(324)} Ω − 1 ≤ c − ∣ ξ ∣ ≤ ∣ ( Y − Y ′ ) ξ ∣ ≤ c + ∣ ξ ∣ ≤ Ω K − 2 . (324) Therefore Corollary 8.9 gives
D U o f f ( ( Y − Y ′ ) ξ ~ ) ≥ Ω − q k / 2 (325) D^{\mathrm{off}}_{U}\bigl((Y-Y')\widetilde{\xi}\bigr) \ge\Omega^{-q_k}/2 \tag*{(325)} D U off ( ( Y − Y ′ ) ξ ) ≥ Ω − q k /2 (325) on this event. Taking expectations and using (317) completes the proof.
A local limit theorem for sums of increments
We now fix the parameters. Recall that τ = N 1 / 100 \tau=N^{1/100} τ = N 1/100
ε k = 1 4000 q k , Ω = τ 1 / ( 4 q k ) = N 10 ε k , K = ( 2 k ) 1 / 2 N ε k . (326) \varepsilon_k=\frac{1}{4000q_k},\qquad\Omega=\tau^{1/(4q_k)}=N^{10\varepsilon_k},\qquad K=(2k)^{1/2}N^{\varepsilon_k}. \tag*{(326)} ε k = 4000 q k 1 , Ω = τ 1/ ( 4 q k ) = N 10 ε k , K = ( 2 k ) 1/2 N ε k . (326) Let C m o m ≥ 1 C_{\mathrm{mom}}\ge1 C mom ≥ 1 C C C ε = ε k \varepsilon=\varepsilon_k ε = ε k
ϱ N = ( 12 C m o m ) − 1 N − 9 ε k . (327) \varrho_N=(12C_{\mathrm{mom}})^{-1}N^{-9\varepsilon_k}. \tag*{(327)} ϱ N = ( 12 C mom ) − 1 N − 9 ε k . (327) In the proof of Lemma 8.13, ϱ N \varrho_N ϱ N Φ H \Phi_H Φ H Ω \Omega Ω Ω − q k = τ − 1 / 4 \Omega^{-q_k}=\tau^{-1/4} Ω − q k = τ − 1/4 m Ω − q k m\Omega^{-q_k} m Ω − q k τ \tau τ Ω \Omega Ω K 2 K^2 K 2 ϱ N − 1 \varrho_N^{-1} ϱ N − 1 N 10 ε k N^{10\varepsilon_k} N 10 ε k N 2 ε k N^{2\varepsilon_k} N 2 ε k N 9 ε k N^{9\varepsilon_k} N 9 ε k N 0 N_0 N 0 N ≥ N 0 N\ge N_0 N ≥ N 0
Ω ≥ max { Ω 0 ( k ) , K 2 } , Ω q k ≤ N 4 , c + + K 2 ≤ Ω 1 / 2 , 1 c − Ω ≤ ϱ N . (328) \Omega\ge\max\{\Omega_0(k),K^2\},\qquad\Omega^{q_k}\le\frac{N}{4},\qquad c_++K^2\le\Omega^{1/2},\qquad\frac{1}{c_-\Omega}\le\varrho_N. \tag*{(328)} Ω ≥ max { Ω 0 ( k ) , K 2 } , Ω q k ≤ 4 N , c + + K 2 ≤ Ω 1/2 , c − Ω 1 ≤ ϱ N . (328) The last condition dictates the choice of ε k \varepsilon_k ε k Ω = N 1 / ( 400 q k ) \Omega=N^{1/(400q_k)} Ω = N 1/ ( 400 q k ) τ \tau τ N N N 9 ε k ≤ 1 / ( 400 q k ) 9\varepsilon_k\le1/(400q_k) 9 ε k ≤ 1/ ( 400 q k ) 9 ε k < 10 ε k = 1 / ( 400 q k ) 9\varepsilon_k<10\varepsilon_k=1/(400q_k) 9 ε k < 10 ε k = 1/ ( 400 q k ) 10 ε k p k < 1 10\varepsilon_kp_k<1 10 ε k p k < 1 10 ε k ( a k + 1 ) < 1 / 2 10\varepsilon_k(a_k+1)<1/2 10 ε k ( a k + 1 ) < 1/2
Ω p k = o ( N ) , Ω a k + 1 = o ( N 1 / 2 ) . (329) \Omega^{p_k}=o(N),\qquad\Omega^{a_k+1}=o(N^{1/2}). \tag*{(329)} Ω p k = o ( N ) , Ω a k + 1 = o ( N 1/2 ) . (329) Definition 8.11. We let A N , k A_{N,k} A N , k A N r e g ( ε k ) \mathcal A^{\mathrm{reg}}_N(\varepsilon_k) A N reg ( ε k ) w w w H H H
inf v ∈ V ⌊ Ω p k ⌋ ∥ N 1 / 2 w − v ∥ 2 , N > Ω − a k . (330) \inf_{v\in\mathcal V_{\lfloor\Omega^{p_k}\rfloor}}\|N^{1/2}w-v\|_{2,N}>\Omega^{-a_k}. \tag*{(330)} v ∈ V ⌊ Ω p k ⌋ inf ∥ N 1/2 w − v ∥ 2 , N > Ω − a k . (330) The event A N , k A_{N,k} A N , k i \mathbf{i} i H H H
Lemma 8.12. The event A N , k \mathcal{A}_{N,k} A N , k N ≥ N 0 N \ge N_0 N ≥ N 0 A N , k \mathcal{A}_{N,k} A N , k k k k i \mathbf{i} i
1 − ∣ Φ H ( ξ ) ∣ 2 ≥ p Y 2 τ − 1 / 4 for all ξ ∈ R 2 k such that ϱ N ≤ ∣ ξ ∣ ≤ Ω 1 / 2 . (331) 1-\lvert\Phi_H(\xi)\rvert^2 \ge\frac{p_Y}{2}\tau^{-1/4}\qquad\text{for all }\xi\in\mathbb{R}^{2k}\text{ such that }\varrho_N\le\lvert\xi\rvert\le\Omega^{1/2}. \tag*{(331)} 1 − ∣ Φ H ( ξ ) ∣ 2 ≥ 2 p Y τ − 1/4 for all ξ ∈ R 2 k such that ϱ N ≤ ∣ ξ ∣ ≤ Ω 1/2 . (331) Proof. Recall that ∥ H ∥ ≤ 3 \lVert H\rVert\le3 ∥ H ∥ ≤ 3 w w w H H H N 1 / 2 ∥ w ∥ ∞ ≤ N ε k ≤ K N^{1/2}\lVert w\rVert_\infty\le N^{\varepsilon_k}\le K N 1/2 ∥ w ∥ ∞ ≤ N ε k ≤ K A N r e g ( ε k ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon_k) A N reg ( ε k ) A N , k \mathcal{A}_{N,k} A N , k A N r e g ( ε k ) \mathcal{A}^{\mathrm{reg}}_N(\varepsilon_k) A N reg ( ε k ) p = p k p=p_k p = p k a = a k a=a_k a = a k Ω \Omega Ω K K K (326) . The hypotheses of Lemma 8.5 hold by (329) , and K ≤ Ω K\le\Omega K ≤ Ω N N N A N , k \mathcal{A}_{N,k} A N , k
Now let N ≥ N 0 N\ge N_0 N ≥ N 0 H H H A N , k \mathcal{A}_{N,k} A N , k i \mathbf{i} i k k k N 1 / 2 U i N^{1/2}U_{\mathbf{i}} N 1/2 U i ∣ r b ∣ ≤ K \lvert r_b\rvert\le K ∣ r b ∣ ≤ K (255) . The columns of U i U_{\mathbf{i}} U i H H H A N , k \mathcal{A}_{N,k} A N , k (297) . Together with the first two conditions in (328) , this verifies the hypotheses of Lemma 8.10. The last two conditions in (328) show that every ξ \xi ξ ϱ N ≤ ∣ ξ ∣ ≤ Ω 1 / 2 \varrho_N\le\lvert\xi\rvert\le\Omega^{1/2} ϱ N ≤ ∣ ξ ∣ ≤ Ω 1/2 (318) . Since Ω − q k = τ − 1 / 4 \Omega^{-q_k}=\tau^{-1/4} Ω − q k = τ − 1/4 (318) gives (331) .
For a positive definite 2 k × 2 k 2k\times2k 2 k × 2 k Σ \Sigma Σ g Σ g_\Sigma g Σ R 2 k \mathbb{R}^{2k} R 2 k Σ \Sigma Σ
g Σ ( y ) = ( 2 π ) − k ( det Σ ) − 1 / 2 exp ( − 1 2 y T Σ − 1 y ) , y ∈ R 2 k . (332) g_\Sigma(y)=(2\pi)^{-k}(\det\Sigma)^{-1/2}\exp\left(-\frac{1}{2}y^{\mathsf T}\Sigma^{-1}y\right),\qquad y\in\mathbb{R}^{2k}. \tag*{(332)} g Σ ( y ) = ( 2 π ) − k ( det Σ ) − 1/2 exp ( − 2 1 y T Σ − 1 y ) , y ∈ R 2 k . (332) This extends the notation (69) on R 2 k \mathbb{R}^{2k} R 2 k g ν = g ν Id 2 k g_\nu=g_{\nu\operatorname{Id}_{2k}} g ν = g ν Id 2 k ν > 0 \nu>0 ν > 0 g 4 n Id 2 k ( y ) = ( 8 π n ) − k e − ∣ y ∣ 2 / ( 8 n ) g_{4n\operatorname{Id}_{2k}}(y)=(8\pi n)^{-k}e^{-\lvert y\rvert^2/(8n)} g 4 n Id 2 k ( y ) = ( 8 π n ) − k e − ∣ y ∣ 2 / ( 8 n ) [7 ] ), made uniform in H H H i \mathbf{i} i (331) .
Lemma 8.13. Let Ψ ⊂ C ∞ ( R 2 k ) \Psi\subset C^\infty(\mathbb{R}^{2k}) Ψ ⊂ C ∞ ( R 2 k ) R Ψ > 0 R_\Psi>0 R Ψ > 0 supp ψ ⊂ { y ∈ R 2 k : ∣ y ∣ ≤ R Ψ } \operatorname{supp}\psi\subset\{y\in\mathbb{R}^{2k}:\lvert y\rvert\le R_\Psi\} supp ψ ⊂ { y ∈ R 2 k : ∣ y ∣ ≤ R Ψ } ψ ∈ Ψ \psi\in\Psi ψ ∈ Ψ ψ ∈ Ψ \psi\in\Psi ψ ∈ Ψ A N , k \mathcal{A}_{N,k} A N , k k k k i \mathbf{i} i n n n τ / 10 ≤ n ≤ τ / 3 \tau/10\le n\le\tau/3 τ /10 ≤ n ≤ τ /3 x ∈ R 2 k x\in\mathbb{R}^{2k} x ∈ R 2 k ψ ∈ Ψ \psi\in\Psi ψ ∈ Ψ
E [ ψ ( S ( n ) + x ) ∣ H ] = g 4 n Id 2 k ( x ) ∫ R 2 k ψ ( y ) d y + o ( τ − k ) , (333) \mathbb{E}\left[\psi\left(S^{(n)}+x\right)\mid H\right]=g_{4n\operatorname{Id}_{2k}}(x)\int_{\mathbb{R}^{2k}}\psi(y)\,\mathrm{d}y+o(\tau^{-k}), \tag*{(333)} E [ ψ ( S ( n ) + x ) ∣ H ] = g 4 n Id 2 k ( x ) ∫ R 2 k ψ ( y ) d y + o ( τ − k ) , (333) where the error term is bounded in absolute value by a deterministic quantity that is o ( τ − k ) o(\tau^{-k}) o ( τ − k ) Ψ \Psi Ψ i \mathbf{i} i n n n x x x ψ \psi ψ H H H A N , k \mathcal{A}_{N,k} A N , k B ⊂ R 2 k B\subset\mathbb{R}^{2k} B ⊂ R 2 k C B > 0 C_B>0 C B > 0 A N , k \mathcal{A}_{N,k} A N , k k k k i \mathbf{i} i n n n τ / 10 ≤ n ≤ τ / 3 \tau/10\le n\le\tau/3 τ /10 ≤ n ≤ τ /3
sup x ∈ R 2 k P ( S ( n ) + x ∈ B ∣ H ) ≤ C B τ − k . (334) \sup_{x\in\mathbb{R}^{2k}}\mathbb{P}\left(S^{(n)}+x\in B\mid H\right)\le C_B\tau^{-k}. \tag*{(334)} x ∈ R 2 k sup P ( S ( n ) + x ∈ B ∣ H ) ≤ C B τ − k . (334) Proof. In this proof, constants may depend on Ψ \Psi Ψ k k k X X X Z Z Z N N N Ψ \Psi Ψ R Ψ R_{\Psi} R Ψ
sup ψ ∈ Ψ ∥ ψ ∥ ∞ < ∞ , sup ψ ∈ Ψ ∫ R 2 k ∣ ψ ( y ) ∣ d y ≤ ( 2 R Ψ ) 2 k sup ψ ∈ Ψ ∥ ψ ∥ ∞ < ∞ . (335) \sup_{\psi\in\Psi}\lVert\psi\rVert_{\infty}<\infty,\qquad\sup_{\psi\in\Psi}\int_{\mathbb{R}^{2k}}|\psi(y)|\,\mathrm{d}y\leq(2R_{\Psi})^{2k}\sup_{\psi\in\Psi}\lVert\psi\rVert_{\infty}<\infty. \tag*{(335)} ψ ∈ Ψ sup ∥ ψ ∥ ∞ < ∞ , ψ ∈ Ψ sup ∫ R 2 k ∣ ψ ( y ) ∣ d y ≤ ( 2 R Ψ ) 2 k ψ ∈ Ψ sup ∥ ψ ∥ ∞ < ∞. (335) Every integer n n n τ / 10 ≤ n ≤ τ / 3 \tau/10\leq n\leq\tau/3 τ /10 ≤ n ≤ τ /3 g 4 n I d 2 k ( x ) ≤ ( 8 π n ) − k ≤ ( 8 π ) − k g_{4nI_{d_{2k}}}(x)\leq(8\pi n)^{-k}\leq(8\pi)^{-k} g 4 n I d 2 k ( x ) ≤ ( 8 π n ) − k ≤ ( 8 π ) − k x ∈ R 2 k x\in\mathbb{R}^{2k} x ∈ R 2 k H H H i i i n n n x x x ψ \psi ψ sup ψ ∈ Ψ ∥ ψ ∥ ∞ + ( 8 π ) − k sup ψ ∈ Ψ ∫ ∣ ψ ∣ \sup_{\psi\in\Psi}\lVert\psi\rVert_{\infty}+(8\pi)^{-k}\sup_{\psi\in\Psi}\int|\psi| sup ψ ∈ Ψ ∥ ψ ∥ ∞ + ( 8 π ) − k sup ψ ∈ Ψ ∫ ∣ ψ ∣ N < N 0 N<N_{0} N < N 0 τ < 20 \tau<20 τ < 20 C m o m N 3 ε k − 1 / 2 > 1 C_{\mathrm{mom}}N^{3\varepsilon k-1/2}>1 C mom N 3 ε k − 1/2 > 1 N N N o ( τ − k ) o(\tau^{-k}) o ( τ − k ) N → ∞ N\to\infty N → ∞ N ≥ N 0 N\geq N_{0} N ≥ N 0 τ ≥ 20 \tau\geq20 τ ≥ 20 C m o m N 3 ε k − 1 / 2 ≤ 1 C_{\mathrm{mom}}N^{3\varepsilon k-1/2}\leq1 C mom N 3 ε k − 1/2 ≤ 1 H H H A N , k \mathcal{A}_{N,k} A N , k k k k i \mathbf{i} i n n n τ / 10 ≤ n ≤ τ / 3 \tau/10\leq n\leq\tau/3 τ /10 ≤ n ≤ τ /3 x ∈ R 2 k x\in\mathbb{R}^{2k} x ∈ R 2 k ψ ∈ Ψ \psi\in\Psi ψ ∈ Ψ n ≥ τ / 10 ≥ 2 n\geq\tau/10\geq2 n ≥ τ /10 ≥ 2
We begin with bounds on the Fourier transform
ψ ^ ( ξ ) = ∫ R 2 k e − i ξ ⋅ y ψ ( y ) d y , ξ ∈ R 2 k . (336) \widehat{\psi}(\xi)=\int_{\mathbb{R}^{2k}}e^{-i\xi\cdot y}\psi(y)\,\mathrm{d}y,\qquad\xi\in\mathbb{R}^{2k}. \tag*{(336)} ψ ( ξ ) = ∫ R 2 k e − i ξ ⋅ y ψ ( y ) d y , ξ ∈ R 2 k . (336) Set
s k = 4 k q k + k + 1. (337) s_k=4kq_k+k+1. \tag*{(337)} s k = 4 k q k + k + 1. (337) Integrating by parts 2 s k 2s_k 2 s k Ψ \Psi Ψ
∣ ψ ^ ( ξ ) ∣ ≤ C 1 + ∣ ξ ∣ 2 s k , ∣ ψ ^ ( ξ ) − ψ ^ ( 0 ) ∣ ≤ C ∣ ξ ∣ , ξ ∈ R 2 k . (338) |\widehat{\psi}(\xi)|\leq\frac{C}{1+|\xi|^{2s_k}},\qquad|\widehat{\psi}(\xi)-\widehat{\psi}(0)|\leq C|\xi|,\qquad\xi\in\mathbb{R}^{2k}. \tag*{(338)} ∣ ψ ( ξ ) ∣ ≤ 1 + ∣ ξ ∣ 2 s k C , ∣ ψ ( ξ ) − ψ ( 0 ) ∣ ≤ C ∣ ξ ∣ , ξ ∈ R 2 k . (338) Since 2 s k > 2 k 2s_k>2k 2 s k > 2 k
∫ R 2 k ∣ ψ ^ ( ξ ) ∣ d ξ ≤ C . (339) \int_{\mathbb{R}^{2k}}|\widehat{\psi}(\xi)|\,\mathrm{d}\xi\leq C. \tag*{(339)} ∫ R 2 k ∣ ψ ( ξ ) ∣ d ξ ≤ C . (339) Next, we write both sides of (8.88) as Fourier integrals. Since ψ ^ \widehat{\psi} ψ
ψ ( y ) = ( 2 π ) − 2 k ∫ R 2 k e i ξ ⋅ y ψ ^ ( ξ ) d ξ (340) \psi(y)=(2\pi)^{-2k}\int_{\mathbb{R}^{2k}}e^{i\xi\cdot y}\widehat{\psi}(\xi)\,\mathrm{d}\xi \tag*{(340)} ψ ( y ) = ( 2 π ) − 2 k ∫ R 2 k e i ξ ⋅ y ψ ( ξ ) d ξ (340) holds for all y y y
E [ ψ ( S ( n ) + x ) ∣ H ] = ( 2 π ) − 2 k ∫ R 2 k e i ξ ⋅ x ψ ^ ( ξ ) Φ H ( ξ ) n d ξ . (341) \mathbb{E}\left[\psi\left(S^{(n)}+x\right)\mid H\right]=(2\pi)^{-2k}\int_{\mathbb{R}^{2k}}e^{i\xi\cdot x}\widehat{\psi}(\xi)\Phi_H(\xi)^n\,\mathrm{d}\xi. \tag*{(341)} E [ ψ ( S ( n ) + x ) ∣ H ] = ( 2 π ) − 2 k ∫ R 2 k e i ξ ⋅ x ψ ( ξ ) Φ H ( ξ ) n d ξ . (341) Further, we have
ψ ^ ( 0 ) = ∫ ψ , ∫ R 2 k e i ζ ⃗ ⋅ x − 2 n ∣ ζ ∣ 2 d ζ = ( 2 π ) 2 k g 4 n I d 2 k ( x ) . (342) \widehat{\psi}(0)=\int\psi,\qquad\int_{\mathbb{R}^{2k}}e^{i\vec{\zeta}\cdot x-2n|\zeta|^2}\,d\zeta=(2\pi)^{2k}g_{4nI d_{2k}}(x). \tag*{(342)} ψ ( 0 ) = ∫ ψ , ∫ R 2 k e i ζ ⋅ x − 2 n ∣ ζ ∣ 2 d ζ = ( 2 π ) 2 k g 4 n I d 2 k ( x ) . (342) Subtracting, we find
∣ E [ ψ ( S ( n ) + x ) ∣ H ] − g 4 n I d 2 k ( x ) ∫ ψ ∣ ≤ ( 2 π ) − 2 k ∫ R 2 k ∣ ψ ^ ( ζ ) Φ H ( ζ ) n − ψ ^ ( 0 ) e − 2 n ∣ ζ ∣ 2 ∣ d ζ . (343) \left|\mathbb{E}\left[\psi(S^{(n)}+x)\mid H\right]-g_{4nI d_{2k}}(x)\int\psi\right|\leq(2\pi)^{-2k}\int_{\mathbb{R}^{2k}}\left|\widehat{\psi}(\zeta)\Phi_H(\zeta)^n-\widehat{\psi}(0)e^{-2n|\zeta|^2}\right|\,d\zeta. \tag*{(343)} E [ ψ ( S ( n ) + x ) ∣ H ] − g 4 n I d 2 k ( x ) ∫ ψ ≤ ( 2 π ) − 2 k ∫ R 2 k ψ ( ζ ) Φ H ( ζ ) n − ψ ( 0 ) e − 2 n ∣ ζ ∣ 2 d ζ . (343) The right side of (8.98) does not involve x x x x x x
We split the integral in (8.98) into three ranges of frequencies. Since n ≥ τ / 10 n\geq\tau/10 n ≥ τ /10 C τ − k ( L 1 + L 2 + L 3 + L 4 ) C\tau^{-k}(\mathcal{L}_1+\mathcal{L}_2+\mathcal{L}_3+\mathcal{L}_4) C τ − k ( L 1 + L 2 + L 3 + L 4 )
L 1 = n k ∫ ∣ ζ ∣ ≤ q N ∣ ψ ^ ( ζ ) Φ H ( ζ ) n − ψ ^ ( 0 ) e − 2 n ∣ ζ ∣ 2 ∣ d ζ , L 2 = n k ∫ q N < ∣ ζ ∣ ≤ Ω 1 / 2 ∣ ψ ^ ( ζ ) ∣ ∣ Φ H ( ζ ) ∣ n d ζ , \mathcal{L}_1=n^k\int_{|\zeta|\leq q_N}\left|\widehat{\psi}(\zeta)\Phi_H(\zeta)^n-\widehat{\psi}(0)e^{-2n|\zeta|^2}\right|\,d\zeta,\qquad\mathcal{L}_2=n^k\int_{q_N<|\zeta|\leq\Omega^{1/2}}\left|\widehat{\psi}(\zeta)\right|\left|\Phi_H(\zeta)\right|^n\,d\zeta, L 1 = n k ∫ ∣ ζ ∣ ≤ q N ψ ( ζ ) Φ H ( ζ ) n − ψ ( 0 ) e − 2 n ∣ ζ ∣ 2 d ζ , L 2 = n k ∫ q N < ∣ ζ ∣ ≤ Ω 1/2 ψ ( ζ ) ∣ Φ H ( ζ ) ∣ n d ζ , L 3 = n k ∫ ∣ ζ ∣ > Ω 1 / 2 ∣ ψ ^ ( ζ ) ∣ d ζ , L 4 = n k ∣ ψ ^ ( 0 ) ∣ ∫ ∣ ζ ∣ > q N e − 2 n ∣ ζ ∣ 2 d ζ . (344) \mathcal{L}_3=n^k\int_{|\zeta|>\Omega^{1/2}}\left|\widehat{\psi}(\zeta)\right|\,d\zeta,\qquad\mathcal{L}_4=n^k\left|\widehat{\psi}(0)\right|\int_{|\zeta|>q_N}e^{-2n|\zeta|^2}\,d\zeta. \tag*{(344)} L 3 = n k ∫ ∣ ζ ∣ > Ω 1/2 ψ ( ζ ) d ζ , L 4 = n k ψ ( 0 ) ∫ ∣ ζ ∣ > q N e − 2 n ∣ ζ ∣ 2 d ζ . (344) and where we used ∣ Φ H ∣ ≤ 1 |\Phi_H|\leq1 ∣ Φ H ∣ ≤ 1 L 3 \mathcal{L}_3 L 3 ψ \psi ψ R Ψ R_\Psi R Ψ L 1 + L 2 + L 3 + L 4 \mathcal{L}_1+\mathcal{L}_2+\mathcal{L}_3+\mathcal{L}_4 L 1 + L 2 + L 3 + L 4
We first treat small frequencies. Let
Φ H ∘ ( ζ ) = e − i ζ ˉ ⋅ J ˉ Φ H ( ζ ) = E [ e i ζ ˉ ⋅ ( J − J ˉ ) ∣ H ] , ζ ∈ R 2 k . (345) \Phi_H^\circ(\zeta)=e^{-i\bar{\zeta}\cdot\bar{J}}\Phi_H(\zeta)=\mathbb{E}\left[e^{i\bar{\zeta}\cdot(J-\bar{J})}\mid H\right],\qquad\zeta\in\mathbb{R}^{2k}. \tag*{(345)} Φ H ∘ ( ζ ) = e − i ζ ˉ ⋅ J ˉ Φ H ( ζ ) = E [ e i ζ ˉ ⋅ ( J − J ˉ ) ∣ H ] , ζ ∈ R 2 k . (345) By Lemma 8.4 with ϵ = ϵ k \epsilon=\epsilon_k ϵ = ϵ k C m o m N 3 ϵ k − 1 / 2 ≤ 1 C_{\mathrm{mom}}N^{3\epsilon_k-1/2}\leq1 C mom N 3 ϵ k − 1/2 ≤ 1
3 I d 2 k ≤ Σ H ≤ 5 I d 2 k (346) 3I d_{2k}\leq\Sigma_H\leq5I d_{2k} \tag*{(346)} 3 I d 2 k ≤ Σ H ≤ 5 I d 2 k (346) and
E [ ∣ J − J ˉ ∣ 3 ∣ H ] ≤ 4 ( E [ ∣ J ∣ 3 ∣ H ] + ∣ J ˉ ∣ 3 ) ≤ 8 E [ ∣ J ∣ 3 ∣ H ] ≤ 8 C m o m N 9 ϵ k . (347) \mathbb{E}\left[|J-\bar{J}|^3\mid H\right]\leq4\left(\mathbb{E}\left[|J|^3\mid H\right]+|\bar{J}|^3\right)\leq8\mathbb{E}\left[|J|^3\mid H\right]\leq8C_{\mathrm{mom}}N^{9\epsilon_k}. \tag*{(347)} E [ ∣ J − J ˉ ∣ 3 ∣ H ] ≤ 4 ( E [ ∣ J ∣ 3 ∣ H ] + ∣ J ˉ ∣ 3 ) ≤ 8 E [ ∣ J ∣ 3 ∣ H ] ≤ 8 C mom N 9 ϵ k . (347) We apply the inequality ∣ e i t ′ − 1 − i t ′ + t ′ 2 / 2 ∣ ≤ ∣ t ′ ∣ 3 / 6 |e^{it'}-1-it'+t'^2/2|\leq|t'|^3/6 ∣ e i t ′ − 1 − i t ′ + t ′2 /2∣ ≤ ∣ t ′ ∣ 3 /6 t ′ t' t ′ t ′ = ζ ⋅ ( J − J ˉ ) t'=\zeta\cdot(J-\bar{J}) t ′ = ζ ⋅ ( J − J ˉ )
∣ Φ H ∘ ( ζ ) − 1 + 1 2 ζ T Σ H ζ ∣ ≤ 4 3 C m o m N 9 ϵ k ∣ ζ ∣ 3 , ζ ∈ R 2 k . (348) \left|\Phi_H^\circ(\zeta)-1+\frac{1}{2}\zeta^T\Sigma_H\zeta\right|\leq\frac{4}{3}C_{\mathrm{mom}}N^{9\epsilon_k}|\zeta|^3,\qquad\zeta\in\mathbb{R}^{2k}. \tag*{(348)} Φ H ∘ ( ζ ) − 1 + 2 1 ζ T Σ H ζ ≤ 3 4 C mom N 9 ϵ k ∣ ζ ∣ 3 , ζ ∈ R 2 k . (348) For ∣ ζ ∣ ≤ q N |\zeta|\leq q_N ∣ ζ ∣ ≤ q N
3 2 ∣ ζ ∣ 2 ≤ 1 2 ζ T Σ H ζ ≤ 5 2 ∣ ζ ∣ 2 ≤ 1 , 4 3 C m o m N 9 ϵ k ∣ ζ ∣ 3 ≤ 1 9 ∣ ζ ∣ 2 . (349) \frac{3}{2}|\zeta|^2\leq\frac{1}{2}\zeta^T\Sigma_H\zeta\leq\frac{5}{2}|\zeta|^2\leq1,\qquad\frac{4}{3}C_{\mathrm{mom}}N^{9\epsilon_k}|\zeta|^3\leq\frac{1}{9}|\zeta|^2. \tag*{(349)} 2 3 ∣ ζ ∣ 2 ≤ 2 1 ζ T Σ H ζ ≤ 2 5 ∣ ζ ∣ 2 ≤ 1 , 3 4 C mom N 9 ϵ k ∣ ζ ∣ 3 ≤ 9 1 ∣ ζ ∣ 2 . (349) Hence (8.103) implies
∣ Φ H ( ζ ) ∣ = ∣ Φ H ∘ ( ζ ) ∣ ≤ 1 − 3 2 ∣ ζ ∣ 2 + 1 9 ∣ ζ ∣ 2 ≤ e − ∣ ζ ∣ 2 , ∣ ζ ∣ ≤ q N . (350) |\Phi_H(\zeta)|=|\Phi_H^\circ(\zeta)|\leq1-\frac{3}{2}|\zeta|^2+\frac{1}{9}|\zeta|^2\leq e^{-|\zeta|^2},\qquad|\zeta|\leq q_N. \tag*{(350)} ∣ Φ H ( ζ ) ∣ = ∣ Φ H ∘ ( ζ ) ∣ ≤ 1 − 2 3 ∣ ζ ∣ 2 + 9 1 ∣ ζ ∣ 2 ≤ e − ∣ ζ ∣ 2 , ∣ ζ ∣ ≤ q N . (350) We now change variables ζ = n − 1 / 2 ζ ′ \zeta=n^{-1/2}\zeta' ζ = n − 1/2 ζ ′ I 1 \mathcal{I}_1 I 1
I 1 = ∫ ∣ ζ ′ ∣ ≤ Q n 1 / 2 ∣ ψ ^ ( n − 1 / 2 ζ ′ ) Φ H ( n − 1 / 2 ζ ′ ) n − ψ ^ ( 0 ) e − 2 ∣ ζ ′ ∣ 2 ∣ d ζ ′ . (351) \mathcal{I}_1=\int_{|\zeta'|\leq Qn^{1/2}}\left|\hat{\psi}(n^{-1/2}\zeta')\Phi_H(n^{-1/2}\zeta')^n-\hat{\psi}(0)e^{-2|\zeta'|^2}\right|\,\mathrm{d}\zeta'. \tag*{(351)} I 1 = ∫ ∣ ζ ′ ∣ ≤ Q n 1/2 ψ ^ ( n − 1/2 ζ ′ ) Φ H ( n − 1/2 ζ ′ ) n − ψ ^ ( 0 ) e − 2∣ ζ ′ ∣ 2 d ζ ′ . (351) By (338) and (350) , the integrand is bounded by
C n − 1 / 2 ∣ ζ ′ ∣ e − ∣ ζ ′ ∣ 2 + C ∣ Φ H ( n − 1 / 2 ζ ′ ) n − e − 2 ∣ ζ ′ ∣ 2 ∣ . (352) Cn^{-1/2}|\zeta'|e^{-|\zeta'|^2}+C\left|\Phi_H(n^{-1/2}\zeta')^n-e^{-2|\zeta'|^2}\right|. \tag*{(352)} C n − 1/2 ∣ ζ ′ ∣ e − ∣ ζ ′ ∣ 2 + C Φ H ( n − 1/2 ζ ′ ) n − e − 2∣ ζ ′ ∣ 2 . (352) We have Φ H ( ζ ) n = e i n ζ ⋅ J ˉ Φ H ∘ ( ζ ) n \Phi_H(\zeta)^n=e^{in\zeta\cdot\bar{J}}\Phi_H^\circ(\zeta)^n Φ H ( ζ ) n = e in ζ ⋅ J ˉ Φ H ∘ ( ζ ) n
∣ Φ H ( n − 1 / 2 ζ ′ ) n − e − 2 ∣ ζ ′ ∣ 2 ∣ ≤ ∣ e i n 1 / 2 ζ ′ ⋅ J ˉ − 1 ∣ e − ∣ ζ ′ ∣ 2 + ∣ Φ H ∘ ( n − 1 / 2 ζ ′ ) n − e − ζ ′ T Σ H ζ ′ / 2 ∣ + ∣ e − ζ ′ T Σ H ζ ′ / 2 − e − 2 ∣ ζ ′ ∣ 2 ∣ . (353) \left|\Phi_H(n^{-1/2}\zeta')^n-e^{-2|\zeta'|^2}\right|
\leq\left|e^{in^{1/2}\zeta'\cdot\bar{J}}-1\right|e^{-|\zeta'|^2}
+\left|\Phi_H^\circ(n^{-1/2}\zeta')^n-e^{-\zeta'^{T}\Sigma_H\zeta'/2}\right|
+\left|e^{-\zeta'^{T}\Sigma_H\zeta'/2}-e^{-2|\zeta'|^2}\right|. \tag*{(353)} Φ H ( n − 1/2 ζ ′ ) n − e − 2∣ ζ ′ ∣ 2 ≤ e i n 1/2 ζ ′ ⋅ J ˉ − 1 e − ∣ ζ ′ ∣ 2 + Φ H ∘ ( n − 1/2 ζ ′ ) n − e − ζ ′ T Σ H ζ ′ /2 + e − ζ ′ T Σ H ζ ′ /2 − e − 2∣ ζ ′ ∣ 2 . (353) We bound the three terms on the right side in turn. By Lemma 8.4, the first term satisfies
∣ e i n 1 / 2 ζ ′ ⋅ J ˉ − 1 ∣ e − ∣ ζ ′ ∣ 2 ≤ n 1 / 2 ∣ ζ ′ ∣ ∥ J ˉ ∥ e − ∣ ζ ′ ∣ 2 ≤ C τ 1 / 2 N − 1 / 2 ∣ ζ ′ ∣ e − ∣ ζ ′ ∣ 2 . (354) \left|e^{in^{1/2}\zeta'\cdot\bar{J}}-1\right|e^{-|\zeta'|^2}
\leq n^{1/2}|\zeta'|\lVert\bar{J}\rVert e^{-|\zeta'|^2}
\leq C\tau^{1/2}N^{-1/2}|\zeta'|e^{-|\zeta'|^2}. \tag*{(354)} e i n 1/2 ζ ′ ⋅ J ˉ − 1 e − ∣ ζ ′ ∣ 2 ≤ n 1/2 ∣ ζ ′ ∣ ∥ J ˉ ∥ e − ∣ ζ ′ ∣ 2 ≤ C τ 1/2 N − 1/2 ∣ ζ ′ ∣ e − ∣ ζ ′ ∣ 2 . (354) The two exponents in the third term are at least 3 ∣ ζ ′ ∣ 2 / 2 3|\zeta'|^2/2 3∣ ζ ′ ∣ 2 /2
∣ e − ζ ′ T Σ H ζ ′ / 2 − e − 2 ∣ ζ ′ ∣ 2 ∣ ≤ 1 2 ∣ Σ H − 4 I d 2 k ∣ ∣ ζ ′ ∣ 2 e − 3 ∣ ζ ′ ∣ 2 / 2 ≤ C N 3 ε k − 1 / 2 ∣ ζ ′ ∣ 2 e − 3 ∣ ζ ′ ∣ 2 / 2 . (355) \left|e^{-\zeta'^{T}\Sigma_H\zeta'/2}-e^{-2|\zeta'|^2}\right|
\leq\frac{1}{2}\left|\Sigma_H-4I\mathrm{d}_{2k}\right||\zeta'|^2e^{-3|\zeta'|^2/2}
\leq CN^{3\varepsilon_k-1/2}|\zeta'|^2e^{-3|\zeta'|^2/2}. \tag*{(355)} e − ζ ′ T Σ H ζ ′ /2 − e − 2∣ ζ ′ ∣ 2 ≤ 2 1 ∣ Σ H − 4 I d 2 k ∣ ∣ ζ ′ ∣ 2 e − 3∣ ζ ′ ∣ 2 /2 ≤ C N 3 ε k − 1/2 ∣ ζ ′ ∣ 2 e − 3∣ ζ ′ ∣ 2 /2 . (355) To bound the second term in (353) , we set
z ′ = Φ H ∘ ( n − 1 / 2 ζ ′ ) , z ′ ′ = e − ζ ′ T Σ H ζ ′ / ( 2 n ) . z'=\Phi_H^\circ(n^{-1/2}\zeta'),\qquad z''=e^{-\zeta'^{T}\Sigma_H\zeta'/(2n)}. z ′ = Φ H ∘ ( n − 1/2 ζ ′ ) , z ′′ = e − ζ ′ T Σ H ζ ′ / ( 2 n ) . The bound (350) and Σ H ≥ 3 I d 2 k \Sigma_H\geq3I\mathrm{d}_{2k} Σ H ≥ 3 I d 2 k
max { ∣ z ′ ∣ , ∣ z ′ ′ ∣ } ≤ e − ∣ ζ ′ ∣ 2 / n . (356) \max\{|z'|,|z''|\}\leq e^{-|\zeta'|^2/n}. \tag*{(356)} max { ∣ z ′ ∣ , ∣ z ′′ ∣ } ≤ e − ∣ ζ ′ ∣ 2 / n . (356) Using (348) and ∣ e − t ′ − 1 + t ′ ∣ ≤ t ′ 2 / 2 |e^{-t'}-1+t'|\leq t'^2/2 ∣ e − t ′ − 1 + t ′ ∣ ≤ t ′2 /2 t ′ ≥ 0 t'\geq0 t ′ ≥ 0
∣ z ′ − z ′ ′ ∣ ≤ 1 2 ( 5 ∣ ζ ′ ∣ 2 2 n ) 2 + 4 3 C m o m N 9 ε k ∣ ζ ′ ∣ 3 n 3 / 2 ≤ 4 ∣ ζ ′ ∣ 4 n 2 + 2 C m o m N 9 ε k ∣ ζ ′ ∣ 3 n 3 / 2 . (357) |z'-z''|\leq\frac{1}{2}\left(\frac{5|\zeta'|^2}{2n}\right)^2+\frac{4}{3}C_{\mathrm{mom}}N^{9\varepsilon_k}\frac{|\zeta'|^3}{n^{3/2}}
\leq\frac{4|\zeta'|^4}{n^2}+\frac{2C_{\mathrm{mom}}N^{9\varepsilon_k}|\zeta'|^3}{n^{3/2}}. \tag*{(357)} ∣ z ′ − z ′′ ∣ ≤ 2 1 ( 2 n 5∣ ζ ′ ∣ 2 ) 2 + 3 4 C mom N 9 ε k n 3/2 ∣ ζ ′ ∣ 3 ≤ n 2 4∣ ζ ′ ∣ 4 + n 3/2 2 C mom N 9 ε k ∣ ζ ′ ∣ 3 . (357) We now use the elementary inequality
∣ z ′ n − z ′ ′ n ∣ ≤ n ∣ z ′ − z ′ ′ ∣ max { ∣ z ′ ∣ , ∣ z ′ ′ ∣ } n − 1 ≤ n ∣ z ′ − z ′ ′ ∣ e − ∣ ζ ′ ∣ 2 ( n − 1 ) / n ≤ n ∣ z ′ − z ′ ′ ∣ e − ∣ ζ ′ ∣ 2 / 2 , (358) |z'{}^n-z''{}^n|\leq n|z'-z''|\max\{|z'|,|z''|\}^{n-1}\leq n|z'-z''|e^{-|\zeta'|^2(n-1)/n}\leq n|z'-z''|e^{-|\zeta'|^2/2}, \tag*{(358)} ∣ z ′ n − z ′′ n ∣ ≤ n ∣ z ′ − z ′′ ∣ max { ∣ z ′ ∣ , ∣ z ′′ ∣ } n − 1 ≤ n ∣ z ′ − z ′′ ∣ e − ∣ ζ ′ ∣ 2 ( n − 1 ) / n ≤ n ∣ z ′ − z ′′ ∣ e − ∣ ζ ′ ∣ 2 /2 , (358) where we used n ≥ 2 n\geq2 n ≥ 2
∣ Φ H ∘ ( n − 1 / 2 ζ ′ ) n − e − ζ ′ T Σ H ζ ′ / 2 ∣ ≤ ( 4 ∣ ζ ′ ∣ 4 n + 2 C m o m N 9 ε k ∣ ζ ′ ∣ 3 n 1 / 2 ) e − ∣ ζ ′ ∣ 2 / 2 . (359) \left|\Phi_H^\circ(n^{-1/2}\zeta')^n-e^{-\zeta'^{T}\Sigma_H\zeta'/2}\right|
\leq\left(\frac{4|\zeta'|^4}{n}+\frac{2C_{\mathrm{mom}}N^{9\varepsilon_k}|\zeta'|^3}{n^{1/2}}\right)e^{-|\zeta'|^2/2}. \tag*{(359)} Φ H ∘ ( n − 1/2 ζ ′ ) n − e − ζ ′ T Σ H ζ ′ /2 ≤ ( n 4∣ ζ ′ ∣ 4 + n 1/2 2 C mom N 9 ε k ∣ ζ ′ ∣ 3 ) e − ∣ ζ ′ ∣ 2 /2 . (359) We now collect these bounds. Using τ / 10 ≤ n ≤ τ / 3 \tau/10 \le n \le\tau/3 τ /10 ≤ n ≤ τ /3 τ = N 1 / 100 \tau= N^{1/100} τ = N 1/100 τ − 1 / 2 N − 1 / 2 + N 3 ε k − 1 / 2 ≤ 2 τ − 1 / 2 \tau^{-1/2}N^{-1/2} + N^{3\varepsilon_k-1/2} \le2\tau^{-1/2} τ − 1/2 N − 1/2 + N 3 ε k − 1/2 ≤ 2 τ − 1/2 (352) is bounded by
C N 9 ε k τ − 1 / 2 ( 1 + ∣ ζ ′ ∣ ) 4 e − ∣ ζ ′ ∣ 2 / 2 = C N 9 ε k − 1 / 200 ( 1 + ∣ ζ ′ ∣ ) 4 e − ∣ ζ ′ ∣ 2 / 2 . (360) CN^{9\varepsilon_k}\tau^{-1/2}(1+|\zeta'|)^4e^{-|\zeta'|^2/2}=CN^{9\varepsilon_k-1/200}(1+|\zeta'|)^4e^{-|\zeta'|^2/2}. \tag*{(360)} C N 9 ε k τ − 1/2 ( 1 + ∣ ζ ′ ∣ ) 4 e − ∣ ζ ′ ∣ 2 /2 = C N 9 ε k − 1/200 ( 1 + ∣ ζ ′ ∣ ) 4 e − ∣ ζ ′ ∣ 2 /2 . (360) Integrating over ζ ′ ∈ R 2 k \zeta' \in\mathbb{R}^{2k} ζ ′ ∈ R 2 k
I 1 ≤ C N 9 ε k − 1 / 200 , (361) \mathcal{I}_1 \le CN^{9\varepsilon_k-1/200}, \tag*{(361)} I 1 ≤ C N 9 ε k − 1/200 , (361) which tends to zero by (326) .
Next, we bound I 2 \mathcal{I}_2 I 2 Lemma 8.12 . For ϱ N < ∣ ζ ∣ ≤ Ω 1 / 2 \varrho_N < |\zeta| \le\Omega^{1/2} ϱ N < ∣ ζ ∣ ≤ Ω 1/2 (331) gives
∣ Φ H ( ζ ) ∣ n ≤ ( 1 − p Y 2 τ − 1 / 4 ) n / 2 ≤ exp ( − p Y 4 n τ − 1 / 4 ) ≤ exp ( − p Y 40 τ 3 / 4 ) . (362) |\Phi_H(\zeta)|^n \le\left(1-\frac{p_Y}{2}\tau^{-1/4}\right)^{n/2}\le\exp\left(-\frac{p_Y}{4}n\tau^{-1/4}\right)\le\exp\left(-\frac{p_Y}{40}\tau^{3/4}\right). \tag*{(362)} ∣ Φ H ( ζ ) ∣ n ≤ ( 1 − 2 p Y τ − 1/4 ) n /2 ≤ exp ( − 4 p Y n τ − 1/4 ) ≤ exp ( − 40 p Y τ 3/4 ) . (362) Using (339) and n ≤ τ n \le\tau n ≤ τ
I 2 ≤ C τ k exp ( − p Y τ 3 / 4 / 40 ) . (363) \mathcal{I}_2 \le C\tau^k\exp(-p_Y\tau^{3/4}/40). \tag*{(363)} I 2 ≤ C τ k exp ( − p Y τ 3/4 /40 ) . (363) For large frequencies, we use the first bound in (338) and n ≤ τ n \le\tau n ≤ τ
I 3 ≤ C τ k ∫ ∣ ζ ∣ > Ω 1 / 2 ∣ ζ ∣ − 2 s k d ζ = C τ k Ω k − s k = C τ k + ( k − s k ) / ( 4 q k ) = C τ − 1 / ( 4 q k ) , (364) \mathcal{I}_3 \le C\tau^k\int_{|\zeta|>\Omega^{1/2}}|\zeta|^{-2s_k}\,d\zeta=C\tau^k\Omega^{k-s_k}=C\tau^{k+(k-s_k)/(4q_k)}=C\tau^{-1/(4q_k)}, \tag*{(364)} I 3 ≤ C τ k ∫ ∣ ζ ∣ > Ω 1/2 ∣ ζ ∣ − 2 s k d ζ = C τ k Ω k − s k = C τ k + ( k − s k ) / ( 4 q k ) = C τ − 1/ ( 4 q k ) , (364) where the last equality follows from the definition of s k s_k s k I 1 \mathcal{I}_1 I 1
I 4 = ∣ ψ ^ ( 0 ) ∣ ∫ ∣ ζ ′ ∣ > ϱ N n 1 / 2 e − 2 ∣ ζ ′ ∣ 2 d ζ ′ ≤ C e − ϱ N 2 n ≤ C exp ( − c N 1 / 100 − 18 ε k ) . (365) \mathcal{I}_4=|\widehat{\psi}(0)|\int_{|\zeta'|>\varrho_N n^{1/2}}e^{-2|\zeta'|^2}\,d\zeta'\le Ce^{-\varrho_N^2n}\le C\exp\left(-cN^{1/100-18\varepsilon_k}\right). \tag*{(365)} I 4 = ∣ ψ ( 0 ) ∣ ∫ ∣ ζ ′ ∣ > ϱ N n 1/2 e − 2∣ ζ ′ ∣ 2 d ζ ′ ≤ C e − ϱ N 2 n ≤ C exp ( − c N 1/100 − 18 ε k ) . (365) The four bounds tend to zero, which proves (333) .
It remains to prove (334) . Since B B B ψ B ∈ C ∞ ( R 2 k ) \psi_B\in C^\infty(\mathbb{R}^{2k}) ψ B ∈ C ∞ ( R 2 k ) ψ B ≥ 1 \psi_B\ge1 ψ B ≥ 1 B B B (333) with Ψ = { ψ B } \Psi=\{\psi_B\} Ψ = { ψ B }
P ( S ( n ) + x ∈ B ∣ H ) ≤ E [ ψ B ( S ( n ) + x ) ∣ H ] ≤ ( 8 π n ) − k ∫ ψ B + o ( τ − k ) . (366) \mathbb{P}(S^{(n)}+x\in B\mid H)\le\mathbb{E}[\psi_B(S^{(n)}+x)\mid H]\le(8\pi n)^{-k}\int\psi_B+o(\tau^{-k}). \tag*{(366)} P ( S ( n ) + x ∈ B ∣ H ) ≤ E [ ψ B ( S ( n ) + x ) ∣ H ] ≤ ( 8 π n ) − k ∫ ψ B + o ( τ − k ) . (366) The deterministic quantity that bounds the error term for the family { ψ B } \{\psi_B\} { ψ B } N N N o ( τ − k ) o(\tau^{-k}) o ( τ − k ) C τ − k C\tau^{-k} C τ − k N N N n ≥ τ / 10 n\ge\tau/10 n ≥ τ /10 C B τ − k C_B\tau^{-k} C B τ − k □ \square □
Pairs with large gaps
The second claim of Proposition 4.4 concerns a single pair. In Lemma 8.14 and its proof, we apply the preceding results with k = 1 k=1 k = 1 A N , 1 \mathcal{A}_{N,1} A N , 1 k = 1 k=1 k = 1 i ∈ [ N − 1 ] i \in[N-1] i ∈ [ N − 1 ] J J J J ˉ \bar{J} J ˉ S ( η ) S^{(\eta)} S ( η ) i = ( i ) \mathbf{i}=(i) i = ( i ) z i = ( − g i / ( 2 d ) , 0 ) z_i=(-g_i/(2d),0) z i = ( − g i / ( 2 d ) , 0 ) ℓ = d τ 1 / 2 \ell=d\tau^{1/2} ℓ = d τ 1/2
∣ z i ∣ = g i 2 d = g i 2 ℓ τ 1 / 2 . (367) |z_i|=\frac{g_i}{2d}=\frac{g_i}{2\ell}\tau^{1/2}. \tag*{(367)} ∣ z i ∣ = 2 d g i = 2 ℓ g i τ 1/2 . (367) When g i ≥ ℓ g_i\ge\ell g i ≥ ℓ z i z_i z i τ 1 / 2 / 2 \tau^{1/2}/2 τ 1/2 /2 z i + S i z_i+S_i z i + S i S i S_i S i
Lemma 8.14. There exists c > 0 c>0 c > 0 x 0 > 0 x_0>0 x 0 > 0 C x 0 > 0 C_{x_0}>0 C x 0 > 0 A N , 1 \mathcal{A}_{N,1} A N , 1 i ∈ [ N − 1 ] i\in[N-1] i ∈ [ N − 1 ] g i ≥ ℓ g_i\ge\ell g i ≥ ℓ
P ( ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C x 0 τ − 1 e − c g i / ℓ . (368) \mathbb{P}\left(|z_i+S_i|\le x_0\mid H\right)\le C_{x_0}\tau^{-1}e^{-cg_i/\ell}. \tag*{(368)} P ( ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ C x 0 τ − 1 e − c g i / ℓ . (368) Proof. Let N 1 ≥ N 0 N_1\ge N_0 N 1 ≥ N 0 x 0 x_0 x 0 N ≥ N 1 N\ge N_1 N ≥ N 1 Y Y Y E e ∣ Y ∣ / 2 < ∞ \mathbb{E}e^{|Y|/2}<\infty E e ∣ Y ∣/2 < ∞ β 0 = N − 3 ε 1 / 4 \beta_0=N^{-3\varepsilon_1/4} β 0 = N − 3 ε 1 /4 β ∈ R \beta\in\mathbb{R} β ∈ R ∣ β ∣ ≤ β 0 |\beta|\le\beta_0 ∣ β ∣ ≤ β 0 k = 1 k=1 k = 1 A N , 1 \mathcal{A}_{N,1} A N , 1 η ∈ { 1 , 2 } \eta\in\{1,2\} η ∈ { 1 , 2 }
∣ J ( η ) ∣ ≤ ∣ J ∣ ≤ 2 N 2 ε 1 ( ∣ Y ∣ + N ε 1 ) , and hence ∣ β J ( η ) ∣ ≤ ∣ Y ∣ + 1 2 . (369) |J(\eta)|\le|J|\le2N^{2\varepsilon_1}(|Y|+N^{\varepsilon_1}),\qquad\text{and hence}\qquad|\beta J(\eta)|\le\frac{|Y|+1}{2}. \tag*{(369)} ∣ J ( η ) ∣ ≤ ∣ J ∣ ≤ 2 N 2 ε 1 ( ∣ Y ∣ + N ε 1 ) , and hence ∣ β J ( η ) ∣ ≤ 2 ∣ Y ∣ + 1 . (369) Using e t ′ ≤ 1 + t ′ + t ′ 2 e ∣ t ′ ∣ / 2 e^{t'}\le1+t'+t'^2e^{|t'|}/2 e t ′ ≤ 1 + t ′ + t ′2 e ∣ t ′ ∣ /2 t ′ t' t ′
E [ e β J ( η ) ∣ H ] ≤ 1 + β J ˉ ( η ) + e 1 / 2 2 β 2 E [ J ( η ) 2 e ∣ Y ∣ / 2 ∣ H ] . (370) \mathbb{E}\left[e^{\beta J(\eta)}\mid H\right]\le1+\beta\bar{J}(\eta)+\frac{e^{1/2}}{2}\beta^2\mathbb{E}\left[J(\eta)^2e^{|Y|/2}\mid H\right]. \tag*{(370)} E [ e β J ( η ) ∣ H ] ≤ 1 + β J ˉ ( η ) + 2 e 1/2 β 2 E [ J ( η ) 2 e ∣ Y ∣/2 ∣ H ] . (370) To bound the last expectation, recall from (272) that
E [ J ( η ) 2 ∣ H ] = ( N 2 ) − 1 ∑ a < b ( 1 + X a b 2 ) ( D a b ( i ) ( η ) ) 2 , (371) \mathbb{E}\left[J(\eta)^2\mid H\right]=\binom{N}{2}^{-1}\sum_{a<b}(1+X_{ab}^2)(D_{ab}^{(i)}(\eta))^2, \tag*{(371)} E [ J ( η ) 2 ∣ H ] = ( 2 N ) − 1 a < b ∑ ( 1 + X ab 2 ) ( D ab ( i ) ( η ) ) 2 , (371) and that this quantity is bounded by 5 5 5 N 1 N_1 N 1 ( a , b ) (a,b) ( a , b ) ( Y − X a b ) 2 ≤ 2 Y 2 + 2 X a b 2 (Y-X_{ab})^2\le2Y^2+2X_{ab}^2 ( Y − X ab ) 2 ≤ 2 Y 2 + 2 X ab 2
E [ J ( η ) 2 e ∣ Y ∣ / 2 ∣ H ] ≤ 2 ( N 2 ) − 1 ∑ a < b ( D a b ( i ) ( η ) ) 2 ( E [ Y 2 e ∣ Y ∣ / 2 ] + X a b 2 E [ e ∣ Y ∣ / 2 ] ) ≤ C E [ J ( η ) 2 ∣ H ] ≤ C , (372) \begin{aligned}
\mathbb{E}\left[J(\eta)^2e^{|Y|/2}\mid H\right]
&\le2\binom{N}{2}^{-1}\sum_{a<b}(D_{ab}^{(i)}(\eta))^2\left(\mathbb{E}[Y^2e^{|Y|/2}]+X_{ab}^2\mathbb{E}[e^{|Y|/2}]\right)\\
&\le C\mathbb{E}\left[J(\eta)^2\mid H\right]\le C, \tag*{(372)}
\end{aligned} E [ J ( η ) 2 e ∣ Y ∣/2 ∣ H ] ≤ 2 ( 2 N ) − 1 a < b ∑ ( D ab ( i ) ( η ) ) 2 ( E [ Y 2 e ∣ Y ∣/2 ] + X ab 2 E [ e ∣ Y ∣/2 ] ) ≤ C E [ J ( η ) 2 ∣ H ] ≤ C , (372) where in the second inequality we used that Y Y Y 1 + t ′ ≤ e t ′ 1+t'\le e^{t'} 1 + t ′ ≤ e t ′ C ∗ > 0 C_*>0 C ∗ > 0
E [ e β ( J ( η ) − J ˉ ( η ) ) ∣ H ] ≤ e − β J ˉ ( η ) ( 1 + β J ˉ ( η ) + C ∗ β 2 ) ≤ e C ∗ β 2 , ∣ β ∣ ≤ β 0 . (373) \mathbb{E}\left[e^{\beta(J(\eta)-\bar{J}(\eta))}\mid H\right]\le e^{-\beta\bar{J}(\eta)}\left(1+\beta\bar{J}(\eta)+C_*\beta^2\right)\le e^{C_*\beta^2},\qquad|\beta|\le\beta_0. \tag*{(373)} E [ e β ( J ( η ) − J ˉ ( η )) ∣ H ] ≤ e − β J ˉ ( η ) ( 1 + β J ˉ ( η ) + C ∗ β 2 ) ≤ e C ∗ β 2 , ∣ β ∣ ≤ β 0 . (373) We next derive a tail bound for sums. Let n ≥ 1 n \ge1 n ≥ 1 y > 0 y > 0 y > 0 H H H S ( n ) S^{(n)} S ( n ) n n n J J J (373) and Markov’s inequality give for 0 < β ≤ β 0 0 < \beta\le\beta_0 0 < β ≤ β 0 η ∈ { 1 , 2 } \eta\in\{1,2\} η ∈ { 1 , 2 }
P ( ± ( S ( n ) − n J ˉ ) ( η ) ≥ y 2 ∣ H ) ≤ exp ( − β y 2 + C ∗ n β 2 ) . (374) \mathbb{P}\left(\pm(S^{(n)}-n\bar{J})(\eta)\ge\frac{y}{\sqrt{2}}\mathrel{\Big|}H\right)\le\exp\left(-\frac{\beta y}{\sqrt{2}}+C_*n\beta^2\right). \tag*{(374)} P ( ± ( S ( n ) − n J ˉ ) ( η ) ≥ 2 y H ) ≤ exp ( − 2 β y + C ∗ n β 2 ) . (374) We choose
β = { y / ( 2 2 C ∗ n ) if y / ( 2 2 C ∗ n ) ≤ β 0 , β 0 otherwise . (375) \beta=\begin{cases}
y/(2\sqrt{2}C_*n) & \text{if }y/(2\sqrt{2}C_*n)\le\beta_0,\\
\beta_0 & \text{otherwise}.
\end{cases} \tag*{(375)} β = { y / ( 2 2 C ∗ n ) β 0 if y / ( 2 2 C ∗ n ) ≤ β 0 , otherwise . (375) In the second case, we have C ∗ n β 0 2 ≤ β 0 y / ( 2 2 ) C_*n\beta_0^2\le\beta_0y/(2\sqrt{2}) C ∗ n β 0 2 ≤ β 0 y / ( 2 2 ) S ( n ) − n J ˉ S^{(n)}-n\bar{J} S ( n ) − n J ˉ ∣ S ( n ) − n J ˉ ∣ / 2 |S^{(n)}-n\bar{J}|/\sqrt{2} ∣ S ( n ) − n J ˉ ∣/ 2
P ( ∣ S ( n ) − n J ˉ ∣ ≥ y ∣ H ) ≤ 4 exp ( − min { y 2 8 C ∗ n , β 0 y 2 2 } ) . (376) \mathbb{P}\left(|S^{(n)}-n\bar{J}|\ge y\mathrel{\Big|}H\right)\le4\exp\left(-\min\left\{\frac{y^2}{8C_*n},\frac{\beta_0y}{2\sqrt{2}}\right\}\right). \tag*{(376)} P ( ∣ S ( n ) − n J ˉ ∣ ≥ y H ) ≤ 4 exp ( − min { 8 C ∗ n y 2 , 2 2 β 0 y } ) . (376) We now split the m m m m m m S i = S ′ + S ′ ′ S_i=S'+S'' S i = S ′ + S ′′ S ′ S' S ′ m / 2 m/2 m /2 (85) and S ′ ′ S'' S ′′ m / 2 m/2 m /2 H H H S ′ S' S ′ S ′ ′ S'' S ′′ S ( m / 2 ) S^{(m/2)} S ( m /2 ) N 1 N_1 N 1 N ≥ N 1 N\ge N_1 N ≥ N 1 τ / 10 ≤ m / 2 ≤ τ / 3 \tau/10\le m/2\le\tau/3 τ /10 ≤ m /2 ≤ τ /3 m ∣ J ˉ ∣ ≤ 1 m|\bar{J}|\le1 m ∣ J ˉ ∣ ≤ 1 A N , 1 \mathcal{A}_{N,1} A N , 1 τ 1 / 2 ≥ 4 ( x 0 + 1 ) \tau^{1/2}\ge4(x_0+1) τ 1/2 ≥ 4 ( x 0 + 1 )
Suppose that ∣ z i + S i ∣ ≤ x 0 |z_i+S_i|\le x_0 ∣ z i + S i ∣ ≤ x 0 (367) and g i ≥ ℓ g_i\ge\ell g i ≥ ℓ ∣ z i ∣ ≥ τ 1 / 2 / 2 |z_i|\ge\tau^{1/2}/2 ∣ z i ∣ ≥ τ 1/2 /2
∣ S ′ − m 2 J ˉ ∣ + ∣ S ′ ′ − m 2 J ˉ ∣ ≥ ∣ S ′ + S ′ ′ − m J ˉ ∣ ≥ ∣ z i ∣ − x 0 − 1 ≥ ∣ z i ∣ 2 , (377) \left|S'-\frac{m}{2}\bar{J}\right|+\left|S''-\frac{m}{2}\bar{J}\right|\ge|S'+S''-m\bar{J}|\ge|z_i|-x_0-1\ge\frac{|z_i|}{2}, \tag*{(377)} S ′ − 2 m J ˉ + S ′′ − 2 m J ˉ ≥ ∣ S ′ + S ′′ − m J ˉ ∣ ≥ ∣ z i ∣ − x 0 − 1 ≥ 2 ∣ z i ∣ , (377) where the last inequality follows from ∣ z i ∣ ≥ τ 1 / 2 / 2 |z_i|\ge\tau^{1/2}/2 ∣ z i ∣ ≥ τ 1/2 /2 τ 1 / 2 ≥ 4 ( x 0 + 1 ) \tau^{1/2}\ge4(x_0+1) τ 1/2 ≥ 4 ( x 0 + 1 )
∣ S ′ − m 2 J ˉ ∣ ≥ ∣ z i ∣ 4 or ∣ S ′ ′ − m 2 J ˉ ∣ ≥ ∣ z i ∣ 4 . (378) \left|S'-\frac{m}{2}\bar{J}\right|\ge\frac{|z_i|}{4}\quad\text{or}\quad\left|S''-\frac{m}{2}\bar{J}\right|\ge\frac{|z_i|}{4}. \tag*{(378)} S ′ − 2 m J ˉ ≥ 4 ∣ z i ∣ or S ′′ − 2 m J ˉ ≥ 4 ∣ z i ∣ . (378) Let B x 0 ⊂ R 2 B_{x_0}\subset\mathbb{R}^2 B x 0 ⊂ R 2 x 0 x_0 x 0 0 0 0 C B x 0 C_{B_{x_0}} C B x 0 (334) for k = 1 k=1 k = 1 S ′ S' S ′ ∣ z i + S i ∣ ≤ x 0 |z_i+S_i|\le x_0 ∣ z i + S i ∣ ≤ x 0 S ′ ′ + z i + S ′ ∈ B x 0 S''+z_i+S'\in B_{x_0} S ′′ + z i + S ′ ∈ B x 0 S ′ ′ S'' S ′′ S ′ S' S ′ (334) with n = m / 2 n=m/2 n = m /2 (376) , gives
P ( ∣ z i + S i ∣ ≤ x 0 , ∣ S ′ − m 2 J ˉ ∣ ≥ ∣ z i ∣ 4 ∣ H ) ≤ C B x 0 τ P ( ∣ S ( m / 2 ) − m 2 J ˉ ∣ ≥ ∣ z i ∣ 4 ∣ H ) . (379) \mathbb{P}\left(|z_i+S_i|\le x_0,\left|S'-\frac{m}{2}\bar{J}\right|\ge\frac{|z_i|}{4}\mathrel{\Big|}H\right)\le\frac{C_{B_{x_0}}}{\tau}\mathbb{P}\left(\left|S^{(m/2)}-\frac{m}{2}\bar{J}\right|\ge\frac{|z_i|}{4}\mathrel{\Big|}H\right). \tag*{(379)} P ( ∣ z i + S i ∣ ≤ x 0 , S ′ − 2 m J ˉ ≥ 4 ∣ z i ∣ H ) ≤ τ C B x 0 P ( S ( m /2 ) − 2 m J ˉ ≥ 4 ∣ z i ∣ H ) . (379) The second event is treated in the same way, with the roles of S ′ S' S ′ S ′ ′ S'' S ′′ (376) with n = m / 2 ≤ τ / 8 n=m/2\le\tau/8 n = m /2 ≤ τ /8
y = ∣ z i ∣ 4 = g i τ 1 / 2 8 ℓ . (380) y=\frac{|z_i|}{4}=\frac{g_i\tau^{1/2}}{8\ell}. \tag*{(380)} y = 4 ∣ z i ∣ = 8 ℓ g i τ 1/2 . (380) Our choice of ε 1 \varepsilon_1 ε 1 ε 1 ≤ 1 / 1200 \varepsilon_1 \le1/1200 ε 1 ≤ 1/1200 τ 1 / 2 N − 3 ε 1 ≥ 1 \tau^{1/2}N^{-3\varepsilon_1} \ge1 τ 1/2 N − 3 ε 1 ≥ 1
y 2 8 C ∗ n ≥ 1 64 C ∗ ( g i ℓ ) 2 , β 0 y 2 2 = τ 1 / 2 N − 3 ε 1 64 2 g i ℓ ≥ 1 64 2 g i ℓ . (381) \frac{y^2}{8C_*n} \ge\frac{1}{64C_*}\left(\frac{g_i}{\ell}\right)^2,\qquad\frac{\beta_0y}{2\sqrt{2}}=\frac{\tau^{1/2}N^{-3\varepsilon_1}}{64\sqrt{2}}\frac{g_i}{\ell}\ge\frac{1}{64\sqrt{2}}\frac{g_i}{\ell}. \tag*{(381)} 8 C ∗ n y 2 ≥ 64 C ∗ 1 ( ℓ g i ) 2 , 2 2 β 0 y = 64 2 τ 1/2 N − 3 ε 1 ℓ g i ≥ 64 2 1 ℓ g i . (381) Since g i / ℓ ≥ 1 g_i/\ell\ge1 g i / ℓ ≥ 1 c g i / ℓ cg_i/\ell c g i / ℓ
c = min { ( 64 C ∗ ) − 1 , ( 64 2 ) − 1 } . (382) c=\min\left\{(64C_*)^{-1},(64\sqrt{2})^{-1}\right\}. \tag*{(382)} c = min { ( 64 C ∗ ) − 1 , ( 64 2 ) − 1 } . (382) Combining the two events, we conclude that
P ( ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ 8 C B x 0 τ e − c g i / ℓ , N ≥ N 1 . (383) \mathbb{P}\left(\lvert z_i+S_i\rvert\le x_0\mid H\right)\le\frac{8C_{B x_0}}{\tau}e^{-cg_i/\ell},\qquad N\ge N_1. \tag*{(383)} P ( ∣ z i + S i ∣ ≤ x 0 ∣ H ) ≤ τ 8 C B x 0 e − c g i / ℓ , N ≥ N 1 . (383) Finally, let N < N 1 N<N_1 N < N 1 A N , 1 \mathcal{A}_{N,1} A N , 1 g i ≤ 2 ∥ H ∥ ≤ 6 g_i\le2\lVert H\rVert\le6 g i ≤ 2 ∥ H ∥ ≤ 6 ℓ ≥ N − 3 / 2 ≥ N 1 − 3 / 2 \ell\ge N^{-3/2}\ge N_1^{-3/2} ℓ ≥ N − 3/2 ≥ N 1 − 3/2
τ e c g i / ℓ ≤ N 1 exp ( 6 c N 1 3 / 2 ) . (384) \tau e^{cg_i/\ell}\le N_1\exp(6cN_1^{3/2}). \tag*{(384)} τ e c g i / ℓ ≤ N 1 exp ( 6 c N 1 3/2 ) . (384) Hence (8.123) holds for all N N N C x 0 = max { 8 C B x 0 , N 1 exp ( 6 c N 1 3 / 2 ) } C_{x_0}=\max\{8C_{B x_0},N_1\exp(6cN_1^{3/2})\} C x 0 = max { 8 C B x 0 , N 1 exp ( 6 c N 1 3/2 )}
Proof of Proposition 4.4. We show that the events A N , k \mathcal{A}_{N,k} A N , k c c c P ( A N , k ) → 1 \mathbb{P}(\mathcal{A}_{N,k})\to1 P ( A N , k ) → 1 H H H G ε k \mathcal{G}_{\varepsilon_k} G ε k H H H A A A w w w ( A , w ) (A,w) ( A , w ) ∥ w ∥ 2 = 1 \lVert w\rVert_2=1 ∥ w ∥ 2 = 1 A w = ( w T A w ) w Aw=(w^{T}Aw)w A w = ( w T A w ) w w w w
We now prove the local limit theorem. Let Ψ \Psi Ψ { y ∈ R 2 k : ∣ y ∣ ≤ R Ψ } \{y\in\mathbb{R}^{2k}:|y|\le R_\Psi\} { y ∈ R 2 k : ∣ y ∣ ≤ R Ψ } Ψ \Psi Ψ ϵ N ′ \epsilon'_N ϵ N ′ Ψ \Psi Ψ ϵ N ′ \epsilon'_N ϵ N ′ N N N τ k ϵ N ′ → 0 \tau^k\epsilon'_N\to0 τ k ϵ N ′ → 0
ϵ N = { τ k ϵ N ′ + 16 ( k + 1 ) π − k τ − 1 sup ψ ∈ Ψ ∫ R 2 k ∣ ψ ( y ) ∣ d y if τ ≥ 16 , sup ψ ∈ Ψ ( k k ∥ ψ ∥ ∞ + ( 2 π ) − k ∫ R 2 k ∣ ψ ( y ) ∣ d y ) if τ < 16. (385) \epsilon_N=
\begin{cases}
\tau^k\epsilon'_N+16(k+1)\pi^{-k}\tau^{-1}\sup_{\psi\in\Psi}\int_{\mathbb{R}^{2k}}|\psi(y)|\,dy & \text{if }\tau\ge16,\\
\sup_{\psi\in\Psi}\left(k^k\lVert\psi\rVert_\infty+(2\pi)^{-k}\int_{\mathbb{R}^{2k}}|\psi(y)|\,dy\right) & \text{if }\tau<16.
\end{cases} \tag*{(385)} ϵ N = { τ k ϵ N ′ + 16 ( k + 1 ) π − k τ − 1 sup ψ ∈ Ψ ∫ R 2 k ∣ ψ ( y ) ∣ d y sup ψ ∈ Ψ ( k k ∥ ψ ∥ ∞ + ( 2 π ) − k ∫ R 2 k ∣ ψ ( y ) ∣ d y ) if τ ≥ 16 , if τ < 16. (385) Since ϵ N ′ \epsilon'_N ϵ N ′ ϵ N \epsilon_N ϵ N k k k Ψ \Psi Ψ τ k ϵ N ′ → 0 \tau^k\epsilon'_N\to0 τ k ϵ N ′ → 0 A N , k \mathcal{A}_{N,k} A N , k τ < 16 \tau<16 τ < 16 H H H ∣ E [ ψ ( S i + x ) ∣ H ] ∣ ≤ ∥ ψ ∥ ∞ \lvert\mathbb{E}[\psi(S_i+x)\mid H]\rvert\le\lVert\psi\rVert_\infty ∣ E [ ψ ( S i + x ) ∣ H ]∣ ≤ ∥ ψ ∥ ∞ e − ∣ x ∣ 2 / ( 2 τ ) ≤ 1 e^{-|x|^2/(2\tau)}\le1 e − ∣ x ∣ 2 / ( 2 τ ) ≤ 1 τ ≥ 16 \tau\ge16 τ ≥ 16 S i S_i S i H H H S ( m ) S^{(m)} S ( m ) m = 2 ⌊ τ / 8 ⌋ m=2\lfloor\tau/8\rfloor m = 2 ⌊ τ /8 ⌋ τ − 8 < 4 m ≤ τ \tau-8<4m\le\tau τ − 8 < 4 m ≤ τ τ / 10 ≤ m ≤ τ / 3 \tau/10 \le m \le\tau/3 τ /10 ≤ m ≤ τ /3 τ / 2 ≤ 4 m ≤ τ \tau/2 \le4m \le\tau τ /2 ≤ 4 m ≤ τ Lemma 8.13 with n = m n=m n = m A N , k \mathcal{A}_{N,k} A N , k
∣ E [ ψ ( S i + x ) ∣ H ] − g 4 m I d 2 k ( x ) ∣ ∣ ψ ∣ ≤ ϵ N ′ (386) \left|\mathbb{E}\left[\psi(S_i+x)\mid H\right]-g_{4mI_{d_{2k}}}(x)\right|\lvert\psi\rvert\le\epsilon'_N \tag*{(386)} E [ ψ ( S i + x ) ∣ H ] − g 4 m I d 2 k ( x ) ∣ ψ ∣ ≤ ϵ N ′ (386) for all k k k x ∈ R 2 k x\in\mathbb{R}^{2k} x ∈ R 2 k ψ ∈ Ψ \psi\in\Psi ψ ∈ Ψ 4 m 4m 4 m τ \tau τ g r ′ I d 2 k ( x ) g_{r'I_{d_{2k}}}(x) g r ′ I d 2 k ( x ) r ′ > 0 r'>0 r ′ > 0 y ′ e − y ′ ≤ 1 y'e^{-y'}\le1 y ′ e − y ′ ≤ 1 y ′ ≥ 0 y'\ge0 y ′ ≥ 0
∣ ∂ r ′ g r ′ I d 2 k ( x ) ∣ ≤ k + 1 r ′ ( 2 π r ′ ) − k , x ∈ R 2 k . \left|\partial_{r'}g_{r'I_{d_{2k}}}(x)\right|\le\frac{k+1}{r'}(2\pi r')^{-k},\qquad x\in\mathbb{R}^{2k}. ∂ r ′ g r ′ I d 2 k ( x ) ≤ r ′ k + 1 ( 2 π r ′ ) − k , x ∈ R 2 k . The numbers 4 m 4m 4 m τ \tau τ [ τ / 2 , τ ] [\tau/2,\tau] [ τ /2 , τ ]
sup x ∈ R 2 k ∣ g 4 m I d 2 k ( x ) − g τ I d 2 k ( x ) ∣ ≤ 16 ( k + 1 ) τ ( π τ ) − k . (387) \sup_{x\in\mathbb{R}^{2k}}\left|g_{4mI_{d_{2k}}}(x)-g_{\tau I_{d_{2k}}}(x)\right|\le\frac{16(k+1)}{\tau}(\pi\tau)^{-k}. \tag*{(387)} x ∈ R 2 k sup g 4 m I d 2 k ( x ) − g τ I d 2 k ( x ) ≤ τ 16 ( k + 1 ) ( π τ ) − k . (387) Inserting this into (386) , multiplying by τ k \tau^k τ k ∣ ∫ ψ ∣ ≤ ∫ ∣ ψ ∣ \lvert\int\psi\rvert\le\int\lvert\psi\rvert ∣ ∫ ψ ∣ ≤ ∫ ∣ ψ ∣
τ k g τ I d 2 k ( x ) = ( 2 π ) − k e − ∣ x ∣ 2 / ( 2 τ ) , (388) \tau^k g_{\tau I_{d_{2k}}}(x)=(2\pi)^{-k}e^{-\lvert x\rvert^2/(2\tau)}, \tag*{(388)} τ k g τ I d 2 k ( x ) = ( 2 π ) − k e − ∣ x ∣ 2 / ( 2 τ ) , (388) we obtain (4.15) with ϵ N \epsilon_N ϵ N Proposition 4.4 concerns the events A N , 1 \mathcal{A}_{N,1} A N , 1 Lemma 8.14 , applied with x 0 / 2 x_0/2 x 0 /2 x 0 x_0 x 0
Complex Hermitian matrices
In this section, we sketch the proof of Theorem 1.5 . The argument uses the comparison estimate of § 6 and the induction on the scale of Section 6, but not the resampling argument of Sections 7 and 8; we explain the reason at the beginning of § 11.2 . We indicate the changes to the arguments of the previous sections and give the details of the two estimates that are new. The first is a relative bound on the number of triples of nearby eigenvalues, Proposition 9.2, and the second is a bound on the probability that a Gaussian perturbation of a 3 × 3 3 \times3 3 × 3 Lemma 9.3 . We prove both in full, given the inputs collected in § 11.3 . Where we only sketch an argument, we name the result of the earlier sections whose proof we follow and the properties that this proof uses, and we explain why these properties hold in the complex Hermitian case.
Setup
Throughout this section, H H H λ 1 ≤ ⋯ ≤ λ N \lambda_1 \le\cdots\le\lambda_N λ 1 ≤ ⋯ ≤ λ N g i g_i g i c 2 ( E ) = ( π 2 / 3 ) ρ s c ( E ) 4 c_2(E) = (\pi^2/3)\rho_{\mathrm{sc}}(E)^4 c 2 ( E ) = ( π 2 /3 ) ρ sc ( E ) 4 Theorem 1.5 ; we also let E i = ( λ i + λ i + 1 ) / 2 E_i = (\lambda_i + \lambda_{i+1})/2 E i = ( λ i + λ i + 1 ) /2 G G G H H H G a b G_{ab} G ab a < b a < b a < b E G a b = E G a b 2 = 0 \mathbb{E}G_{ab} = \mathbb{E}G_{ab}^2 = 0 E G ab = E G ab 2 = 0 E ∣ G a b ∣ 2 = 1 / N \mathbb{E}|G_{ab}|^2 = 1/N E ∣ G ab ∣ 2 = 1/ N 1 / N 1/N 1/ N H t H_t H t (36) with this G G G
d 2 = N − 4 / 3 , M α C = N 4 β α C = N 1 − 3 α ( α ∈ ( 0 , 1 / 3 ) ) . (389) d_2 = N^{-4/3}, \qquad M_\alpha^{\mathrm{C}} = N^{4\beta_\alpha^{\mathrm{C}}} = N^{1-3\alpha} \qquad(\alpha\in(0,1/3)). \tag*{(389)} d 2 = N − 4/3 , M α C = N 4 β α C = N 1 − 3 α ( α ∈ ( 0 , 1/3 )) . (389) Theorem 1.5 states that the smallest gaps in the bulk are of order d 2 d_2 d 2 d 2 d_2 d 2 critical scale . By Lemma 9.1 below, for α ∈ ( 0 , 1 / 4 ) \alpha\in(0,1/4) α ∈ ( 0 , 1/4 ) ℓ α = N − 1 − α \ell_\alpha= N^{-1-\alpha} ℓ α = N − 1 − α M α C M_\alpha^{\mathrm{C}} M α C
Let p ≥ 1 p \ge1 p ≥ 1 χ ∈ C c ∞ ( ( − 2 , 2 ) ) \chi\in C_c^\infty((-2,2)) χ ∈ C c ∞ (( − 2 , 2 )) q > 0 q > 0 q > 0 A A A
U q ( 3 ) ( A ) = ∑ i < j < k χ N ( λ i ( A ) + λ j ( A ) + λ k ( A ) 3 ) ∏ { i ′ , i ′ ′ } ⊂ { i , j , k } f p ( λ i ′ ( A ) − λ i ′ ′ ( A ) q ) , U_q^{(3)}(A)=\sum_{i<j<k}\chi_N\left(\frac{\lambda_i(A)+\lambda_j(A)+\lambda_k(A)}{3}\right)\prod_{\{i',i''\}\subset\{i,j,k\}}f_p\left(\frac{\lambda_{i'}(A)-\lambda_{i''}(A)}{q}\right), U q ( 3 ) ( A ) = i < j < k ∑ χ N ( 3 λ i ( A ) + λ j ( A ) + λ k ( A ) ) { i ′ , i ′′ } ⊂ { i , j , k } ∏ f p ( q λ i ′ ( A ) − λ i ′′ ( A ) ) , U q ( 4 ) ( A ) = ∑ i < j < k < l χ N ( λ i ( A ) + λ j ( A ) + λ k ( A ) + λ l ( A ) 4 ) ∏ { i ′ , i ′ ′ } ⊂ { i , j , k , l } f p ( λ i ′ ( A ) − λ i ′ ′ ( A ) q ) , (390) U_q^{(4)}(A)=\sum_{i<j<k<l}\chi_N\left(\frac{\lambda_i(A)+\lambda_j(A)+\lambda_k(A)+\lambda_l(A)}{4}\right)\prod_{\{i',i''\}\subset\{i,j,k,l\}}f_p\left(\frac{\lambda_{i'}(A)-\lambda_{i''}(A)}{q}\right), \tag*{(390)} U q ( 4 ) ( A ) = i < j < k < l ∑ χ N ( 4 λ i ( A ) + λ j ( A ) + λ k ( A ) + λ l ( A ) ) { i ′ , i ′′ } ⊂ { i , j , k , l } ∏ f p ( q λ i ′ ( A ) − λ i ′′ ( A ) ) , (390) where χ N \chi_N χ N χ \chi χ p p p χ \chi χ λ k ( A ) − λ i ( A ) ≤ 4 q \lambda_k(A)-\lambda_i(A)\le4q λ k ( A ) − λ i ( A ) ≤ 4 q χ N ≥ 1 \chi_N\ge1 χ N ≥ 1 λ i ( A ) \lambda_i(A) λ i ( A ) λ j ( A ) \lambda_j(A) λ j ( A ) λ k ( A ) \lambda_k(A) λ k ( A ) ( i , j , k ) (i,j,k) ( i , j , k ) U q ( 3 ) ( A ) U_q^{(3)}(A) U q ( 3 ) ( A ) 17 − 3 p 17^{-3p} 1 7 − 3 p f p ( 4 ) = 17 − p f_p(4)=17^{-p} f p ( 4 ) = 1 7 − p λ l ( A ) − λ i ( A ) ≤ 6 q \lambda_l(A)-\lambda_i(A)\le6q λ l ( A ) − λ i ( A ) ≤ 6 q χ N ≥ 1 \chi_N\ge1 χ N ≥ 1 ( i , j , k , l ) (i,j,k,l) ( i , j , k , l ) U q ( 4 ) ( A ) U_q^{(4)}(A) U q ( 4 ) ( A ) 37 − 6 p 37^{-6p} 3 7 − 6 p f p ( 6 ) = 37 − p f_p(6)=37^{-p} f p ( 6 ) = 3 7 − p
Strategy
The smallest gaps of H H H d 2 = N − 4 / 3 d_2=N^{-4/3} d 2 = N − 4/3 N − 3 / 2 = o ( d 2 ) N^{-3/2}=o(d_2) N − 3/2 = o ( d 2 ) Lemma 5.2 holds at the scale d 2 d_2 d 2 α = 1 / 3 < 1 / 2 \alpha=1/3<1/2 α = 1/3 < 1/2 d 2 d_2 d 2
C t N 1 / 2 ( N d 2 ) − 3 N C ε = C t N 3 / 2 + C ε . (391) CtN^{1/2}(Nd_2)^{-3}N^{C\varepsilon}=CtN^{3/2+C\varepsilon}. \tag*{(391)} C t N 1/2 ( N d 2 ) − 3 N C ε = C t N 3/2 + C ε . (391) This is small if t = t α 1 = N − 1 − 2 α 1 t=t_{\alpha_1}=N^{-1-2\alpha_1} t = t α 1 = N − 1 − 2 α 1 α 1 > 1 / 4 \alpha_1>1/4 α 1 > 1/4 t t t H t H_t H t ℓ α 1 = N − 1 − α 1 \ell_{\alpha_1}=N^{-1-\alpha_1} ℓ α 1 = N − 1 − α 1 2 × 2 2\times2 2 × 2 H t H_t H t d 2 d_2 d 2 H H H ℓ α 1 \ell_{\alpha_1} ℓ α 1 H H H q = N − 1 − κ q=N^{-1-\kappa} q = N − 1 − κ d 2 d_2 d 2 α 1 = 8 / 25 \alpha_1=8/25 α 1 = 8/25
We do not use the resampling argument, which would not be available in the same form. Heuristically, a resampling step moves the traceless part of the 2 × 2 2\times2 2 × 2 u i u_i u i u i + 1 u_{i+1} u i + 1 L i L_i L i (81) , with a third coordinate in the complex Hermitian case) by O ( N − 3 / 2 ) = o ( d 2 ) O(N^{-3/2})=o(d_2) O ( N − 3/2 ) = o ( d 2 ) N 1 / 3 N^{1/3} N 1/3 d 2 d_2 d 2 (18) .
The main difference with Section 6 concerns clusters of three eigenvalues. In Section 6, we used the bound of Nguyen, Tao, and Vu on three consecutive eigenvalues, Proposition 3.4. In the complex Hermitian case, the expected number of triples of span N − 1 − κ N^{-1-\kappa} N − 1 − κ N ( N − κ ) 8 N(N^{-\kappa})^8 N ( N − κ ) 8 κ > 1 / 8 \kappa> 1/8 κ > 1/8 N − 1 − κ N^{-1-\kappa} N − 1 − κ H H H T T T
N − 1 + κ ≪ T ≪ N − 1 / 2 − 3 κ . N^{-1+\kappa} \ll T \ll N^{-1/2-3\kappa}. N − 1 + κ ≪ T ≪ N − 1/2 − 3 κ . The lower bound makes the coupling error N ε / ( N 2 T ) N^\varepsilon/(N^2T) N ε / ( N 2 T ) N − 1 − κ N^{-1-\kappa} N − 1 − κ T N 1 / 2 + 3 κ TN^{1/2+3\kappa} T N 1/2 + 3 κ κ < 1 / 8 \kappa< 1/8 κ < 1/8
We therefore use relative bounds. For κ < 1 / 8 \kappa< 1/8 κ < 1/8 4 N − 1 − κ 4N^{-1-\kappa} 4 N − 1 − κ N 1 − 8 κ + ε N^{1-8\kappa+\varepsilon} N 1 − 8 κ + ε M α 1 ′ C = N 1 − 3 α ′ M_{\alpha'_1}^{\mathrm{C}} = N^{1-3\alpha'} M α 1 ′ C = N 1 − 3 α ′ 8 κ > 3 α ′ 8\kappa> 3\alpha' 8 κ > 3 α ′ d 2 d_2 d 2 N C ε ( d 2 / ℓ α 1 ) 3 = N − 1 + 3 α 1 + C ε N^{C\varepsilon}(d_2/\ell_{\alpha_1})^3 = N^{-1+3\alpha_1+C\varepsilon} N C ε ( d 2 / ℓ α 1 ) 3 = N − 1 + 3 α 1 + C ε N 3 α 1 − 8 κ + C ε N^{3\alpha_1-8\kappa+C\varepsilon} N 3 α 1 − 8 κ + C ε 8 κ > 3 α 1 8\kappa> 3\alpha_1 8 κ > 3 α 1 α 1 < 1 / 3 \alpha_1 < 1/3 α 1 < 1/3 κ < 1 / 8 \kappa< 1/8 κ < 1/8
Inputs
Regularity and coupling. In this section, G ε \mathcal{G}_\varepsilon G ε C N \mathbb{C}^N C N H t H_t H t G G G [4 Definition 2.2] concerns complex Hermitian matrices and [29 Theorem 2.2] covers complex Hermitian generalized Wigner matrices. These proofs use the entries only through their independence, their centering, their sub-Gaussian tails, and their variances, which are now E ∣ ( H t ) a b ∣ 2 = 1 / N \mathbb{E}|(H_t)_{ab}|^2 = 1/N E ∣ ( H t ) ab ∣ 2 = 1/ N a ≠ b a \ne b a = b E ( H t ) a a 2 = ( e − t E Z 2 + 1 − e − t ) / N \mathbb{E}(H_t)_{aa}^2 = (e^{-t}\mathbb{E}Z^2 + 1 - e^{-t})/N E ( H t ) aa 2 = ( e − t E Z 2 + 1 − e − t ) / N
The coupling of Proposition 3.5 holds with GUE in place of GOE. In this case, B B B E ∣ B a b ( t ) ∣ 2 = t \mathbb{E}|B_{ab}(t)|^2 = t E ∣ B ab ( t ) ∣ 2 = t E β i ( t ) 2 = t \mathbb{E}\beta_i(t)^2 = t E β i ( t ) 2 = t [10 (2.2)] , whose equilibrium measure is the GUE, and [10 Theorem 3.1 and Corollary 3.3] are stated for this dynamics as well; by [10 Remark 3.2] , they require only optimal rigidity estimates for the initial conditions. The verification of these estimates in the proof of Proposition 3.5 applies without change, since it uses only Lemma 3.3 and Proposition 3.2.
One step requires a modification. In the proof of Proposition 3.5, the dynamics is started from H H H [57 Section 2] assumes that the real and imaginary parts of the entries are independent and identically distributed, which we do not assume. We therefore start the dynamics from H t ∗ H_{t_*} H t ∗ t ∗ = N − 10 t_* = N^{-10} t ∗ = N − 10 H H H H t ∗ H_{t_*} H t ∗ H t ∗ H_{t_*} H t ∗
Moreover, for t ′ ≥ 0 t' \ge0 t ′ ≥ 0
H t ∗ + t ′ = d e − t ′ / 2 H t ∗ + ( 1 − e − t ′ ) 1 / 2 G ~ , H_{t_*+t'} \mathrel{\stackrel{d}{=}} e^{-t'/2}H_{t_*} + (1-e^{-t'})^{1/2}\widetilde{G}, H t ∗ + t ′ = d e − t ′ /2 H t ∗ + ( 1 − e − t ′ ) 1/2 G , where G ~ \widetilde{G} G H t ∗ H_{t_*} H t ∗ Lemma 3.3 , whose proof applies for all times in [ 0 , 2 ] [0,2] [ 0 , 2 ] [10 Corollary 3.3] with the initial condition λ ( H t ∗ ) \lambda(H_{t_*}) λ ( H t ∗ ) t − t ∗ t-t_* t − t ∗ t − t ∗ ≥ t / 2 t-t_* \ge t/2 t − t ∗ ≥ t /2 (47) with 2 N ε / ( N 2 t ) 2N^\varepsilon/(N^2t) 2 N ε / ( N 2 t ) N ε / ( N 2 t ) N^\varepsilon/(N^2t) N ε / ( N 2 t ) t t t N − 1 + ε ≤ t < 1 N^{-1+\varepsilon} \le t < 1 N − 1 + ε ≤ t < 1 H t H_t H t
Comparison. The definition of φ N \varphi_N φ N Lemma 5.2 , the holomorphic extension of Definition 5.4 and Lemma 5.5 , and the derivative bound of Lemma 5.6 carry over verbatim to complex Hermitian matrices, since their proofs use only the spectral decomposition of a Hermitian matrix, Sylvester’s determinant identity, and the properties of G ε \mathcal{G}_\varepsilon G ε a < b a<b a < b V ~ a b = i ( e a e b T − e b e a T ) \widetilde{V}^{ab} = i(e_a e_b^T-e_b e_a^T) V ab = i ( e a e b T − e b e a T ) ( H t ) a b (H_t)_{ab} ( H t ) ab Re ( H t ) a b V a b + Im ( H t ) a b V ~ a b \operatorname{Re}(H_t)_{ab}V^{ab}+\operatorname{Im}(H_t)_{ab}\widetilde{V}^{ab} Re ( H t ) ab V ab + Im ( H t ) ab V ab H t H_t H t ζ 1 \zeta_1 ζ 1 ζ 2 \zeta_2 ζ 2
ζ 1 V a b + ζ 2 V ~ a b = ( ζ 1 + i ζ 2 ) e a e b T + ( ζ 1 − i ζ 2 ) e b e a T , \zeta_1V^{ab}+\zeta_2\widetilde{V}^{ab}=(\zeta_1+i\zeta_2)e_a e_b^T+(\zeta_1-i\zeta_2)e_b e_a^T, ζ 1 V ab + ζ 2 V ab = ( ζ 1 + i ζ 2 ) e a e b T + ( ζ 1 − i ζ 2 ) e b e a T , and the proof of Lemma 5.2 , which uses only the factorization of a perturbation supported on the rows and columns a a a b b b ζ e a e b T + ζ ′ e b e a T \zeta e_a e_b^T+\zeta'e_b e_a^T ζ e a e b T + ζ ′ e b e a T ∣ ζ ∣ , ∣ ζ ′ ∣ ≤ 8 c p N − α − 4 ε |\zeta|,|\zeta'|\le8c_pN^{-\alpha-4\varepsilon} ∣ ζ ∣ , ∣ ζ ′ ∣ ≤ 8 c p N − α − 4 ε
In Lemma 5.7 , the sum runs over the diagonal entries, with v a a G = 1 / N v^G_{aa}=1/N v aa G = 1/ N V a b V^{ab} V ab V ~ a b \widetilde{V}^{ab} V ab v ~ G = 1 / ( 2 N ) \widetilde{v}^G=1/(2N) v G = 1/ ( 2 N ) G G G
The real and imaginary parts of X C X_C X C t ′ ∈ ( 0 , 1 ] t'\in(0,1] t ′ ∈ ( 0 , 1 ] a < b a<b a < b H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] H t ′ H_{t'} H t ′ ( a , b ) (a,b) ( a , b ) ( b , a ) (b,a) ( b , a ) ( Re ( H t ′ ) a b , Im ( H t ′ ) a b ) (\operatorname{Re}(H_{t'})_{ab},\operatorname{Im}(H_{t'})_{ab}) ( Re ( H t ′ ) ab , Im ( H t ′ ) ab ) E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) Spec ( H t ′ [ a b ] + ζ 1 V a b + ζ 2 V ~ a b ) ⊂ R α , p \operatorname{Spec}(H_{t'}^{[ab]}+\zeta_1V^{ab}+\zeta_2\widetilde{V}^{ab})\subset\mathcal{R}_{\alpha,p} Spec ( H t ′ [ ab ] + ζ 1 V ab + ζ 2 V ab ) ⊂ R α , p ζ 1 , ζ 2 \zeta_1,\zeta_2 ζ 1 , ζ 2 ∣ ζ 1 ∣ , ∣ ζ 2 ∣ ≤ 2 r N |\zeta_1|,|\zeta_2|\le2r_N ∣ ζ 1 ∣ , ∣ ζ 2 ∣ ≤ 2 r N H t ′ [ a b ] H_{t'}^{[ab]} H t ′ [ ab ] { H t ′ ∈ G ε } ∩ { ∣ ( H t ′ ) a b ∣ ≤ r N } \{H_{t'}\in\mathcal{G}_\varepsilon\}\cap\{|(H_{t'})_{ab}|\le r_N\} { H t ′ ∈ G ε } ∩ { ∣ ( H t ′ ) ab ∣ ≤ r N } H t ′ H_{t'} H t ′ 6 r N ≤ 8 c p N − α − 4 ε 6r_N\le8c_pN^{-\alpha-4\varepsilon} 6 r N ≤ 8 c p N − α − 4 ε H t ′ [ a b ] + ζ ~ 1 V a b + ζ ~ 2 V ~ a b H_{t'}^{[ab]}+\widetilde{\zeta}_1V^{ab}+\widetilde{\zeta}_2\widetilde{V}^{ab} H t ′ [ ab ] + ζ 1 V ab + ζ 2 V ab H t ′ H_{t'} H t ′ ζ ~ 1 − Re ( H t ′ ) a b \widetilde{\zeta}_1-\operatorname{Re}(H_{t'})_{ab} ζ 1 − Re ( H t ′ ) ab V a b V^{ab} V ab ζ ~ 2 − Im ( H t ′ ) a b \widetilde{\zeta}_2-\operatorname{Im}(H_{t'})_{ab} ζ 2 − Im ( H t ′ ) ab V ~ a b \widetilde{V}^{ab} V ab 3 r N 3r_N 3 r N E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) C D N − D C_DN^{-D} C D N − D (177) .
On E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) ( ζ 1 , ζ 2 ) ↦ F ( H t ′ [ a b ] + ζ 1 V a b + ζ 2 V ~ a b ) (\zeta_1,\zeta_2)\mapsto F(H_{t'}^{[ab]}+\zeta_1V^{ab}+\zeta_2\widetilde{V}^{ab}) ( ζ 1 , ζ 2 ) ↦ F ( H t ′ [ ab ] + ζ 1 V ab + ζ 2 V ab ) 2 r N 2r_N 2 r N Lemma 5.5 . Cauchy’s estimates on the polydisks of radius r N r_N r N [ − r N , r N ] 2 [-r_N,r_N]^2 [ − r N , r N ] 2 n n n [ − r N , r N ] 2 [-r_N,r_N]^2 [ − r N , r N ] 2 C n r N − n C_nr_N^{-n} C n r N − n x = ( Re ( H t ′ ) a b , Im ( H t ′ ) a b ) \mathbf{x}=(\operatorname{Re}(H_{t'})_{ab},\operatorname{Im}(H_{t'})_{ab}) x = ( Re ( H t ′ ) ab , Im ( H t ′ ) ab )
Υ ( x 1 , x 2 ) = F ( H t ′ [ a b ] + x 1 V a b + x 2 V ~ a b ) , ( x 1 , x 2 ) ∈ R 2 . \Upsilon(x_1,x_2)=F(H_{t'}^{[ab]}+x_1V^{ab}+x_2\widetilde{V}^{ab}),\qquad(x_1,x_2)\in\mathbb{R}^2. Υ ( x 1 , x 2 ) = F ( H t ′ [ ab ] + x 1 V ab + x 2 V ab ) , ( x 1 , x 2 ) ∈ R 2 . The contribution of the entry ( a , b ) (a,b) ( a , b ) (161) is then E [ ( 2 N ) − 1 ( ∂ 1 2 + ∂ 2 2 ) Υ ( x ) − x ⋅ ∇ Υ ( x ) ] \mathbb{E}[(2N)^{-1}(\partial_1^2+\partial_2^2)\Upsilon(\mathbf{x})-\mathbf{x}\cdot\nabla\Upsilon(\mathbf{x})] E [( 2 N ) − 1 ( ∂ 1 2 + ∂ 2 2 ) Υ ( x ) − x ⋅ ∇Υ ( x )] ∇ \nabla ∇ ( x 1 , x 2 ) (x_1,x_2) ( x 1 , x 2 ) ∂ 1 \partial_1 ∂ 1 ∂ 2 \partial_2 ∂ 2 x 1 x_1 x 1 x 2 x_2 x 2 ( Re X C , Im X C ) (\operatorname{Re}X_C,\operatorname{Im}X_C) ( Re X C , Im X C ) 2 − 1 I d 2 2^{-1}\mathrm{Id}_2 2 − 1 Id 2 E X C 2 = 0 \mathbb{E}X_C^2=0 E X C 2 = 0 E ∣ X C ∣ 2 = 1 \mathbb{E}|X_C|^2=1 E ∣ X C ∣ 2 = 1 N 1 / 2 ( Re G a b , Im G a b ) N^{1/2}(\operatorname{Re}G_{ab},\operatorname{Im}G_{ab}) N 1/2 ( Re G ab , Im G ab ) E x = 0 \mathbb{E}\mathbf{x}=0 E x = 0 E x x T = ( 2 N ) − 1 I d 2 \mathbb{E}\mathbf{x}\mathbf{x}^{T}=(2N)^{-1}\mathrm{Id}_2 E x x T = ( 2 N ) − 1 Id 2
We expand ∇ Υ \nabla\Upsilon ∇Υ ( ∂ 1 2 + ∂ 2 2 ) Υ (\partial_1^2+\partial_2^2)\Upsilon ( ∂ 1 2 + ∂ 2 2 ) Υ { ∣ x ∣ ≤ r N } \{|\mathbf{x}|\le r_N\} { ∣ x ∣ ≤ r N } Lemma 5.8 . By the last two identities, the terms of order zero and one cancel, and we obtain, on E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ )
∣ E [ ∂ 1 2 + ∂ 2 2 2 N Υ ( x ) − x ⋅ ∇ Υ ( x ) ∣ H t ′ [ a b ] ] ∣ ≤ C r N − 3 E ∣ x ∣ 3 + C D N − D ≤ C N − 3 / 2 r N − 3 + C D N − D . \left|\mathbb{E}\left[\left.\frac{\partial_1^2+\partial_2^2}{2N}\Upsilon(\mathbf{x})-\mathbf{x}\cdot\nabla\Upsilon(\mathbf{x})\right|H_{t'}^{[ab]}\right]\right|\le Cr_N^{-3}\mathbb{E}|\mathbf{x}|^3+C_DN^{-D}\le CN^{-3/2}r_N^{-3}+C_DN^{-D}. E [ 2 N ∂ 1 2 + ∂ 2 2 Υ ( x ) − x ⋅ ∇Υ ( x ) H t ′ [ ab ] ] ≤ C r N − 3 E ∣ x ∣ 3 + C D N − D ≤ C N − 3/2 r N − 3 + C D N − D . Here the term C D N − D C_DN^{-D} C D N − D { ∣ x ∣ > r N } \{|\mathbf{x}|>r_N\} { ∣ x ∣ > r N } Lemma 5.6 as in Step 3 of the proof of Proposition 5.9; the complement of E a b ( t ′ ) \mathcal{E}_{ab}(t') E ab ( t ′ ) ∣ E ( H t ′ ) a a 2 − 1 / N ∣ ≤ C N − 1 |\mathbb{E}(H_{t'})_{aa}^2-1/N|\le CN^{-1} ∣ E ( H t ′ ) aa 2 − 1/ N ∣ ≤ C N − 1 r N − 2 r_N^{-2} r N − 2 (172) for complex Hermitian matrices,
∣ E F ( H t ) − E F ( H ) ∣ ≤ C t ( N 1 / 2 r N − 3 + r N − 2 ) + C D N − D ≤ C t N 1 / 2 + 3 α + 12 ε + C D N − D . (392) |\mathbb{E}F(H_t)-\mathbb{E}F(H)|\le Ct(N^{1/2}r_N^{-3}+r_N^{-2})+C_DN^{-D}\le CtN^{1/2+3\alpha+12\varepsilon}+C_DN^{-D}. \tag*{(392)} ∣ E F ( H t ) − E F ( H ) ∣ ≤ C t ( N 1/2 r N − 3 + r N − 2 ) + C D N − D ≤ C t N 1/2 + 3 α + 12 ε + C D N − D . (392) for F F F Definition 5.4 , α + 5 ε < 1 / 2 \alpha+5\varepsilon<1/2 α + 5 ε < 1/2 r N = c p N − α − 4 ε r_N=c_pN^{-\alpha-4\varepsilon} r N = c p N − α − 4 ε
Triple and quadruple statistics. The comparison estimate also applies to the statistics (9.2). We first write them as traces. Let s 3 ( x 1 , x 2 , x 3 ) s_3(x_1,x_2,x_3) s 3 ( x 1 , x 2 , x 3 ) s 4 ( x 1 , x 2 , x 3 , x 4 ) s_4(x_1,x_2,x_3,x_4) s 4 ( x 1 , x 2 , x 3 , x 4 ) U q ( 3 ) U_q^{(3)} U q ( 3 ) U q ( 4 ) U_q^{(4)} U q ( 4 ) s 3 ( λ i , λ j , λ k ) s_3(\lambda_i,\lambda_j,\lambda_k) s 3 ( λ i , λ j , λ k ) s 4 ( λ i , λ j , λ k , λ l ) s_4(\lambda_i,\lambda_j,\lambda_k,\lambda_l) s 4 ( λ i , λ j , λ k , λ l ) λ i = λ i ( A ) \lambda_i=\lambda_i(A) λ i = λ i ( A )
U q ( 3 ) ( A ) = 1 6 ( ∑ i , j , k s 3 ( λ i , λ j , λ k ) − 3 ∑ i , j s 3 ( λ i , λ i , λ j ) + 2 ∑ i s 3 ( λ i , λ i , λ i ) ) . (393) U_q^{(3)}(A)=\frac{1}{6}\left(\sum_{i,j,k}s_3(\lambda_i,\lambda_j,\lambda_k)-3\sum_{i,j}s_3(\lambda_i,\lambda_i,\lambda_j)+2\sum_i s_3(\lambda_i,\lambda_i,\lambda_i)\right). \tag*{(393)} U q ( 3 ) ( A ) = 6 1 i , j , k ∑ s 3 ( λ i , λ j , λ k ) − 3 i , j ∑ s 3 ( λ i , λ i , λ j ) + 2 i ∑ s 3 ( λ i , λ i , λ i ) . (393) and
24 U q ( 4 ) ( A ) = ∑ i , j , k , l s 4 ( λ i , λ j , λ k , λ l ) − 6 ∑ i , j , k s 4 ( λ i , λ i , λ j , λ k ) + 3 ∑ i , j s 4 ( λ i , λ i , λ j , λ j ) + 8 ∑ i , j s 4 ( λ i , λ i , λ i , λ j ) − 6 ∑ i s 4 ( λ i , λ i , λ i , λ i ) . (394) 24U_q^{(4)}(A)=\sum_{i,j,k,l}s_4(\lambda_i,\lambda_j,\lambda_k,\lambda_l)-6\sum_{i,j,k}s_4(\lambda_i,\lambda_i,\lambda_j,\lambda_k)+3\sum_{i,j}s_4(\lambda_i,\lambda_i,\lambda_j,\lambda_j)
+8\sum_{i,j}s_4(\lambda_i,\lambda_i,\lambda_i,\lambda_j)-6\sum_i s_4(\lambda_i,\lambda_i,\lambda_i,\lambda_i). \tag*{(394)} 24 U q ( 4 ) ( A ) = i , j , k , l ∑ s 4 ( λ i , λ j , λ k , λ l ) − 6 i , j , k ∑ s 4 ( λ i , λ i , λ j , λ k ) + 3 i , j ∑ s 4 ( λ i , λ i , λ j , λ j ) + 8 i , j ∑ s 4 ( λ i , λ i , λ i , λ j ) − 6 i ∑ s 4 ( λ i , λ i , λ i , λ i ) . (394) The coefficients are the values ∏ B ( − 1 ) ∣ B ∣ − 1 ( ∣ B ∣ − 1 ) ! \prod_B(-1)^{|B|-1}(|B|-1)! ∏ B ( − 1 ) ∣ B ∣ − 1 ( ∣ B ∣ − 1 )! B B B
Each sum on the right sides of (393) and (394) is the trace of a function of commuting matrices obtained from A A A Definition 5.4 . For example, the first sum in (393) is the trace over ( C N ) ⊗ 3 (\mathbb{C}^{N})^{\otimes3} ( C N ) ⊗ 3
χ N ( A 1 + A 2 + A 3 3 ) ∏ 1 ≤ i < j ≤ 3 ( Id + q − 2 ( A i − A j ) 2 ) − p , \chi_N\left(\frac{A_1+A_2+A_3}{3}\right)\prod_{1\leq i<j\leq3}\left(\operatorname{Id}+q^{-2}(A_i-A_j)^2\right)^{-p}, χ N ( 3 A 1 + A 2 + A 3 ) 1 ≤ i < j ≤ 3 ∏ ( Id + q − 2 ( A i − A j ) 2 ) − p , where A 1 = A ⊗ Id ⊗ Id A_1=A\otimes\operatorname{Id}\otimes\operatorname{Id} A 1 = A ⊗ Id ⊗ Id A 2 = Id ⊗ A ⊗ Id A_2=\operatorname{Id}\otimes A\otimes\operatorname{Id} A 2 = Id ⊗ A ⊗ Id A 3 = Id ⊗ Id ⊗ A A_3=\operatorname{Id}\otimes\operatorname{Id}\otimes A A 3 = Id ⊗ Id ⊗ A Lemma 5.5 , these traces are holomorphic functions of the entries of A A A w w w ∣ Im w ∣ < ω p q |\operatorname{Im}w|<\omega_{pq} ∣ Im w ∣ < ω pq (393) and (394) hold there with λ 1 , … , λ N \lambda_1,\ldots,\lambda_N λ 1 , … , λ N A A A
Next, we bound the arguments of the summands. Suppose now that Spec A \operatorname{Spec} A Spec A R κ , p \mathcal{R}_{\kappa,p} R κ , p (135) , where q = ℓ κ = N − 1 − κ q=\ell_\kappa=N^{-1-\kappa} q = ℓ κ = N − 1 − κ Lemma 5.5 shows that every rational factor in (390) has argument of absolute value at most 1 / 375 1/375 1/375 χ N \chi_N χ N U q ( 3 ) ( A ) U_q^{(3)}(A) U q ( 3 ) ( A ) U q ( 4 ) ( A ) U_q^{(4)}(A) U q ( 4 ) ( A ) 1 / 200 + 6 / 375 < 1 / 40 1/200+6/375<1/40 1/200 + 6/375 < 1/40 Re U q ( r ) ( A ) ≥ 0 \operatorname{Re}U_q^{(r)}(A)\geq0 Re U q ( r ) ( A ) ≥ 0 ∣ exp ( − θ U q ( r ) ( A ) / M ) ∣ ≤ 1 |\exp(-\theta U_q^{(r)}(A)/M)|\leq1 ∣ exp ( − θ U q ( r ) ( A ) / M ) ∣ ≤ 1 θ , M > 0 \theta,M>0 θ , M > 0 r ∈ { 3 , 4 } r\in\{3,4\} r ∈ { 3 , 4 } Lemma 5.6 applies to these observables with ℓ α \ell_\alpha ℓ α q q q exp ( − θ U q ( r ) / M ) \exp(-\theta U_q^{(r)}/M) exp ( − θ U q ( r ) / M ) q = N − 1 − κ q=N^{-1-\kappa} q = N − 1 − κ κ ∈ ( 0 , 1 / 2 ) \kappa\in(0,1/2) κ ∈ ( 0 , 1/2 ) M > 0 M>0 M > 0 θ > 0 \theta>0 θ > 0 D > 0 D>0 D > 0 t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] ε > 0 \varepsilon>0 ε > 0 κ + 5 ε < 1 / 2 \kappa+5\varepsilon<1/2 κ + 5 ε < 1/2
∣ E e − θ U q ( r ) ( H t ) / M − E e − θ U q ( r ) ( H ) / M ∣ ≤ C t N 1 / 2 + 3 κ + 12 ε + C D N − D , r ∈ { 3 , 4 } . (395) \left|\mathbb{E}e^{-\theta U_q^{(r)}(H_t)/M}-\mathbb{E}e^{-\theta U_q^{(r)}(H)/M}\right|\leq CtN^{1/2+3\kappa+12\varepsilon}+C_DN^{-D},\qquad r\in\{3,4\}. \tag*{(395)} E e − θ U q ( r ) ( H t ) / M − E e − θ U q ( r ) ( H ) / M ≤ C t N 1/2 + 3 κ + 12 ε + C D N − D , r ∈ { 3 , 4 } . (395) The Gaussian unitary ensemble. The following estimates replace Proposition 3.6 .
Lemma 9.1. The following statements hold.
(a) Let α ∈ ( 0 , 1 / 4 ) \alpha\in(0,1/4) α ∈ ( 0 , 1/4 ) p > 3 / 2 p > 3/2 p > 3/2 f : [ 0 , ∞ ) × R → R f : [0,\infty) \times\mathbb{R} \to\mathbb{R} f : [ 0 , ∞ ) × R → R I I I C f > 0 C_f > 0 C f > 0 f ( r , E ) = 0 f(r,E) = 0 f ( r , E ) = 0 E ∉ I E \notin I E ∈ / I ∣ f ( r , E ) ∣ ≤ C f ( 1 + r 2 ) − p |f(r,E)| \le C_f(1+r^2)^{-p} ∣ f ( r , E ) ∣ ≤ C f ( 1 + r 2 ) − p ( r , E ) (r,E) ( r , E ) L 1 L^1 L 1
1 M α α ∑ 1 ≤ i < j ≤ N f ( λ j ( G ) − λ i ( G ) ℓ α , λ i ( G ) + λ j ( G ) 2 ) ⟶ ∫ − 2 2 ∫ 0 ∞ c 2 ( E ) r 2 f ( r , E ) d r d E . (396) \frac{1}{M_\alpha^{\alpha}}\sum_{1 \le i < j \le N} f\left(\frac{\lambda_j(G)-\lambda_i(G)}{\ell_\alpha},\frac{\lambda_i(G)+\lambda_j(G)}{2}\right) \longrightarrow\int_{-2}^{2}\int_0^\infty c_2(E)r^2f(r,E)\,dr\,dE. \tag*{(396)} M α α 1 1 ≤ i < j ≤ N ∑ f ( ℓ α λ j ( G ) − λ i ( G ) , 2 λ i ( G ) + λ j ( G ) ) ⟶ ∫ − 2 2 ∫ 0 ∞ c 2 ( E ) r 2 f ( r , E ) d r d E . (396) (b) Let κ > 0 \kappa> 0 κ > 0 q = N − 1 − κ q = N^{-1-\kappa} q = N − 1 − κ p ≥ 4 p \ge4 p ≥ 4
E U q ( 3 ) ( G ) ≤ C N ( N q ) 8 + o ( 1 ) , E U q ( 4 ) ( G ) ≤ C N ( N q ) 15 + o ( 1 ) . (397) \mathbb{E}U_q^{(3)}(G) \le CN(Nq)^8 + o(1), \qquad\mathbb{E}U_q^{(4)}(G) \le CN(Nq)^{15} + o(1). \tag*{(397)} E U q ( 3 ) ( G ) ≤ C N ( N q ) 8 + o ( 1 ) , E U q ( 4 ) ( G ) ≤ C N ( N q ) 15 + o ( 1 ) . (397) Sketch of proof. We use the kernel K N K_N K N Definition A.2 and the estimates for it proved in Appendix A. Recall that the eigenvalues of G G G K N K_N K N n n n
R n , N ( x 1 , … , x n ) = det [ K N ( x i , x j ) ] i , j = 1 n , n ∈ [ N ] , (398) R_{n,N}(x_1,\ldots,x_n)=\det[K_N(x_i,x_j)]_{i,j=1}^n,\qquad n\in[N], \tag*{(398)} R n , N ( x 1 , … , x n ) = det [ K N ( x i , x j ) ] i , j = 1 n , n ∈ [ N ] , (398) (see [2 Section 3.2] ), and (A.12) holds.
The proof follows Appendix § 12 .§ 6 . The determinantal structure makes the correlation estimates simpler than for the GOE, and we give them below; we omit the routine verification of the dominated convergence arguments.
Throughout the proof, I ~ \widetilde I I I ~ \widetilde I I Lemma A.5 and Lemma A.7 for x , y ∈ I ~ x,y \in\widetilde I x , y ∈ I
h ( x ) = ( N 1 / 4 h j ( N x ) ) j = 0 N − 1 ∈ R N , so that K N ( x , y ) = h ( x ) ⋅ h ( y ) . \mathbf{h}(x)=\left(N^{1/4}h_j(\sqrt{N}x)\right)_{j=0}^{N-1}\in\mathbb{R}^N,\qquad\text{so that}\qquad K_N(x,y)=\mathbf{h}(x)\cdot\mathbf{h}(y). h ( x ) = ( N 1/4 h j ( N x ) ) j = 0 N − 1 ∈ R N , so that K N ( x , y ) = h ( x ) ⋅ h ( y ) . The two-point function. We begin with the expansion of R 2 , N R_{2,N} R 2 , N K N K_N K N n = 2 n=2 n = 2
R 2 , N ( x , y ) = K N ( x , x ) K N ( y , y ) − K N ( x , y ) 2 . R_{2,N}(x,y)=K_N(x,x)K_N(y,y)-K_N(x,y)^2. R 2 , N ( x , y ) = K N ( x , x ) K N ( y , y ) − K N ( x , y ) 2 . Using the symmetry of K N K_N K N
R 2 , N ( x , x ) = ∂ y R 2 , N ( x , y ) ∣ y = x = 0 , R_{2,N}(x,x)=\left.\partial_yR_{2,N}(x,y)\right|_{y=x}=0, R 2 , N ( x , x ) = ∂ y R 2 , N ( x , y ) ∣ y = x = 0 , ∂ y 2 R 2 , N ( x , y ) ∣ y = x = K N ( x , x ) d 2 d x 2 K N ( x , x ) − 2 ( ∂ 2 K N ( x , x ) ) 2 − 2 K N ( x , x ) ∂ y 2 K N ( x , y ) ∣ y = x . \left.\partial_y^2R_{2,N}(x,y)\right|_{y=x}=K_N(x,x)\left.\frac{d^2}{dx^2}K_N(x,x)\right. -2\bigl(\partial_2K_N(x,x)\bigr)^2-2K_N(x,x)\left.\partial_y^2K_N(x,y)\right|_{y=x}. ∂ y 2 R 2 , N ( x , y ) y = x = K N ( x , x ) d x 2 d 2 K N ( x , x ) − 2 ( ∂ 2 K N ( x , x ) ) 2 − 2 K N ( x , x ) ∂ y 2 K N ( x , y ) y = x . We now bound the first two terms. By the proof of Lemma A.5 (c), we have
d d x K N ( x , x ) = 2 ∂ 2 K N ( x , x ) = − N 3 / 2 h N − 1 ( N x ) h N ( N x ) . \frac{d}{dx}K_N(x,x)=2\partial_2K_N(x,x)=-N^{3/2}h_{N-1}(\sqrt{N}x)h_N(\sqrt{N}x). d x d K N ( x , x ) = 2 ∂ 2 K N ( x , x ) = − N 3/2 h N − 1 ( N x ) h N ( N x ) . Differentiating once more and using (A.20) and K N ( x , x ) ≤ C N K_N(x,x) \le CN K N ( x , x ) ≤ C N
K N ( x , x ) d 2 d x 2 K N ( x , x ) = O ( N 3 ) , ( ∂ 2 K N ( x , x ) ) 2 = O ( N 2 ) . K_N(x,x)\frac{\mathrm{d}^2}{\mathrm{d}x^2}K_N(x,x)=O(N^3),\qquad(\partial_2K_N(x,x))^2=O(N^2). K N ( x , x ) d x 2 d 2 K N ( x , x ) = O ( N 3 ) , ( ∂ 2 K N ( x , x ) ) 2 = O ( N 2 ) . so the first two terms on the right side of the expression for ∂ y 2 R 2 , N ( x , y ) ∣ y = x \partial_y^2R_{2,N}(x,y)|_{y=x} ∂ y 2 R 2 , N ( x , y ) ∣ y = x O ( N 3 ) O(N^3) O ( N 3 ) Lemma A.7 , and the identity 1 − x 2 / 4 = π 2 ρ s c ( x ) 2 1-x^2/4=\pi^2\rho_{\mathrm{sc}}(x)^2 1 − x 2 /4 = π 2 ρ sc ( x ) 2
− 2 K N ( x , x ) ∂ y 2 K N ( x , y ) ∣ y = x = 2 π 2 3 N 2 ρ s c ( x ) 2 K N ( x , x ) 2 + O ( N 3 ) = 2 π 2 3 N 4 ρ s c ( x ) 4 + O ( N 31 / 8 ) . \begin{aligned}
-2K_N(x,x)\left.\partial_y^2K_N(x,y)\right|_{y=x}
=\frac{2\pi^2}{3}N^2\rho_{\mathrm{sc}}(x)^2K_N(x,x)^2+O(N^3) \\
&=\frac{2\pi^2}{3}N^4\rho_{\mathrm{sc}}(x)^4+O(N^{31/8}).
\end{aligned} − 2 K N ( x , x ) ∂ y 2 K N ( x , y ) y = x = 3 2 π 2 N 2 ρ sc ( x ) 2 K N ( x , x ) 2 + O ( N 3 ) = 3 2 π 2 N 4 ρ sc ( x ) 4 + O ( N 31/8 ) . Further, Lemma A.5 (a) gives
∣ ∂ y 3 R 2 , N ( x , y ) ∣ ≤ C N 5 , x , y ∈ I ~ . \left|\partial_y^3R_{2,N}(x,y)\right|\le CN^5,\qquad x,y\in\widetilde I. ∂ y 3 R 2 , N ( x , y ) ≤ C N 5 , x , y ∈ I . Taylor’s theorem in y y y y = x y=x y = x
∣ R 2 , N ( x , x + h ) − π 2 3 N 4 ρ s c ( x ) 4 h 2 ∣ ≤ C ( N 31 / 8 h 2 + N 5 h 3 ) , x , x + h ∈ I ~ , 0 < h ≤ N − 1 . (399) \left|R_{2,N}(x,x+h)-\frac{\pi^2}{3}N^4\rho_{\mathrm{sc}}(x)^4h^2\right|
\le C\left(N^{31/8}h^2+N^5h^3\right),\qquad x,x+h\in\widetilde I,\quad0<h\le N^{-1}. \tag*{(399)} R 2 , N ( x , x + h ) − 3 π 2 N 4 ρ sc ( x ) 4 h 2 ≤ C ( N 31/8 h 2 + N 5 h 3 ) , x , x + h ∈ I , 0 < h ≤ N − 1 . (399) This is the source of the function c 2 ( E ) c_2(E) c 2 ( E )
R 2 , N ( x , y ) ≤ K N ( x , x ) K N ( y , y ) ≤ C N 2 , ∣ K N ( x , y ) ∣ ≤ C min { N , ∣ x − y ∣ − 1 } , (400) R_{2,N}(x,y)\le K_N(x,x)K_N(y,y)\le CN^2,\qquad
|K_N(x,y)|\le C\min\{N,|x-y|^{-1}\}, \tag*{(400)} R 2 , N ( x , y ) ≤ K N ( x , x ) K N ( y , y ) ≤ C N 2 , ∣ K N ( x , y ) ∣ ≤ C min { N , ∣ x − y ∣ − 1 } , (400) which follow from (398) and Lemma A.5 (a) and (b).
Correlation functions of nearby points. We claim that for n ∈ { 2 , 3 , 4 } n\in\{2,3,4\} n ∈ { 2 , 3 , 4 } x 1 , … , x n ∈ I ~ x_1,\ldots,x_n\in\widetilde I x 1 , … , x n ∈ I
R n , N ( x 1 , … , x n ) ≤ C N n min { 1 , ∏ 1 ≤ i < j ≤ n ( N ∣ x i − x j ∣ ) 2 } . (401) R_{n,N}(x_1,\ldots,x_n)\le CN^n\min\left\{1,\prod_{1\le i<j\le n}(N|x_i-x_j|)^2\right\}. \tag*{(401)} R n , N ( x 1 , … , x n ) ≤ C N n min { 1 , 1 ≤ i < j ≤ n ∏ ( N ∣ x i − x j ∣ ) 2 } . (401) The matrix [ K N ( x i , x j ) ] i , j = 1 n [K_N(x_i,x_j)]_{i,j=1}^n [ K N ( x i , x j ) ] i , j = 1 n h ( x 1 ) , … , h ( x n ) \mathbf{h}(x_1),\ldots,\mathbf{h}(x_n) h ( x 1 ) , … , h ( x n )
R n , N ( x 1 , … , x n ) ≤ ∏ i = 1 n K N ( x i , x i ) ≤ C N n . R_{n,N}(x_1,\ldots,x_n)\le\prod_{i=1}^nK_N(x_i,x_i)\le CN^n. R n , N ( x 1 , … , x n ) ≤ i = 1 ∏ n K N ( x i , x i ) ≤ C N n . To prove the second bound in (401) , we may assume that the points are distinct. We replace h ( x 1 ) , … , h ( x n ) \mathbf{h}(x_1),\ldots,\mathbf{h}(x_n) h ( x 1 ) , … , h ( x n ) h [ x 1 ] , h [ x 1 , x 2 ] , … , h [ x 1 , … , x n ] \mathbf{h}[x_1],\mathbf{h}[x_1,x_2],\ldots,\mathbf{h}[x_1,\ldots,x_n] h [ x 1 ] , h [ x 1 , x 2 ] , … , h [ x 1 , … , x n ] ∏ i < j ( x j − x i ) − 1 \prod_{i<j}(x_j-x_i)^{-1} ∏ i < j ( x j − x i ) − 1
R n , N ( x 1 , … , x n ) = ∏ 1 ≤ i < j ≤ n ( x j − x i ) 2 det [ h [ x 1 , … , x i ] ⋅ h [ x 1 , … , x j ] ] i , j = 1 n . R_{n,N}(x_1,\ldots,x_n)=\prod_{1\le i<j\le n}(x_j-x_i)^2\det\left[\mathbf{h}[x_1,\ldots,x_i]\cdot\mathbf{h}[x_1,\ldots,x_j]\right]_{i,j=1}^n. R n , N ( x 1 , … , x n ) = 1 ≤ i < j ≤ n ∏ ( x j − x i ) 2 det [ h [ x 1 , … , x i ] ⋅ h [ x 1 , … , x j ] ] i , j = 1 n . We now bound the divided differences. By the Hermite–Genocchi formula, h [ x 1 , … , x k ] \mathbf{h}[x_1,\ldots,x_k] h [ x 1 , … , x k ] h ( k − 1 ) \mathbf{h}^{(k-1)} h ( k − 1 ) 1 / ( k − 1 ) ! 1/(k-1)! 1/ ( k − 1 )! x 1 , … , x k x_1,\ldots,x_k x 1 , … , x k Lemma A.5 , with k − 1 k-1 k − 1 k k k
∣ h [ x 1 , … , x k ] ∣ 2 ≤ sup y ∈ I ~ ∣ h ( k − 1 ) ( y ) ∣ 2 = sup y ∈ I ~ N k − 1 / 2 ∑ j = 0 N − 1 h j ( k − 1 ) ( N y ) 2 ≤ C N 2 k − 1 , k ≤ 4. \left|h[x_1,\ldots,x_k]\right|^2 \le\sup_{y\in\widetilde I}\left|h^{(k-1)}(y)\right|^2
= \sup_{y\in\widetilde I} N^{k-1/2}\sum_{j=0}^{N-1}h_j^{(k-1)}\left(\sqrt{N}y\right)^2 \le C N^{2k-1}, \qquad k\le4. ∣ h [ x 1 , … , x k ] ∣ 2 ≤ y ∈ I sup h ( k − 1 ) ( y ) 2 = y ∈ I sup N k − 1/2 j = 0 ∑ N − 1 h j ( k − 1 ) ( N y ) 2 ≤ C N 2 k − 1 , k ≤ 4. Applying Hadamard’s inequality to the Gram determinant of the divided differences, we obtain
R n , N ( x 1 , … , x n ) ≤ ∏ i < j ( x j − x i ) 2 ∏ k = 1 n ∣ h [ x 1 , … , x k ] ∣ 2 ≤ C ∏ i < j ( x j − x i ) 2 ∏ k = 1 n N 2 k − 1 = C N n 2 ∏ i < j ∣ x i − x j ∣ 2 , \begin{aligned}
R_{n,N}(x_1,\ldots,x_n)\le\prod_{i<j}(x_j-x_i)^2\prod_{k=1}^{n}\left|h[x_1,\ldots,x_k]\right|^2 \\
&\le C\prod_{i<j}(x_j-x_i)^2\prod_{k=1}^{n}N^{2k-1}
=CN^{n^2}\prod_{i<j}|x_i-x_j|^2,
\end{aligned} R n , N ( x 1 , … , x n ) ≤ i < j ∏ ( x j − x i ) 2 k = 1 ∏ n ∣ h [ x 1 , … , x k ] ∣ 2 ≤ C i < j ∏ ( x j − x i ) 2 k = 1 ∏ n N 2 k − 1 = C N n 2 i < j ∏ ∣ x i − x j ∣ 2 , which is the second bound in (401) , since n + n ( n − 1 ) = n 2 n+n(n-1)=n^2 n + n ( n − 1 ) = n 2
Proof of (a). We first compute the mean. By (A.12), we have
E ∑ 1 ≤ i < j ≤ N f ( λ j ( G ) − λ i ( G ) ℓ α , λ i ( G ) + λ j ( G ) 2 ) = ∫ R ∫ 0 ∞ f ( h ℓ α , x + h 2 ) R 2 , N ( x , x + h ) d h d x . \mathbb{E}\sum_{1\le i<j\le N}f\left(\frac{\lambda_j(G)-\lambda_i(G)}{\ell_\alpha},\frac{\lambda_i(G)+\lambda_j(G)}{2}\right)
=\int_{\mathbb{R}}\int_0^\infty f\left(\frac{h}{\ell_\alpha},x+\frac{h}{2}\right)R_{2,N}(x,x+h)\,dh\,dx. E 1 ≤ i < j ≤ N ∑ f ( ℓ α λ j ( G ) − λ i ( G ) , 2 λ i ( G ) + λ j ( G ) ) = ∫ R ∫ 0 ∞ f ( ℓ α h , x + 2 h ) R 2 , N ( x , x + h ) d h d x . Let δ > 0 \delta>0 δ > 0 ∣ f ( r , E ) ∣ ≤ C f r − 2 p |f(r,E)|\le C_f r^{-2p} ∣ f ( r , E ) ∣ ≤ C f r − 2 p R 2 , N R_{2,N} R 2 , N R 2 \mathbb{R}^2 R 2 N ( N − 1 ) N(N-1) N ( N − 1 ) δ \delta δ
C f ( ℓ α δ ) 2 p ∫ R ∫ 0 ∞ R 2 , N ( x , x + h ) d h d x ≤ C N 2 ( ℓ α δ ) 2 p = o ( 1 ) . C_f\left(\frac{\ell_\alpha}{\delta}\right)^{2p}\int_{\mathbb{R}}\int_0^\infty R_{2,N}(x,x+h)\,dh\,dx
\le CN^2\left(\frac{\ell_\alpha}{\delta}\right)^{2p}=o(1). C f ( δ ℓ α ) 2 p ∫ R ∫ 0 ∞ R 2 , N ( x , x + h ) d h d x ≤ C N 2 ( δ ℓ α ) 2 p = o ( 1 ) . For the remaining pairs, both points lie in a bulk interval I ~ \widetilde I I f ( r , E ) = 0 f(r,E)=0 f ( r , E ) = 0 E ∉ I E\notin I E ∈ / I (399) for h ≤ N − 1 h\le N^{-1} h ≤ N − 1 h = ℓ α r h=\ell_\alpha r h = ℓ α r
N 4 ℓ α 3 = M α C , N 31 / 8 ℓ α 3 = N − 1 / 8 M α C , N 5 ℓ α 4 = N − α M α C . N^4\ell_\alpha^3=M_\alpha^{\mathrm C},\qquad
N^{31/8}\ell_\alpha^3=N^{-1/8}M_\alpha^{\mathrm C},\qquad
N^5\ell_\alpha^4=N^{-\alpha}M_\alpha^{\mathrm C}. N 4 ℓ α 3 = M α C , N 31/8 ℓ α 3 = N − 1/8 M α C , N 5 ℓ α 4 = N − α M α C . By the first identity, the main term is
M α C ∫ R ∫ 0 N α f ( r , x + ℓ α r 2 ) π 2 3 ρ s c ( x ) 4 r 2 d r d x , M_\alpha^{\mathrm C}\int_{\mathbb{R}}\int_0^{N^\alpha}f\left(r,x+\frac{\ell_\alpha r}{2}\right)\frac{\pi^2}{3}\rho_{\mathrm{sc}}(x)^4r^2\,dr\,dx, M α C ∫ R ∫ 0 N α f ( r , x + 2 ℓ α r ) 3 π 2 ρ sc ( x ) 4 r 2 d r d x , and since p > 3 / 2 p>3/2 p > 3/2 M α C M_\alpha^{\mathrm C} M α C (396) by dominated convergence. Using the other two identities, we find that the error terms in (399) contribute at most
C M α C ( N − 1 / 8 + N − α ∫ 0 N α r 3 d r ( 1 + r 2 ) p ) ≤ C M α C ( N − 1 / 8 + N − α + N − α ( 2 p − 3 ) log N ) . CM_\alpha^{\mathrm C}\left(N^{-1/8}+N^{-\alpha}\int_0^{N^\alpha}\frac{r^3\,dr}{(1+r^2)^p}\right)
\le CM_\alpha^{\mathrm C}\left(N^{-1/8}+N^{-\alpha}+N^{-\alpha(2p-3)}\log N\right). C M α C ( N − 1/8 + N − α ∫ 0 N α ( 1 + r 2 ) p r 3 d r ) ≤ C M α C ( N − 1/8 + N − α + N − α ( 2 p − 3 ) log N ) . Finally, by (400) , the pairs with N − 1 < h ≤ δ N^{-1}<h\le\delta N − 1 < h ≤ δ
C N 2 ∫ N − 1 δ ( ℓ α h ) 2 p d h ≤ C N 2 ℓ α ∫ N α ∞ r − 2 p d r = C M α C N − α ( 2 p − 3 ) . CN^2\int_{N^{-1}}^\delta\left(\frac{\ell_\alpha}{h}\right)^{2p}\,dh\le CN^2\ell_\alpha\int_{N^\alpha}^{\infty}r^{-2p}\,dr=CM_\alpha^C N^{-\alpha(2p-3)}. C N 2 ∫ N − 1 δ ( h ℓ α ) 2 p d h ≤ C N 2 ℓ α ∫ N α ∞ r − 2 p d r = C M α C N − α ( 2 p − 3 ) . Hence the mean of the sum truncated to ∣ λ j ( G ) − λ i ( G ) ∣ ≤ δ |\lambda_j(G)-\lambda_i(G)|\le\delta ∣ λ j ( G ) − λ i ( G ) ∣ ≤ δ M α C M_\alpha^C M α C (396) .
We now bound the variance of the truncated sum. We set
f ♯ ( x , y ) = f ( ∣ y − x ∣ ℓ α , x + y 2 ) 1 ∣ y − x ∣ ≤ δ , x , y ∈ R . f^\sharp(x,y)=f\left(\frac{|y-x|}{\ell_\alpha},\frac{x+y}{2}\right)\mathbf{1}_{|y-x|\le\delta},\qquad x,y\in\mathbb{R}. f ♯ ( x , y ) = f ( ℓ α ∣ y − x ∣ , 2 x + y ) 1 ∣ y − x ∣ ≤ δ , x , y ∈ R . As in (A.35), with f ♯ f^\sharp f ♯ g ♯ g^\sharp g ♯ R 4 , N − R 2 , N ⊗ R 2 , N R_{4,N}-R_{2,N}\otimes R_{2,N} R 4 , N − R 2 , N ⊗ R 2 , N R 3 , N R_{3,N} R 3 , N R 2 , N R_{2,N} R 2 , N R 4 , N ( x 1 , x 2 , x 3 , x 4 ) R_{4,N}(x_1,x_2,x_3,x_4) R 4 , N ( x 1 , x 2 , x 3 , x 4 ) R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) R_{2,N}(x_1,x_2)R_{2,N}(x_3,x_4) R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) { 1 , 2 } \{1,2\} { 1 , 2 } K N ( x i , x j ) K_N(x_i,x_j) K N ( x i , x j ) { 1 , 2 } \{1,2\} { 1 , 2 } { 3 , 4 } \{3,4\} { 3 , 4 } K N K_N K N (400) , they are bounded by C min { N 4 , N 2 dist ( { x 1 , x 2 } , { x 3 , x 4 } ) − 2 } C\min\{N^4,N^2\operatorname{dist}(\{x_1,x_2\},\{x_3,x_4\})^{-2}\} C min { N 4 , N 2 dist ({ x 1 , x 2 } , { x 3 , x 4 } ) − 2 }
To integrate this bound, we write x 2 = x 1 + h 1 x_2=x_1+h_1 x 2 = x 1 + h 1 x 4 = x 3 + h 2 x_4=x_3+h_2 x 4 = x 3 + h 2 x 3 x_3 x 3 ∣ x 3 − x 1 ∣ ≥ 2 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) |x_3-x_1|\ge2(|h_1|+|h_2|) ∣ x 3 − x 1 ∣ ≥ 2 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) { x 1 , x 2 } \{x_1,x_2\} { x 1 , x 2 } { x 3 , x 4 } \{x_3,x_4\} { x 3 , x 4 } ∣ x 3 − x 1 ∣ / 2 |x_3-x_1|/2 ∣ x 3 − x 1 ∣/2 x 3 x_3 x 3 4 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) 4(|h_1|+|h_2|) 4 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) x 3 ∈ R x_3\in\mathbb{R} x 3 ∈ R
C ∫ R min { N 4 , 4 N 2 ∣ x 3 − x 1 ∣ 2 } d x 3 + 4 C N 4 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) ≤ C N 3 + C N 4 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) . C\int_{\mathbb{R}}\min\left\{N^4,\frac{4N^2}{|x_3-x_1|^2}\right\}\,dx_3+4CN^4(|h_1|+|h_2|)\le CN^3+CN^4(|h_1|+|h_2|). C ∫ R min { N 4 , ∣ x 3 − x 1 ∣ 2 4 N 2 } d x 3 + 4 C N 4 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) ≤ C N 3 + C N 4 ( ∣ h 1 ∣ + ∣ h 2 ∣ ) . We now integrate over h 1 h_1 h 1 h 2 h_2 h 2 p > 3 / 2 p>3/2 p > 3/2
∫ R ∣ f ♯ ( x 1 , x 1 + h ) ∣ ( 1 + N ∣ h ∣ ) d h ≤ C ℓ α ( 1 + N ℓ α ) ≤ C ℓ α , \int_{\mathbb{R}}|f^\sharp(x_1,x_1+h)|(1+N|h|)\,dh\le C\ell_\alpha(1+N\ell_\alpha)\le C\ell_\alpha, ∫ R ∣ f ♯ ( x 1 , x 1 + h ) ∣ ( 1 + N ∣ h ∣ ) d h ≤ C ℓ α ( 1 + N ℓ α ) ≤ C ℓ α , and we find that the first integral is bounded by
C ℓ α 2 ( N 3 + N 4 ℓ α ) ≤ C N 3 ℓ α 2 = C N 1 − 2 α . C\ell_\alpha^2(N^3+N^4\ell_\alpha)\le CN^3\ell_\alpha^2=CN^{1-2\alpha}. C ℓ α 2 ( N 3 + N 4 ℓ α ) ≤ C N 3 ℓ α 2 = C N 1 − 2 α . By Fischer’s inequality, we have
R 3 , N ( x 1 , x 2 , x 3 ) ≤ K N ( x 3 , x 3 ) R 2 , N ( x 1 , x 2 ) ≤ C N R 2 , N ( x 1 , x 2 ) , R_{3,N}(x_1,x_2,x_3)\le K_N(x_3,x_3)R_{2,N}(x_1,x_2)\le CNR_{2,N}(x_1,x_2), R 3 , N ( x 1 , x 2 , x 3 ) ≤ K N ( x 3 , x 3 ) R 2 , N ( x 1 , x 2 ) ≤ C N R 2 , N ( x 1 , x 2 ) , so the second integral is bounded by C N ℓ α M α C CN\ell_\alpha M_\alpha^C C N ℓ α M α C C M α C CM_\alpha^C C M α C ∣ f ∣ |f| ∣ f ∣ ( M α C ) 2 (M_\alpha^C)^2 ( M α C ) 2
C N 1 − 2 α + N ℓ α M α C + M α C ( M α C ) 2 = C ( N 4 α − 1 + N 2 α − 1 + N 3 α − 1 ) . C\frac{N^{1-2\alpha}+N\ell_\alpha M_\alpha^C+M_\alpha^C}{(M_\alpha^C)^2}
=C\left(N^{4\alpha-1}+N^{2\alpha-1}+N^{3\alpha-1}\right). C ( M α C ) 2 N 1 − 2 α + N ℓ α M α C + M α C = C ( N 4 α − 1 + N 2 α − 1 + N 3 α − 1 ) . which tends to zero since α < 1 / 4 \alpha< 1/4 α < 1/4 L 2 L^2 L 2 L 1 L^1 L 1
Proof of (b). Let δ > 0 \delta> 0 δ > 0 δ \delta δ δ \delta δ supp χ \operatorname{supp}\chi supp χ
∥ χ ∥ ∞ N 4 ( q δ ) 2 p = o ( 1 ) and C N 4 exp ( − δ 2 N 1 / 5 2 ) = o ( 1 ) , \|\chi\|_{\infty}N^4\left(\frac{q}{\delta}\right)^{2p}=o(1)
\quad\text{and}\quad
CN^4\exp\left(-\frac{\delta^2N^{1/5}}{2}\right)=o(1), ∥ χ ∥ ∞ N 4 ( δ q ) 2 p = o ( 1 ) and C N 4 exp ( − 2 δ 2 N 1/5 ) = o ( 1 ) , respectively. For the first bound, we used that the rational factor of the two distant points is at most ( q / δ ) 2 p (q/\delta)^{2p} ( q / δ ) 2 p Lemma 5.1 . For the remaining tuples, all points lie in a bulk interval I ~ \widetilde I I
We now bound the remaining tuples. Using (A.12), the second bound in (9.13), and the substitution x i = x 1 + q a i x_i=x_1+qa_i x i = x 1 + q a i i ≥ 2 i\ge2 i ≥ 2
E U q ( 3 ) ( G ) ≤ C N 9 q 8 ∫ R 2 ( a 2 a 3 ( a 3 − a 2 ) ) 2 f p ( a 2 ) f p ( a 3 ) f p ( a 3 − a 2 ) d a 2 d a 3 + o ( 1 ) ≤ C N ( N q ) 8 + o ( 1 ) . \begin{aligned}
\mathbb{E}U_q^{(3)}(G)\le CN^9q^8\int_{\mathbb{R}^2}(a_2a_3(a_3-a_2))^2f_p(a_2)f_p(a_3)f_p(a_3-a_2)\,\mathrm{d}a_2\mathrm{d}a_3+o(1) \\
&\le CN(Nq)^8+o(1).
\end{aligned} E U q ( 3 ) ( G ) ≤ C N 9 q 8 ∫ R 2 ( a 2 a 3 ( a 3 − a 2 ) ) 2 f p ( a 2 ) f p ( a 3 ) f p ( a 3 − a 2 ) d a 2 d a 3 + o ( 1 ) ≤ C N ( N q ) 8 + o ( 1 ) . where the integral is finite since p ≥ 3 p\ge3 p ≥ 3
( a 2 a 3 ( a 3 − a 2 ) ) 2 ≤ 2 ( a 2 4 a 3 2 + a 2 2 a 3 4 ) . (a_2a_3(a_3-a_2))^2\le2(a_2^4a_3^2+a_2^2a_3^4). ( a 2 a 3 ( a 3 − a 2 ) ) 2 ≤ 2 ( a 2 4 a 3 2 + a 2 2 a 3 4 ) . Similarly, with a 1 = 0 a_1=0 a 1 = 0
E U q ( 4 ) ( G ) ≤ C N 16 q 15 ∫ R 3 ∏ 1 ≤ i < j ≤ 4 ( a i − a j ) 2 f p ( a i − a j ) d a 2 d a 3 d a 4 + o ( 1 ) . \mathbb{E}U_q^{(4)}(G)\le CN^{16}q^{15}\int_{\mathbb{R}^3}\prod_{1\le i<j\le4}(a_i-a_j)^2f_p(a_i-a_j)\,\mathrm{d}a_2\mathrm{d}a_3\mathrm{d}a_4+o(1). E U q ( 4 ) ( G ) ≤ C N 16 q 15 ∫ R 3 1 ≤ i < j ≤ 4 ∏ ( a i − a j ) 2 f p ( a i − a j ) d a 2 d a 3 d a 4 + o ( 1 ) . To bound the integral, we bound the three factors with 2 ≤ i < j ≤ 4 2\le i<j\le4 2 ≤ i < j ≤ 4 y 2 f p ( y ) ≤ 1 y^2f_p(y)\le1 y 2 f p ( y ) ≤ 1 y ∈ R y\in\mathbb{R} y ∈ R a j 2 f p ( a j ) a_j^2f_p(a_j) a j 2 f p ( a j ) j ∈ { 2 , 3 , 4 } j\in\{2,3,4\} j ∈ { 2 , 3 , 4 } ( ∫ R a 2 f p ( a ) d a ) 3 \left(\int_{\mathbb{R}}a^2f_p(a)\,\mathrm{d}a\right)^3 ( ∫ R a 2 f p ( a ) d a ) 3 2 p − 2 > 1 2p-2>1 2 p − 2 > 1 N 16 q 15 = N ( N q ) 15 N^{16}q^{15}=N(Nq)^{15} N 16 q 15 = N ( N q ) 15 U q ( 4 ) U_q^{(4)} U q ( 4 )
Triples of nearby eigenvalues
Proposition 9.2. Let I I I κ ∈ ( 0 , 1 / 8 ) \kappa\in(0,1/8) κ ∈ ( 0 , 1/8 ) q = N − 1 − κ q=N^{-1-\kappa} q = N − 1 − κ ε > 0 \varepsilon>0 ε > 0
lim N → ∞ P ( # { i ∈ [ N − 2 ] : λ i ∈ I , λ i + 2 − λ i ≤ 4 q } > N 1 − 8 κ + ε ) = 0. (402) \lim_{N\to\infty}\mathbb{P}\left(\#\{i\in[N-2]:\lambda_i\in I,\ \lambda_{i+2}-\lambda_i\le4q\}>N^{1-8\kappa+\varepsilon}\right)=0. \tag*{(402)} N → ∞ lim P ( # { i ∈ [ N − 2 ] : λ i ∈ I , λ i + 2 − λ i ≤ 4 q } > N 1 − 8 κ + ε ) = 0. (402) If moreover κ > 1 / 15 \kappa>1/15 κ > 1/15 i ∈ [ N − 3 ] i\in[N-3] i ∈ [ N − 3 ] λ i ∈ I \lambda_i\in I λ i ∈ I λ i + 3 − λ i ≤ 6 q \lambda_{i+3}-\lambda_i\le6q λ i + 3 − λ i ≤ 6 q
Proof. We first reduce both claims to bounds on the statistics U q ( 3 ) U_q^{(3)} U q ( 3 ) U q ( 4 ) U_q^{(4)} U q ( 4 ) (390) . Let I + I_+ I + I I I χ ∈ C c ∞ ( ( − 2 , 2 ) ) \chi\in C^\infty_c((-2,2)) χ ∈ C c ∞ (( − 2 , 2 )) χ ≥ 2 \chi\ge2 χ ≥ 2 I + I_+ I + χ N ≥ 1 \chi_N \ge1 χ N ≥ 1 I + I_+ I + N ≥ N 0 N \ge N_0 N ≥ N 0 Lemma 5.1 , and fix an integer p ≥ 4 p \ge4 p ≥ 4
Let i i i λ i ∈ I \lambda_i \in I λ i ∈ I λ i + 2 − λ i ≤ 4 q \lambda_{i+2}-\lambda_i \le4q λ i + 2 − λ i ≤ 4 q λ i \lambda_i λ i λ i + 1 \lambda_{i+1} λ i + 1 λ i + 2 \lambda_{i+2} λ i + 2 I + I_+ I + N ≥ N 0 N \ge N_0 N ≥ N 0 ( i , i + 1 , i + 2 ) (i,i+1,i+2) ( i , i + 1 , i + 2 ) U q ( 3 ) ( H ) U_q^{(3)}(H) U q ( 3 ) ( H ) 17 − 3 p 17^{-3p} 1 7 − 3 p (390) . Distinct labels i i i U q ( 3 ) ( H ) U_q^{(3)}(H) U q ( 3 ) ( H )
# { i ∈ [ N − 2 ] : λ i ∈ I , λ i + 2 − λ i ≤ 4 q } ≤ 17 3 p U q ( 3 ) ( H ) . \#\{i \in[N-2]: \lambda_i \in I,\ \lambda_{i+2}-\lambda_i \le4q\} \le17^{3p}U_q^{(3)}(H). # { i ∈ [ N − 2 ] : λ i ∈ I , λ i + 2 − λ i ≤ 4 q } ≤ 1 7 3 p U q ( 3 ) ( H ) . Similarly, a label i i i λ i ∈ I \lambda_i \in I λ i ∈ I λ i + 3 − λ i ≤ 6 q \lambda_{i+3}-\lambda_i \le6q λ i + 3 − λ i ≤ 6 q U q ( 4 ) ( H ) ≥ 37 − 6 p U_q^{(4)}(H) \ge37^{-6p} U q ( 4 ) ( H ) ≥ 3 7 − 6 p
It therefore suffices to show that
U q ( 3 ) ( H ) N 1 − 8 κ + ε → 0 , U q ( 4 ) ( H ) → 0 ( κ > 1 / 15 ) , (403) \frac{U_q^{(3)}(H)}{N^{1-8\kappa+\varepsilon}} \to0,\qquad U_q^{(4)}(H) \to0\quad(\kappa> 1/15), \tag*{(403)} N 1 − 8 κ + ε U q ( 3 ) ( H ) → 0 , U q ( 4 ) ( H ) → 0 ( κ > 1/15 ) , (403) in probability. We prove (403) for H T H_T H T T T T (395) .
We now choose the time. Let ε ′ ∈ ( 0 , ε / 10 ) \varepsilon' \in(0,\varepsilon/10) ε ′ ∈ ( 0 , ε /10 )
T = N − 1 + κ + 4 ε ′ , 2 N ε ′ N 2 T = 2 N − 3 ε ′ q , T N 1 / 2 + 3 κ + 12 ε ′ = N − 1 / 2 + 4 κ + 16 ε ′ → 0. (404) T=N^{-1+\kappa+4\varepsilon'},\qquad\frac{2N^{\varepsilon'}}{N^2T}=2N^{-3\varepsilon'}q,\qquad TN^{1/2+3\kappa+12\varepsilon'}=N^{-1/2+4\kappa+16\varepsilon'}\to0. \tag*{(404)} T = N − 1 + κ + 4 ε ′ , N 2 T 2 N ε ′ = 2 N − 3 ε ′ q , T N 1/2 + 3 κ + 12 ε ′ = N − 1/2 + 4 κ + 16 ε ′ → 0. (404) Then T ≥ N − 1 + ε ′ T \ge N^{-1+\varepsilon'} T ≥ N − 1 + ε ′ ε ′ \varepsilon' ε ′ κ < 1 / 8 \kappa< 1/8 κ < 1/8 (404) is the coupling error at time T T T (395) .
Next, we compare H T H_T H T δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) supp χ ⊂ [ − 2 + 3 δ , 2 − 3 δ ] \operatorname{supp}\chi\subset[-2+3\delta,2-3\delta] supp χ ⊂ [ − 2 + 3 δ , 2 − 3 δ ] c ∗ c_* c ∗ J \mathcal{J} J Lemma 6.4 , defined with this δ \delta δ Proposition 3.5 at time T T T § 11.3 and applied with ε ′ \varepsilon' ε ′ ε \varepsilon ε G ′ G' G ′
max c ∗ N ≤ i ≤ ( 1 − c ∗ ) N ∣ ( λ i + 1 ( H T ) − λ i ( H T ) ) − ( λ i + 1 ( G ′ ) − λ i ( G ′ ) ) ∣ ≤ 2 N − 3 ε ′ q . \max_{c_*N\le i\le(1-c_*)N}\left|(\lambda_{i+1}(H_T)-\lambda_i(H_T))-(\lambda_{i+1}(G')-\lambda_i(G'))\right|\le2N^{-3\varepsilon'}q. c ∗ N ≤ i ≤ ( 1 − c ∗ ) N max ∣ ( λ i + 1 ( H T ) − λ i ( H T )) − ( λ i + 1 ( G ′ ) − λ i ( G ′ )) ∣ ≤ 2 N − 3 ε ′ q . We intersect this event with the events of Proposition 3.2 , for H T H_T H T G ′ G' G ′ ε ′ \varepsilon' ε ′ ε \varepsilon ε
U q ( r ) ( H T ) ≤ ( 1 + o ( 1 ) ) U q ( r ) ( G ′ ) + o ( 1 ) , r ∈ { 3 , 4 } . (405) U_q^{(r)}(H_T)\le(1+o(1))U_q^{(r)}(G')+o(1),\qquad r\in\{3,4\}. \tag*{(405)} U q ( r ) ( H T ) ≤ ( 1 + o ( 1 )) U q ( r ) ( G ′ ) + o ( 1 ) , r ∈ { 3 , 4 } . (405) To prove (405) , we follow the proof of Lemma 6.4 , applied to each of the three, respectively six, rational factors of the summands and to the weight χ N \chi_N χ N x i = λ i ( H T ) x_i=\lambda_i(H_T) x i = λ i ( H T ) x i ′ = λ i ( G ′ ) x'_i=\lambda_i(G') x i ′ = λ i ( G ′ ) J \mathcal{J} J i < j i<j i < j J \mathcal{J} J N ε ′ ( 1 + N ( x j − x i ) ) N^{\varepsilon'}(1+N(x_j-x_i)) N ε ′ ( 1 + N ( x j − x i )) x i x_i x i x j x_j x j
∣ ( x j ′ − x i ′ ) − ( x j − x i ) ∣ ≤ N ε ′ ( 1 + N ( x j − x i ) ) ⋅ 2 N − 3 ε ′ q ≤ 2 N − 2 ε ′ ( q + x j − x i ) , \left|(x'_j-x'_i)-(x_j-x_i)\right|\le N^{\varepsilon'}(1+N(x_j-x_i))\cdot2N^{-3\varepsilon'}q\le2N^{-2\varepsilon'}(q+x_j-x_i), ( x j ′ − x i ′ ) − ( x j − x i ) ≤ N ε ′ ( 1 + N ( x j − x i )) ⋅ 2 N − 3 ε ′ q ≤ 2 N − 2 ε ′ ( q + x j − x i ) , where we used N q ≤ 1 Nq\le1 N q ≤ 1
We now compare the terms. Let y = ( x j − x i ) / q y=(x_j-x_i)/q y = ( x j − x i ) / q y ′ = ( x j ′ − x i ′ ) / q y'=(x'_j-x'_i)/q y ′ = ( x j ′ − x i ′ ) / q ∣ y − y ′ ∣ ≤ 2 N − 2 ε ′ ( 1 + y ) |y-y'|\le2N^{-2\varepsilon'}(1+y) ∣ y − y ′ ∣ ≤ 2 N − 2 ε ′ ( 1 + y )
∣ log f p ( y ) − log f p ( y ′ ) ∣ ≤ 8 p ∣ y − y ′ ∣ 1 + y ≤ 16 p N − 2 ε ′ |\log f_p(y)-\log f_p(y')|\le\frac{8p|y-y'|}{1+y}\le16pN^{-2\varepsilon'} ∣ log f p ( y ) − log f p ( y ′ ) ∣ ≤ 1 + y 8 p ∣ y − y ′ ∣ ≤ 16 p N − 2 ε ′ for N ≥ N 0 N\ge N_0 N ≥ N 0
We next compare the weights. By Proposition 3.2, ∣ x i − γ i ∣ |x_i-\gamma_i| ∣ x i − γ i ∣ ∣ x i ′ − γ i ∣ |x'_i-\gamma_i| ∣ x i ′ − γ i ∣ N − 1 + ε ′ N^{-1+\varepsilon'} N − 1 + ε ′ i ∈ J i\in\mathcal{J} i ∈ J x ˉ \bar{x} x ˉ x ˉ ′ \bar{x}' x ˉ ′ x x x x ′ x' x ′ ∣ x ˉ − x ˉ ′ ∣ ≤ 2 N − 1 + ε ′ |\bar{x}-\bar{x}'|\le2N^{-1+\varepsilon'} ∣ x ˉ − x ˉ ′ ∣ ≤ 2 N − 1 + ε ′ h N = N − 1 / 10 h_N=N^{-1/10} h N = N − 1/10 h N − 1 = N 1 / 5 h_N^{-1}=N^{1/5} h N − 1 = N 1/5
e − 16 N − 4 / 5 + ε ′ χ N ( x ˉ ′ ) ≤ χ N ( x ˉ ) ≤ e 16 N − 4 / 5 + ε ′ χ N ( x ˉ ′ ) . e^{-16N^{-4/5+\varepsilon'}}\chi_N(\bar{x}')\le\chi_N(\bar{x})\le e^{16N^{-4/5+\varepsilon'}}\chi_N(\bar{x}'). e − 16 N − 4/5 + ε ′ χ N ( x ˉ ′ ) ≤ χ N ( x ˉ ) ≤ e 16 N − 4/5 + ε ′ χ N ( x ˉ ′ ) . A summand has at most six rational factors, so the term of every such tuple for x x x e 96 p N − 2 ε ′ + 16 N − 4 / 5 + ε ′ e^{96pN^{-2\varepsilon'}+16N^{-4/5+\varepsilon'}} e 96 p N − 2 ε ′ + 16 N − 4/5 + ε ′ x ′ x' x ′ H T H_T H T ( 1 + o ( 1 ) ) U q ( r ) ( G ′ ) (1+o(1))U_q^{(r)}(G') ( 1 + o ( 1 )) U q ( r ) ( G ′ )
We now consider the tuples with a label outside J \mathcal{J} J x x x x ′ x' x ′ δ \delta δ supp χ \operatorname{supp}\chi supp χ [ − 2 + 2 δ , 2 − 2 δ ] [-2+2\delta,2-2\delta] [ − 2 + 2 δ , 2 − 2 δ ] c ∗ N c_*N c ∗ N − 2 + δ -2+\delta − 2 + δ δ \delta δ ( 1 − c ∗ ) N (1-c_*)N ( 1 − c ∗ ) N δ \delta δ ∥ χ ∥ ∞ ( q / δ ) 2 p \|\chi\|_\infty(q/\delta)^{2p} ∥ χ ∥ ∞ ( q / δ ) 2 p
If instead the mean of such a tuple is at distance more than δ \delta δ supp χ \operatorname{supp}\chi supp χ C exp ( − δ 2 N 1 / 5 / 2 ) C\exp(-\delta^2N^{1/5}/2) C exp ( − δ 2 N 1/5 /2 ) N 4 N^4 N 4
N 4 ( ∥ χ ∥ ∞ ( q δ ) 2 p + C exp ( − δ 2 N 1 / 5 2 ) ) = o ( 1 ) , N^4\left(\|\chi\|_\infty\left(\frac{q}{\delta}\right)^{2p}+C\exp\left(-\frac{\delta^2N^{1/5}}{2}\right)\right)=o(1), N 4 ( ∥ χ ∥ ∞ ( δ q ) 2 p + C exp ( − 2 δ 2 N 1/5 ) ) = o ( 1 ) , since p ≥ 4 p\ge4 p ≥ 4
We now use the estimates for the GUE. By (9.9) and Markov’s inequality, we have
U q ( 3 ) ( G ′ ) N 1 − 8 κ + ε → 0 , U q ( 4 ) ( G ′ ) → 0 ( κ > 1 / 15 ) , \frac{U_q^{(3)}(G')}{N^{1-8\kappa+\varepsilon}}\to0,\qquad U_q^{(4)}(G')\to0\qquad(\kappa>1/15), N 1 − 8 κ + ε U q ( 3 ) ( G ′ ) → 0 , U q ( 4 ) ( G ′ ) → 0 ( κ > 1/15 ) , in probability, where for the second limit we used N ( N q ) 15 = N 1 − 15 κ N(Nq)^{15}=N^{1-15\kappa} N ( N q ) 15 = N 1 − 15 κ H T H_T H T H H H θ > 0 \theta>0 θ > 0
lim N → ∞ E e − θ U q ( 3 ) ( H T ) / N 1 − 8 κ + ε = 1 , lim N → ∞ E e − θ U q ( 4 ) ( H T ) = 1 ( κ > 1 / 15 ) . (406) \lim_{N\to\infty}\mathbb{E}e^{-\theta U_q^{(3)}(H_T)/N^{1-8\kappa+\varepsilon}}=1,\qquad
\lim_{N\to\infty}\mathbb{E}e^{-\theta U_q^{(4)}(H_T)}=1\qquad(\kappa>1/15). \tag*{(406)} N → ∞ lim E e − θ U q ( 3 ) ( H T ) / N 1 − 8 κ + ε = 1 , N → ∞ lim E e − θ U q ( 4 ) ( H T ) = 1 ( κ > 1/15 ) . (406) Finally, we remove the Gaussian component. By (395) with t = T t = T t = T ε ′ \varepsilon' ε ′ ε \varepsilon ε (404) , the same limits hold with H H H H T H_T H T U q ( r ) ( H ) ≥ 0 U_q^{(r)}(H) \ge0 U q ( r ) ( H ) ≥ 0 (403) . This completes the proof.
Gaussian perturbations of 3 × 3 3 \times3 3 × 3
The following lemma controls the gaps produced by a cluster of three eigenvalues under a Gaussian perturbation. We state it for a general 3 × 3 3 \times3 3 × 3 σ > 0 \sigma> 0 σ > 0 W W W 3 × 3 3 \times3 3 × 3 3 × 3 3 \times3 3 × 3 B B B exp ( − Tr B 2 / ( 2 σ 2 ) ) \exp\left(-\operatorname{Tr} B^2/(2\sigma^2)\right) exp ( − Tr B 2 / ( 2 σ 2 ) ) W a b W_{ab} W ab a ≤ b a \le b a ≤ b σ 2 \sigma^2 σ 2 E W a b = E W a b 2 = 0 \mathbb{E}W_{ab} = \mathbb{E}W_{ab}^2 = 0 E W ab = E W ab 2 = 0 E ∣ W a b ∣ 2 = σ 2 \mathbb{E}|W_{ab}|^2 = \sigma^2 E ∣ W ab ∣ 2 = σ 2 3 × 3 3 \times3 3 × 3 B B B
gap ( B ) = min { λ 2 ( B ) − λ 1 ( B ) , λ 3 ( B ) − λ 2 ( B ) } \operatorname{gap}(B) = \min\{\lambda_2(B) - \lambda_1(B), \lambda_3(B) - \lambda_2(B)\} gap ( B ) = min { λ 2 ( B ) − λ 1 ( B ) , λ 3 ( B ) − λ 2 ( B )} denote its smallest eigenvalue gap.
Lemma 9.3. There exists a constant C > 0 C > 0 C > 0 Let y 1 ≤ y 2 ≤ y 3 y_1 \le y_2 \le y_3 y 1 ≤ y 2 ≤ y 3 D y = diag ( y 1 , y 2 , y 3 ) D_y = \operatorname{diag}(y_1,y_2,y_3) D y = diag ( y 1 , y 2 , y 3 ) W W W δ ∈ ( 0 , σ ] \delta\in(0,\sigma] δ ∈ ( 0 , σ ] R ≥ 1 R \ge1 R ≥ 1
P ( gap ( D y + W ) ≤ δ ) ≤ C R 4 ( δ σ ) 3 + C e − R 2 / C . (407) \mathbb{P}\left(\operatorname{gap}(D_y + W) \le\delta\right) \le C R^4 \left(\frac{\delta}{\sigma}\right)^3 + C e^{-R^2/C}. \tag*{(407)} P ( gap ( D y + W ) ≤ δ ) ≤ C R 4 ( σ δ ) 3 + C e − R 2 / C . (407) Proof. Replacing D y D_y D y W W W δ \delta δ D y / σ D_y/\sigma D y / σ W / σ W/\sigma W / σ δ / σ \delta/\sigma δ / σ σ = 1 \sigma= 1 σ = 1 δ ≤ 1 ≤ R \delta\le1 \le R δ ≤ 1 ≤ R W W W
P ( ∥ W ∥ > R ) ≤ C e − R 2 / C . (408) \mathbb{P}(\lVert W\rVert> R) \le C e^{-R^2/C}. \tag*{(408)} P (∥ W ∥ > R ) ≤ C e − R 2 / C . (408) We distinguish three cases according to the configuration of y 1 , y 2 , y 3 y_1,y_2,y_3 y 1 , y 2 , y 3
Case 1: y 2 − y 1 ≥ 10 R y_2 - y_1 \ge10R y 2 − y 1 ≥ 10 R y 3 − y 2 ≥ 10 R y_3 - y_2 \ge10R y 3 − y 2 ≥ 10 R { ∥ W ∥ ≤ R } \{\lVert W\rVert\le R\} {∥ W ∥ ≤ R } D y + W D_y + W D y + W y j y_j y j R R R
λ j + 1 ( D y + W ) − λ j ( D y + W ) ≥ y j + 1 − y j − 2 R ≥ 8 R > δ , j ∈ { 1 , 2 } . \lambda_{j+1}(D_y + W) - \lambda_j(D_y + W) \ge y_{j+1} - y_j - 2R \ge8R > \delta,\qquad j \in\{1,2\}. λ j + 1 ( D y + W ) − λ j ( D y + W ) ≥ y j + 1 − y j − 2 R ≥ 8 R > δ , j ∈ { 1 , 2 } . Hence the probability in (407) is bounded by C e − R 2 / C C e^{-R^2/C} C e − R 2 / C (408) .
Case 2: y 3 − y 1 ≤ 30 R y_3 - y_1 \le30R y 3 − y 1 ≤ 30 R [2 Section 4.1] ). The density of D y + W D_y + W D y + W B B B
exp ( − ∥ B − D y ∥ F 2 2 ) , \exp\left(-\frac{\lVert B - D_y\rVert_F^2}{2}\right), exp ( − 2 ∥ B − D y ∥ F 2 ) , where ∥ ⋅ ∥ F \lVert\cdot\rVert_F ∥ ⋅ ∥ F B = O diag ( μ ) O ∗ B = O \operatorname{diag}(\mu)O^{*} B = O diag ( μ ) O ∗ μ 1 ≤ μ 2 ≤ μ 3 \mu_1 \le\mu_2 \le\mu_3 μ 1 ≤ μ 2 ≤ μ 3 O O O D y + W D_y + W D y + W
μ ↦ c ∏ 1 ≤ i < j ≤ 3 ( μ j − μ i ) 2 ∫ U ( 3 ) exp ( − ∥ O diag ( μ ) O ∗ − D y ∥ F 2 2 ) d O (409) \mu\mapsto c \prod_{1 \le i < j \le3}(\mu_j-\mu_i)^2 \int_{U(3)} \exp\left(-\frac{\lVert O\operatorname{diag}(\mu)O^{*}-D_y\rVert_F^2}{2}\right)\,dO \tag*{(409)} μ ↦ c 1 ≤ i < j ≤ 3 ∏ ( μ j − μ i ) 2 ∫ U ( 3 ) exp ( − 2 ∥ O diag ( μ ) O ∗ − D y ∥ F 2 ) d O (409) on { μ 1 ≤ μ 2 ≤ μ 3 } \{\mu_1 \le\mu_2 \le\mu_3\} { μ 1 ≤ μ 2 ≤ μ 3 } d O dO d O U ( 3 ) U(3) U ( 3 ) c > 0 c > 0 c > 0
By the Hoffman–Wielandt inequality for Hermitian matrices [40 ] , we have
∥ O diag ( μ ) O ∗ − D y ∥ F ≥ ∣ μ − y ∣ , y = ( y 1 , y 2 , y 3 ) . \lVert O\operatorname{diag}(\mu)O^{*}-D_y\rVert_F \ge\lvert\mu-y\rvert,\qquad y=(y_1,y_2,y_3). ∥ O diag ( μ ) O ∗ − D y ∥ F ≥ ∣ μ − y ∣ , y = ( y 1 , y 2 , y 3 ) . Hence the density (409) at μ \mu μ
c ∏ 1 ≤ i < j ≤ 3 ( μ j − μ i ) 2 e − ∣ μ − y ∣ 2 / 2 . (410) c\prod_{1 \le i < j \le3}(\mu_j-\mu_i)^2 e^{-\lvert\mu-y\rvert^2/2}. \tag*{(410)} c 1 ≤ i < j ≤ 3 ∏ ( μ j − μ i ) 2 e − ∣ μ − y ∣ 2 /2 . (410) We now bound the probability that μ 2 − μ 1 ≤ δ \mu_2-\mu_1 \le\delta μ 2 − μ 1 ≤ δ
∏ 1 ≤ i < j ≤ 3 ( μ j − μ i ) 2 ≤ δ 2 ( μ 3 − μ 1 ) 4 , μ 3 − μ 1 ≤ ∣ μ 1 − y 1 ∣ + ∣ μ 3 − y 3 ∣ + 30 R , \prod_{1 \le i < j \le3}(\mu_j-\mu_i)^2 \le\delta^2(\mu_3-\mu_1)^4,\qquad\mu_3-\mu_1 \le\lvert\mu_1-y_1\rvert+\lvert\mu_3-y_3\rvert+30R, 1 ≤ i < j ≤ 3 ∏ ( μ j − μ i ) 2 ≤ δ 2 ( μ 3 − μ 1 ) 4 , μ 3 − μ 1 ≤ ∣ μ 1 − y 1 ∣ + ∣ μ 3 − y 3 ∣ + 30 R , e − ∣ μ − y ∣ 2 / 2 ≤ e − ( ( μ 1 − y 1 ) 2 + ( μ 3 − y 3 ) 2 ) / 2 , e^{-\lvert\mu-y\rvert^2/2}\le e^{-((\mu_1-y_1)^2+(\mu_3-y_3)^2)/2}, e − ∣ μ − y ∣ 2 /2 ≤ e − (( μ 1 − y 1 ) 2 + ( μ 3 − y 3 ) 2 ) /2 , where the second bound uses y 3 − y 1 ≤ 30 R y_3-y_1 \le30R y 3 − y 1 ≤ 30 R μ 2 ∈ [ μ 1 , μ 1 + δ ] \mu_2 \in[\mu_1,\mu_1+\delta] μ 2 ∈ [ μ 1 , μ 1 + δ ] μ 1 \mu_1 μ 1 μ 3 \mu_3 μ 3
P ( μ 2 − μ 1 ≤ δ ) ≤ c δ 3 ∫ R 2 ( ∣ μ 1 − y 1 ∣ + ∣ μ 3 − y 3 ∣ + 30 R ) 4 e − ( ( μ 1 − y 1 ) 2 + ( μ 3 − y 3 ) 2 ) / 2 d μ 1 d μ 3 ≤ C R 4 δ 3 . (411) \begin{aligned}
\mathbb{P}(\mu_2-\mu_1\le\delta)\le c\delta^3\int_{\mathbb{R}^2}(\lvert\mu_1-y_1\rvert+\lvert\mu_3-y_3\rvert+30R)^4e^{-((\mu_1-y_1)^2+(\mu_3-y_3)^2)/2}\,d\mu_1\,d\mu_3 \\
&\le CR^4\delta^3. \tag*{(411)}
\end{aligned} P ( μ 2 − μ 1 ≤ δ ) ≤ c δ 3 ∫ R 2 (∣ μ 1 − y 1 ∣ + ∣ μ 3 − y 3 ∣ + 30 R ) 4 e − (( μ 1 − y 1 ) 2 + ( μ 3 − y 3 ) 2 ) /2 d μ 1 d μ 3 ≤ C R 4 δ 3 . (411) The same bound holds for P ( μ 3 − μ 2 ≤ δ ) \mathbb{P}(\mu_3-\mu_2\le\delta) P ( μ 3 − μ 2 ≤ δ ) (407) is bounded by C R 4 δ 3 CR^4\delta^3 C R 4 δ 3
Case 3: neither Case 1 nor Case 2 holds. Then y 3 − y 1 > 30 R y_3-y_1>30R y 3 − y 1 > 30 R y 2 − y 1 y_2-y_1 y 2 − y 1 y 3 − y 2 y_3-y_2 y 3 − y 2 10 R 10R 10 R 20 R 20R 20 R P P P
− P ( D y + W ) P T = diag ( − y 3 , − y 2 , − y 1 ) − P W P T -P(D_y+W)P^{T}=\operatorname{diag}(-y_3,-y_2,-y_1)-PWP^{T} − P ( D y + W ) P T = diag ( − y 3 , − y 2 , − y 1 ) − P W P T has the gaps of D y + W D_y+W D y + W − P W P T -PWP^{T} − P W P T W W W ( y 1 , y 2 , y 3 ) (y_1,y_2,y_3) ( y 1 , y 2 , y 3 ) ( − y 3 , − y 2 , − y 1 ) (-y_3,-y_2,-y_1) ( − y 3 , − y 2 , − y 1 ) y 2 − y 1 < 10 R y_2-y_1<10R y 2 − y 1 < 10 R y 3 − y 2 > 20 R y_3-y_2>20R y 3 − y 2 > 20 R
We work on the event ∥ W ∥ ≤ R \lVert W\rVert\le R ∥ W ∥ ≤ R (408) , and we set μ j = λ j ( D y + W ) \mu_j=\lambda_j(D_y+W) μ j = λ j ( D y + W )
μ 3 ≥ y 3 − R ≥ y 1 + 19 R , μ 1 , μ 2 ∈ [ y 1 − R , y 2 + R ] ⊂ [ y 1 − R , y 1 + 11 R ] , \mu_3 \ge y_3-R \ge y_1+19R,\qquad\mu_1,\mu_2\in[y_1-R,y_2+R]\subset[y_1-R,y_1+11R], μ 3 ≥ y 3 − R ≥ y 1 + 19 R , μ 1 , μ 2 ∈ [ y 1 − R , y 2 + R ] ⊂ [ y 1 − R , y 1 + 11 R ] , μ 3 − μ 2 ≥ 19 R − 11 R = 8 R > δ . \mu_3-\mu_2\ge19R-11R=8R>\delta. μ 3 − μ 2 ≥ 19 R − 11 R = 8 R > δ . Thus the only possible small gap is μ 2 − μ 1 \mu_2-\mu_1 μ 2 − μ 1 μ 1 \mu_1 μ 1 μ 2 \mu_2 μ 2 D y + W D_y+W D y + W
I R = [ y 1 − 12 R , y 1 + 13 R ] . I_R=[y_1-12R,y_1+13R]. I R = [ y 1 − 12 R , y 1 + 13 R ] . We reduce to a 2 × 2 2\times2 2 × 2 W [ 2 ] W_{[2]} W [ 2 ] 2 × 2 2\times2 2 × 2 W W W b = ( W 13 , W 23 ) T b=(W_{13},W_{23})^{T} b = ( W 13 , W 23 ) T w w w y 3 + W 33 − w ≠ 0 y_3+W_{33}-w\ne0 y 3 + W 33 − w = 0
A ( w ) = diag ( y 1 , y 2 ) + W [ 2 ] − b b ∗ y 3 + W 33 − w . (412) A(w)=\operatorname{diag}(y_1,y_2)+W_{[2]}-\frac{bb^*}{y_3+W_{33}-w}. \tag*{(412)} A ( w ) = diag ( y 1 , y 2 ) + W [ 2 ] − y 3 + W 33 − w b b ∗ . (412) Let w ∈ I R w\in I_R w ∈ I R ∣ W 33 ∣ ≤ R \lvert W_{33}\rvert\le R ∣ W 33 ∣ ≤ R ∣ b ∣ ≤ R \lvert b\rvert\le R ∣ b ∣ ≤ R ∥ W ∥ ≤ R \lVert W\rVert\le R ∥ W ∥ ≤ R y 3 − y 1 > 20 R y_3-y_1>20R y 3 − y 1 > 20 R
y 3 + W 33 − w ≥ 20 R − R − 13 R = 6 R , y_3+W_{33}-w\ge20R-R-13R=6R, y 3 + W 33 − w ≥ 20 R − R − 13 R = 6 R , ∥ b b ∗ y 3 + W 33 − w ∥ ≤ R 6 , ∥ d d w b b ∗ y 3 + W 33 − w ∥ ≤ R 2 ( 6 R ) 2 = 1 36 , \left\lVert\frac{bb^*}{y_3+W_{33}-w}\right\rVert\le\frac{R}{6},\qquad
\left\lVert\frac{\mathrm{d}}{\mathrm{d}w}\frac{bb^*}{y_3+W_{33}-w}\right\rVert\le\frac{R^2}{(6R)^2}=\frac{1}{36}, y 3 + W 33 − w b b ∗ ≤ 6 R , d w d y 3 + W 33 − w b b ∗ ≤ ( 6 R ) 2 R 2 = 36 1 , and the last term in (412) is negative semidefinite.
We now locate the eigenvalues of D y + W D_y+W D y + W I R I_R I R λ 1 ( A ( w ) ) ≤ λ 2 ( A ( w ) ) \lambda_1(A(w))\le\lambda_2(A(w)) λ 1 ( A ( w )) ≤ λ 2 ( A ( w )) A ( w ) A(w) A ( w ) w ∈ I R w\in I_R w ∈ I R 1 / 36 1/36 1/36 w ↦ λ k ( A ( w ) ) − w w\mapsto\lambda_k(A(w))-w w ↦ λ k ( A ( w )) − w I R I_R I R ∥ W [ 2 ] ∥ ≤ R \lVert W_{[2]}\rVert\le R ∥ W [ 2 ] ∥ ≤ R y 2 < y 1 + 10 R y_2<y_1+10R y 2 < y 1 + 10 R
λ k ( A ( w ) ) ∈ [ y 1 − 7 R 6 , y 2 + R ] ⊂ [ y 1 − 7 R 6 , y 1 + 11 R ] , w ∈ I R , \lambda_k(A(w))\in\left[y_1-\frac{7R}{6},y_2+R\right]\subset\left[y_1-\frac{7R}{6},y_1+11R\right],\qquad w\in I_R, λ k ( A ( w )) ∈ [ y 1 − 6 7 R , y 2 + R ] ⊂ [ y 1 − 6 7 R , y 1 + 11 R ] , w ∈ I R , so the functions w ↦ λ k ( A ( w ) ) − w w\mapsto\lambda_k(A(w))-w w ↦ λ k ( A ( w )) − w w = y 1 − 12 R w=y_1-12R w = y 1 − 12 R w = y 1 + 12 R w=y_1+12R w = y 1 + 12 R w k w_k w k I R I_R I R w 1 ≤ w 2 w_1\le w_2 w 1 ≤ w 2 λ 1 ( A ( w ) ) ≤ λ 2 ( A ( w ) ) \lambda_1(A(w))\le\lambda_2(A(w)) λ 1 ( A ( w )) ≤ λ 2 ( A ( w ))
We now identify μ 1 \mu_1 μ 1 μ 2 \mu_2 μ 2 w ∈ I R w\in I_R w ∈ I R D y + W D_y+W D y + W det ( A ( w ) − w ) = 0 \det(A(w)-w)=0 det ( A ( w ) − w ) = 0 w ∈ { w 1 , w 2 } w\in\{w_1,w_2\} w ∈ { w 1 , w 2 } A ( w ) − w A(w)-w A ( w ) − w w 1 < w 2 w_1<w_2 w 1 < w 2 D y + W D_y+W D y + W I R I_R I R w 1 w_1 w 1 w 2 w_2 w 2 w 1 = w 2 w_1=w_2 w 1 = w 2 A ( w 1 ) = w 1 Id 2 A(w_1)=w_1\operatorname{Id}_2 A ( w 1 ) = w 1 Id 2 w 1 w_1 w 1 μ k = w k \mu_k=w_k μ k = w k k ∈ { 1 , 2 } k\in\{1,2\} k ∈ { 1 , 2 }
We now transfer a small gap of D y + W D_y+W D y + W A ( w ) A(w) A ( w ) λ k ( A ( μ k ) ) = μ k \lambda_k(A(\mu_k))=\mu_k λ k ( A ( μ k )) = μ k w ↦ λ 2 ( A ( w ) ) w\mapsto\lambda_2(A(w)) w ↦ λ 2 ( A ( w )) 1 / 36 1/36 1/36 μ 2 − μ 1 ≤ δ \mu_2-\mu_1\le\delta μ 2 − μ 1 ≤ δ
λ 2 ( A ( μ 1 ) ) − λ 1 ( A ( μ 1 ) ) = λ 2 ( A ( μ 1 ) ) − μ 1 ≤ λ 2 ( A ( μ 2 ) ) + μ 2 − μ 1 36 − μ 1 ≤ 2 δ . (413) \lambda_2(A(\mu_1))-\lambda_1(A(\mu_1))=\lambda_2(A(\mu_1))-\mu_1\le\lambda_2(A(\mu_2))+\frac{\mu_2-\mu_1}{36}-\mu_1\le2\delta. \tag*{(413)} λ 2 ( A ( μ 1 )) − λ 1 ( A ( μ 1 )) = λ 2 ( A ( μ 1 )) − μ 1 ≤ λ 2 ( A ( μ 2 )) + 36 μ 2 − μ 1 − μ 1 ≤ 2 δ . (413) We now discretize. We set
w j ′ = y 1 − 12 R + j δ , j ∈ Z , 0 ≤ j ≤ 24 R δ + 1 , w'_j = y_1 - 12R + j\delta,\qquad j \in\mathbb{Z},\quad0 \le j \le\frac{24R}{\delta} + 1, w j ′ = y 1 − 12 R + j δ , j ∈ Z , 0 ≤ j ≤ δ 24 R + 1 , and we note that these points lie in I R I_R I R δ ≤ R \delta\le R δ ≤ R μ 2 − μ 1 ≤ δ \mu_2-\mu_1 \le\delta μ 2 − μ 1 ≤ δ j j j ∣ w j ′ − μ 1 ∣ ≤ δ |w'_j-\mu_1| \le\delta ∣ w j ′ − μ 1 ∣ ≤ δ A A A (413) , together with Weyl’s inequality, give
∥ A ( w j ′ ) − A ( μ 1 ) ∥ ≤ δ 36 , \lVert A(w'_j)-A(\mu_1)\rVert\le\frac{\delta}{36}, ∥ A ( w j ′ ) − A ( μ 1 )∥ ≤ 36 δ , λ 2 ( A ( w j ′ ) ) − λ 1 ( A ( w j ′ ) ) ≤ 2 δ + δ 18 ≤ 3 δ , \lambda_2(A(w'_j))-\lambda_1(A(w'_j)) \le2\delta+\frac{\delta}{18}\le3\delta, λ 2 ( A ( w j ′ )) − λ 1 ( A ( w j ′ )) ≤ 2 δ + 18 δ ≤ 3 δ , ∣ λ 1 ( A ( w j ′ ) ) − w j ′ ∣ ≤ ∣ λ 1 ( A ( w j ′ ) ) − λ 1 ( A ( μ 1 ) ) ∣ + ∣ μ 1 − w j ′ ∣ ≤ δ 36 + δ ≤ 2 δ . |\lambda_1(A(w'_j))-w'_j| \le|\lambda_1(A(w'_j))-\lambda_1(A(\mu_1))|+|\mu_1-w'_j| \le\frac{\delta}{36}+\delta\le2\delta. ∣ λ 1 ( A ( w j ′ )) − w j ′ ∣ ≤ ∣ λ 1 ( A ( w j ′ )) − λ 1 ( A ( μ 1 )) ∣ + ∣ μ 1 − w j ′ ∣ ≤ 36 δ + δ ≤ 2 δ . Thus the matrix A ( w j ′ ) A(w'_j) A ( w j ′ ) 3 δ 3\delta 3 δ 2 δ 2\delta 2 δ w j ′ w'_j w j ′
We bound the probability of this event for fixed j j j W 33 W_{33} W 33 b b b { ∣ W 33 ∣ ≤ R , ∣ b ∣ ≤ R } \{|W_{33}|\le R, |b|\le R\} { ∣ W 33 ∣ ≤ R , ∣ b ∣ ≤ R } A ( w j ′ ) = M j + W [ 2 ] A(w'_j)=M_j+W_{[2]} A ( w j ′ ) = M j + W [ 2 ] M j M_j M j W [ 2 ] W_{[2]} W [ 2 ] ( W 33 , b ) (W_{33},b) ( W 33 , b )
v = ( ( M j ) 11 − ( M j ) 22 2 , Re ( M j ) 12 , Im ( M j ) 12 ) , v=\left(\frac{(M_j)_{11}-(M_j)_{22}}{2},\operatorname{Re}(M_j)_{12},\operatorname{Im}(M_j)_{12}\right), v = ( 2 ( M j ) 11 − ( M j ) 22 , Re ( M j ) 12 , Im ( M j ) 12 ) , ξ = ( W 11 − W 22 2 , Re W 12 , Im W 12 ) . \xi=\left(\frac{W_{11}-W_{22}}{2},\operatorname{Re}W_{12},\operatorname{Im}W_{12}\right). ξ = ( 2 W 11 − W 22 , Re W 12 , Im W 12 ) . Then the eigenvalues of the Hermitian 2 × 2 2\times2 2 × 2 M j + W [ 2 ] M_j+W_{[2]} M j + W [ 2 ] ( Tr M j + W 11 + W 22 ) / 2 ± ∣ v + ξ ∣ (\operatorname{Tr}M_j+W_{11}+W_{22})/2\pm|v+\xi| ( Tr M j + W 11 + W 22 ) /2 ± ∣ v + ξ ∣
The four real variables ( W 11 + W 22 ) / 2 (W_{11}+W_{22})/2 ( W 11 + W 22 ) /2 ( W 11 − W 22 ) / 2 (W_{11}-W_{22})/2 ( W 11 − W 22 ) /2 Re W 12 \operatorname{Re}W_{12} Re W 12 Im W 12 \operatorname{Im}W_{12} Im W 12 1 / 2 1/2 1/2
∣ v + ξ ∣ ≤ 3 δ 2 , ∣ Tr M j + W 11 + W 22 2 − w j ′ ∣ ≤ 4 δ . |v+\xi|\le\frac{3\delta}{2},\qquad\left|\frac{\operatorname{Tr}M_j+W_{11}+W_{22}}{2}-w'_j\right|\le4\delta. ∣ v + ξ ∣ ≤ 2 3 δ , 2 Tr M j + W 11 + W 22 − w j ′ ≤ 4 δ . The densities of ξ \xi ξ ( W 11 + W 22 ) / 2 (W_{11}+W_{22})/2 ( W 11 + W 22 ) /2 C δ 3 ⋅ C δ C\delta^3\cdot C\delta C δ 3 ⋅ C δ 24 R / δ + 2 24R/\delta+2 24 R / δ + 2 j j j δ ≤ 1 ≤ R \delta\le1\le R δ ≤ 1 ≤ R
P ( gap ( D y + W ) ≤ δ ) ≤ C ( R δ + 1 ) δ 4 + C e − R 2 / C ≤ C R δ 3 + C e − R 2 / C . (414) \mathbb{P}\bigl(\operatorname{gap}(D_y+W)\le\delta\bigr)\le C\left(\frac{R}{\delta}+1\right)\delta^4+Ce^{-R^2/C}\le CR\delta^3+Ce^{-R^2/C}. \tag*{(414)} P ( gap ( D y + W ) ≤ δ ) ≤ C ( δ R + 1 ) δ 4 + C e − R 2 / C ≤ C R δ 3 + C e − R 2 / C . (414) Combining the three cases proves (407) .
Gap counts at the scales N − 1 − α N^{-1-\alpha} N − 1 − α
For α ∈ ( 0 , 1 / 3 ) \alpha\in(0,1/3) α ∈ ( 0 , 1/3 ) P C ( α ) \mathcal{P}^{\mathrm{C}}(\alpha) P C ( α ) φ ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi\in C^\infty_{\mathrm{c}}((-2,2)) φ ∈ C c ∞ (( − 2 , 2 )) p ≥ p ∗ p \ge p_\ast p ≥ p ∗
1 M α C X α ( H ; φ N , f p ) ⟶ ∫ − 2 2 φ ( E ) c 2 ( E ) d E ∫ 0 ∞ y 2 f p ( y ) d y (415) \frac{1}{M_{\alpha}^{\mathrm{C}}}\mathcal{X}_{\alpha}(H;\varphi_N,f_p) \longrightarrow\int_{-2}^{2}\varphi(E)c_2(E)\,\mathrm{d}E \int_{0}^{\infty}y^2f_p(y)\,\mathrm{d}y \tag*{(415)} M α C 1 X α ( H ; φ N , f p ) ⟶ ∫ − 2 2 φ ( E ) c 2 ( E ) d E ∫ 0 ∞ y 2 f p ( y ) d y (415) in probability, where X α \mathcal{X}_{\alpha} X α (125) for Hermitian matrices. This is the analogue of the statement P ( α ) \mathcal{P}(\alpha) P ( α ) Definition 6.1 .
We first record the results of Section 6 that carry over. The proofs of Lemma 6.2 , Lemma 6.3 , and Lemma 6.6 apply with M α C M_{\alpha}^{\mathrm{C}} M α C c 2 ( E ) c_2(E) c 2 ( E ) ∫ 0 ∞ y 2 f ( y ) d y \int_{0}^{\infty}y^2f(y)\,\mathrm{d}y ∫ 0 ∞ y 2 f ( y ) d y M α M_{\alpha} M α c ( E ) c(E) c ( E ) ∫ 0 ∞ y f ( y ) d y \int_{0}^{\infty}yf(y)\,\mathrm{d}y ∫ 0 ∞ y f ( y ) d y (392) in place of Proposition 5.9; the integrals ∫ 0 ∞ y 2 f p ( y ) d y \int_{0}^{\infty}y^2f_p(y)\,\mathrm{d}y ∫ 0 ∞ y 2 f p ( y ) d y p ≥ p ∗ p \ge p_\ast p ≥ p ∗ Lemma 6.2 is deterministic and uses only the bound ∥ A ∥ ≤ 3 \lVert A\rVert\le3 ∥ A ∥ ≤ 3 Lemma 5.1 . The proof of Lemma 6.3 uses the comparison estimate only through the bound (172) and the nonnegativity of the pair statistic, which holds for Hermitian matrices as well. The proof of Lemma 6.6 uses only P ( α ) \mathcal{P}(\alpha) P ( α ) Lemma 6.2 , with (184) replaced by the finiteness of the integrals above.
Base case. The proof of Proposition 6.5 applies with the following changes, and gives P C ( α ) \mathcal{P}^{\mathrm{C}}(\alpha) P C ( α ) α ∈ ( 0 , 1 / 8 ) \alpha\in(0,1/8) α ∈ ( 0 , 1/8 ) Lemma 9.1 (a) in place of Proposition 3.6 . The proof of Lemma 6.4 is deterministic, and in it the pairs with a label outside J \mathcal{J} J
C N − 2 α + C N 2 exp ( − δ 2 N 1 / 5 / 4 ) = o ( M α C ) CN^{-2\alpha}+CN^2\exp\left(-\delta^2N^{1/5}/4\right)=o\left(M_{\alpha}^{\mathrm{C}}\right) C N − 2 α + C N 2 exp ( − δ 2 N 1/5 /4 ) = o ( M α C ) by (192) , so its proof gives (189) with M α C M_{\alpha}^{\mathrm{C}} M α C M α M_{\alpha} M α T = N − 3 / 4 − α T=N^{-3/4-\alpha} T = N − 3/4 − α § 11.3 and the comparison error of (392) satisfy
2 N ε N 2 T = 2 N − 5 / 4 + α + ε ≤ N − 2 ε ℓ α , T N 1 / 2 + 3 α + 12 ε = N − 1 / 4 + 2 α + 12 ε ⟶ 0 \frac{2N^{\varepsilon}}{N^2T}=2N^{-5/4+\alpha+\varepsilon}\le N^{-2\varepsilon}\ell_{\alpha},
\qquad
TN^{1/2+3\alpha+12\varepsilon}=N^{-1/4+2\alpha+12\varepsilon}\longrightarrow0 N 2 T 2 N ε = 2 N − 5/4 + α + ε ≤ N − 2 ε ℓ α , T N 1/2 + 3 α + 12 ε = N − 1/4 + 2 α + 12 ε ⟶ 0 for α < 1 / 8 \alpha<1/8 α < 1/8 ε \varepsilon ε
Induction step. Suppose that P C ( α ) \mathcal{P}^{\mathrm{C}}(\alpha) P C ( α ) α ′ ∈ ( α , 1 / 3 ) \alpha' \in(\alpha,1/3) α ′ ∈ ( α , 1/3 ) κ > 0 \kappa>0 κ > 0
3 α ′ 8 < κ < min { α , 2 α − α ′ , 1 8 } , α ′ − κ ≥ 1 30 , α ′ < 1 6 + 2 α 3 . (416) \frac{3\alpha'}{8}<\kappa<\min\left\{\alpha,2\alpha-\alpha',\frac{1}{8}\right\},
\qquad
\alpha'-\kappa\ge\frac{1}{30},
\qquad
\alpha'<\frac{1}{6}+\frac{2\alpha}{3}. \tag*{(416)} 8 3 α ′ < κ < min { α , 2 α − α ′ , 8 1 } , α ′ − κ ≥ 30 1 , α ′ < 6 1 + 3 2 α . (416) We follow § 8.3 , with t α = N ℓ α 2 t_{\alpha}=N\ell_{\alpha}^{2} t α = N ℓ α 2 q = N − 1 − κ q=N^{-1-\kappa} q = N − 1 − κ ε > 0 \varepsilon>0 ε > 0 (416) . The conditions (416) replace the choice of κ \kappa κ (206) and the conditions (207) , and we indicate below where each of them is used.
We begin with the Gaussian blocks. Conditionally on H H H H t α H_{t_{\alpha}} H t α D y + W D_y+W D y + W y i = e − t α / 2 λ i y_i=e^{-t_{\alpha}/2}\lambda_i y i = e − t α /2 λ i W = ( 1 − e − t α ) 1 / 2 ( u i ∗ G u j ) i , j = 1 N W=(1-e^{-t_{\alpha}})^{1/2}(u_i^\ast Gu_j)_{i,j=1}^{N} W = ( 1 − e − t α ) 1/2 ( u i ∗ G u j ) i , j = 1 N u 1 , … , u N u_1,\ldots,u_N u 1 , … , u N H H H W W W H H H ( 1 − e − t α ) 1 / 2 G (1-e^{-t_\alpha})^{1/2}G ( 1 − e − t α ) 1/2 G w 0 = N ε ℓ α w_0=N^\varepsilon\ell_\alpha w 0 = N ε ℓ α (66) is o ( ℓ α ′ ) o(\ell_{\alpha'}) o ( ℓ α ′ )
N ε w 0 ( q − 1 + N log N ) ≤ 2 N 2 ε − α + κ → 0 , N^\varepsilon w_0(q^{-1}+N\log N)\le2N^{2\varepsilon-\alpha+\kappa}\to0, N ε w 0 ( q − 1 + N log N ) ≤ 2 N 2 ε − α + κ → 0 , 24 N 2 ε w 0 2 ( q − 1 + N log N ) ≤ C N 4 ε − 1 − 2 α + κ = o ( ℓ α ′ ) , 24N^{2\varepsilon}w_0^2(q^{-1}+N\log N)\le CN^{4\varepsilon-1-2\alpha+\kappa}=o(\ell_{\alpha'}), 24 N 2 ε w 0 2 ( q − 1 + N log N ) ≤ C N 4 ε − 1 − 2 α + κ = o ( ℓ α ′ ) , where we used κ < α \kappa<\alpha κ < α κ < 2 α − α ′ \kappa<2\alpha-\alpha' κ < 2 α − α ′ K = { i , i + 1 } \mathcal{K}=\{i,i+1\} K = { i , i + 1 } 2 × 2 2\times2 2 × 2 g ^ K \hat{g}_{\mathcal{K}} g ^ K ( D y + W ) K K (D_y+W)_{\mathcal{K}\mathcal{K}} ( D y + W ) KK
g ^ K = ∣ ( e − t α / 2 g i + W i + 1 , i + 1 − W i i , 2 Re W i , i + 1 , 2 Im W i , i + 1 ) ∣ . (417) \hat{g}_{\mathcal{K}}=\left|(e^{-t_\alpha/2}g_i+W_{i+1,i+1}-W_{ii},\;2\operatorname{Re}W_{i,i+1},\;2\operatorname{Im}W_{i,i+1})\right|. \tag*{(417)} g ^ K = ( e − t α /2 g i + W i + 1 , i + 1 − W ii , 2 Re W i , i + 1 , 2 Im W i , i + 1 ) . (417) Next, we compute the conditional mean of the block statistic, following the proof of Lemma 6.10 . The first coordinate of the vector in (417) minus e − t α / 2 g i e^{-t_\alpha/2}g_i e − t α /2 g i v N C ℓ α 2 v_N^{\mathrm{C}}\ell_\alpha^2 v N C ℓ α 2
v N C = 2 ( 1 − e − t α ) N ℓ α 2 = 2 ( 1 − e − t α ) t α ∈ [ 2 − t α , 2 ] v_N^{\mathrm{C}}=\frac{2(1-e^{-t_\alpha})}{N\ell_\alpha^2}=\frac{2(1-e^{-t_\alpha})}{t_\alpha}\in[2-t_\alpha,2] v N C = N ℓ α 2 2 ( 1 − e − t α ) = t α 2 ( 1 − e − t α ) ∈ [ 2 − t α , 2 ] (compare (224) ). The vector in (417) is three-dimensional because degeneracies of Hermitian matrices have codimension three. Thus g ^ K / ℓ α \hat{g}_{\mathcal{K}}/\ell_\alpha g ^ K / ℓ α ∣ G x ∣ |G_x| ∣ G x ∣ Lemma 3.10 with n = 3 n=3 n = 3 x = g i / ℓ α x=g_i/\ell_\alpha x = g i / ℓ α ϱ = e − t α / 2 \varrho=e^{-t_\alpha/2} ϱ = e − t α /2 v = v N C v=v_N^{\mathrm{C}} v = v N C Lemma 3.10 with v 0 = 2 v_0=2 v 0 = 2 N α − α ′ = ℓ α ′ / ℓ α N^{\alpha-\alpha'}=\ell_{\alpha'}/\ell_\alpha N α − α ′ = ℓ α ′ / ℓ α p > 2 p>2 p > 2 p ≥ p ∗ p\ge p_* p ≥ p ∗ (69) with n = 3 n=3 n = 3 g 2 ( x , 0 , 0 ) = ( 4 π ) − 3 / 2 e − x 2 / 4 g_2(x,0,0)=(4\pi)^{-3/2}e^{-x^2/4} g 2 ( x , 0 , 0 ) = ( 4 π ) − 3/2 e − x 2 /4 ∫ R 3 f ( ∣ y ∣ ) d y = 4 π ∫ 0 ∞ r 2 f ( r ) d r \int_{\mathbb{R}^3}f(|y|)\,dy=4\pi\int_0^\infty r^2f(r)\,dr ∫ R 3 f ( ∣ y ∣ ) d y = 4 π ∫ 0 ∞ r 2 f ( r ) d r 4 π ( 4 π ) − 3 / 2 = ( 2 π ) − 1 4\pi(4\pi)^{-3/2}=(2\sqrt{\pi})^{-1} 4 π ( 4 π ) − 3/2 = ( 2 π ) − 1
E [ f ( g ^ K ℓ α ′ ) | H ] = N 3 ( α − α ′ ) [ e − g i 2 / ( 4 ℓ α 2 ) 2 π ∫ 0 ∞ y 2 f p ( y ) d y + o ( 1 ) f p ( g i ℓ α ) ] , (418) \mathbb{E}\left[f\left(\frac{\hat{g}_{\mathcal{K}}}{\ell_{\alpha'}}\right)\middle|H\right]
=N^{3(\alpha-\alpha')}\left[\frac{e^{-g_i^2/(4\ell_\alpha^2)}}{2\sqrt{\pi}}\int_0^\infty y^2f_p(y)\,dy+o(1)f_p\left(\frac{g_i}{\ell_\alpha}\right)\right], \tag*{(418)} E [ f ( ℓ α ′ g ^ K ) H ] = N 3 ( α − α ′ ) [ 2 π e − g i 2 / ( 4 ℓ α 2 ) ∫ 0 ∞ y 2 f p ( y ) d y + o ( 1 ) f p ( ℓ α g i ) ] , (418) where the o ( 1 ) o(1) o ( 1 ) H H H K \mathcal{K} K
Recall the block statistic X ^ α ′ ( φ , p ) \widehat{X}_{\alpha'}(\varphi,p) X α ′ ( φ , p ) (216) , and the set P 2 \mathcal{P}_2 P 2 g K g_{\mathcal{K}} g K E K E_{\mathcal{K}} E K (418) , multiplied by φ N ( E K ) \varphi_N(E_{\mathcal{K}}) φ N ( E K ) K ∈ P 2 \mathcal{K}\in\mathcal{P}_2 K ∈ P 2 P C ( α ) \mathcal{P}^{\mathrm{C}}(\alpha) P C ( α ) N 3 ( α − α ′ ) M α C = M α ′ C N^{3(\alpha-\alpha')}M_\alpha^{\mathrm{C}}=M_{\alpha'}^{\mathrm{C}} N 3 ( α − α ′ ) M α C = M α ′ C
1 M α ′ C E [ X ^ α ′ ( φ , p ) ∣ H ] = ∫ 0 ∞ y 2 f p ( y ) d y 2 π M α C ∑ K ∈ P 2 φ N ( E K ) e − g K 2 / ( 4 ℓ α 2 ) + o P ( 1 ) \frac{1}{M_{\alpha'}^{\mathrm{C}}}\mathbb{E}\left[\widehat{X}_{\alpha'}(\varphi,p)\mid H\right]
=\frac{\int_0^\infty y^2f_p(y)\,dy}{2\sqrt{\pi}M_\alpha^{\mathrm{C}}}
\sum_{\mathcal{K}\in\mathcal{P}_2}\varphi_N(E_{\mathcal{K}})e^{-g_{\mathcal{K}}^2/(4\ell_\alpha^2)}
+o_{\mathbb{P}}(1) M α ′ C 1 E [ X α ′ ( φ , p ) ∣ H ] = 2 π M α C ∫ 0 ∞ y 2 f p ( y ) d y K ∈ P 2 ∑ φ N ( E K ) e − g K 2 / ( 4 ℓ α 2 ) + o P ( 1 ) ⟶ ∫ − 2 2 φ ( E ) c 2 ( E ) d E ∫ 0 ∞ y 2 f p ( y ) d y \longrightarrow\int_{-2}^{2}\varphi(E)c_2(E)\,dE\int_0^\infty y^2f_p(y)\,dy ⟶ ∫ − 2 2 φ ( E ) c 2 ( E ) d E ∫ 0 ∞ y 2 f p ( y ) d y in probability, which is the limit in (415) with α ′ \alpha' α ′ α \alpha α P C ( α ) \mathcal{P}^{\mathrm{C}}(\alpha) P C ( α ) y ↦ e − y 2 / 4 y\mapsto e^{-y^2/4} y ↦ e − y 2 /4
∫ 0 ∞ r 2 e − r 2 / 4 d r = 2 π , \int_{0}^{\infty} r^{2}e^{-r^{2}/4}\,\mathrm{d}r=2\sqrt{\pi}, ∫ 0 ∞ r 2 e − r 2 /4 d r = 2 π , and we removed the pairs that do not form a two-element cluster as explained below. The conditional variance bound is as in the proof of Lemma 6.11, since the blocks of distinct clusters are conditionally independent given H H H
We now treat the pairs of eigenvalues in a common cluster with at least three elements, which appear in (219) and in the evaluation of the conditional mean. Here we use Proposition 9.2, which applies since κ < 1 / 8 \kappa<1/8 κ < 1/8 δ > 0 \delta>0 δ > 0 I I I 2 δ 2\delta 2 δ supp φ \operatorname{supp}\varphi supp φ N ε N^{\varepsilon} N ε i i i λ i + 2 − λ i < 2 e t α / 2 q ≤ 4 q \lambda_{i+2}-\lambda_i<2e^{t_{\alpha}/2}q\leq4q λ i + 2 − λ i < 2 e t α /2 q ≤ 4 q I I I N 2 ε N^{2\varepsilon} N 2 ε i i i λ i ∈ I \lambda_i\in I λ i ∈ I λ i + 2 − λ i ≤ 4 q \lambda_{i+2}-\lambda_i\leq4q λ i + 2 − λ i ≤ 4 q N 1 − 8 κ + 3 ε = o ( M α ′ C ) N^{1-8\kappa+3\varepsilon}=o(M_{\alpha'}^{\mathrm{C}}) N 1 − 8 κ + 3 ε = o ( M α ′ C ) 8 κ > 3 α ′ 8\kappa>3\alpha' 8 κ > 3 α ′ (218) . A cluster has diameter at most 2 N ε q 2N^{\varepsilon}q 2 N ε q I I I δ \delta δ supp φ \operatorname{supp}\varphi supp φ C N 2 exp ( − δ 2 N 1 / 5 / 2 ) = o ( 1 ) CN^{2}\exp(-\delta^{2}N^{1/5}/2)=o(1) C N 2 exp ( − δ 2 N 1/5 /2 ) = o ( 1 )
The pairs in different clusters are at distance at least q / 2 q/2 q /2
4 p ∥ φ ∥ ∞ N 2 ( ℓ α ′ q ) 2 p = 4 p ∥ φ ∥ ∞ M α ′ C N 1 + 3 α ′ − 2 p ( α ′ − κ ) ≤ 4 p ∥ φ ∥ ∞ M α ′ C N 2 − 61 / 15 , 4^{p}\|\varphi\|_{\infty}N^{2}\left(\frac{\ell_{\alpha'}}{q}\right)^{2p}
=4^{p}\|\varphi\|_{\infty}M_{\alpha'}^{\mathrm{C}}N^{1+3\alpha'-2p(\alpha'-\kappa)}
\leq4^{p}\|\varphi\|_{\infty}M_{\alpha'}^{\mathrm{C}}N^{2-61/15}, 4 p ∥ φ ∥ ∞ N 2 ( q ℓ α ′ ) 2 p = 4 p ∥ φ ∥ ∞ M α ′ C N 1 + 3 α ′ − 2 p ( α ′ − κ ) ≤ 4 p ∥ φ ∥ ∞ M α ′ C N 2 − 61/15 , where we used p ≥ p ∗ = 61 p\geq p_{*}=61 p ≥ p ∗ = 61 α ′ − κ ≥ 1 / 30 \alpha'-\kappa\geq1/30 α ′ − κ ≥ 1/30 1 + 3 α ′ < 2 1+3\alpha'<2 1 + 3 α ′ < 2 H H H q q q ∥ φ ∥ ∞ N 2 exp ( − q 2 / ( 4 ℓ α 2 ) ) = o ( 1 ) \|\varphi\|_{\infty}N^{2}\exp(-q^{2}/(4\ell_{\alpha}^{2}))=o(1) ∥ φ ∥ ∞ N 2 exp ( − q 2 / ( 4 ℓ α 2 )) = o ( 1 ) κ < α \kappa<\alpha κ < α 4 ℓ α 2 4\ell_{\alpha}^{2} 4 ℓ α 2 8 ℓ α 2 8\ell_{\alpha}^{2} 8 ℓ α 2 (418) is e − g i 2 / ( 4 ℓ α 2 ) e^{-g_i^{2}/(4\ell_{\alpha}^{2})} e − g i 2 / ( 4 ℓ α 2 ) e − g i 2 / ( 8 ℓ α 2 ) e^{-g_i^{2}/(8\ell_{\alpha}^{2})} e − g i 2 / ( 8 ℓ α 2 ) (392) at the scale ℓ α ′ \ell_{\alpha'} ℓ α ′ t α t_{\alpha} t α t α N 1 / 2 + 3 α ′ + 12 ε = N − 1 / 2 − 2 α + 3 α ′ + 12 ε t_{\alpha}N^{1/2+3\alpha'+12\varepsilon}=N^{-1/2-2\alpha+3\alpha'+12\varepsilon} t α N 1/2 + 3 α ′ + 12 ε = N − 1/2 − 2 α + 3 α ′ + 12 ε (416) . This completes the proof of P C ( α ′ ) \mathcal{P}^{\mathrm{C}}(\alpha') P C ( α ′ )
Iteration. The conditions (416) hold for the triples ( α , α ′ , κ ) (\alpha,\alpha',\kappa) ( α , α ′ , κ )
( 3 25 , 17 100 , 67 1000 ) , ( 17 100 , 6 25 , 19 200 ) , ( 6 25 , 8 25 , 49 400 ) . (419) \left(\frac{3}{25},\frac{17}{100},\frac{67}{1000}\right),\qquad
\left(\frac{17}{100},\frac{6}{25},\frac{19}{200}\right),\qquad
\left(\frac{6}{25},\frac{8}{25},\frac{49}{400}\right). \tag*{(419)} ( 25 3 , 100 17 , 1000 67 ) , ( 100 17 , 25 6 , 200 19 ) , ( 25 6 , 25 8 , 400 49 ) . (419) Indeed, for these three triples, in this order, we have
( 3 α ′ 8 , κ , min { α , 2 α − α ′ , 1 8 } ) = ( 255 4000 , 268 4000 , 280 4000 ) , ( 18 200 , 19 200 , 20 200 ) , ( 48 400 , 49 400 , 50 400 ) , \left(\frac{3\alpha'}{8},\kappa,\min\left\{\alpha,2\alpha-\alpha',\frac{1}{8}\right\}\right)
=
\left(\frac{255}{4000},\frac{268}{4000},\frac{280}{4000}\right),
\left(\frac{18}{200},\frac{19}{200},\frac{20}{200}\right),
\left(\frac{48}{400},\frac{49}{400},\frac{50}{400}\right), ( 8 3 α ′ , κ , min { α , 2 α − α ′ , 8 1 } ) = ( 4000 255 , 4000 268 , 4000 280 ) , ( 200 18 , 200 19 , 200 20 ) , ( 400 48 , 400 49 , 400 50 ) , α ′ − κ = 103 1000 , 29 200 , 79 400 , \alpha'-\kappa=\frac{103}{1000},\frac{29}{200},\frac{79}{400}, α ′ − κ = 1000 103 , 200 29 , 400 79 , ( α ′ , 1 6 + 2 α 3 ) = ( 51 300 , 74 300 ) , ( 36 150 , 42 150 ) , ( 48 150 , 49 150 ) . \left(\alpha',\frac{1}{6}+\frac{2\alpha}{3}\right)
=
\left(\frac{51}{300},\frac{74}{300}\right),
\left(\frac{36}{150},\frac{42}{150}\right),
\left(\frac{48}{150},\frac{49}{150}\right). ( α ′ , 6 1 + 3 2 α ) = ( 300 51 , 300 74 ) , ( 150 36 , 150 42 ) , ( 150 48 , 150 49 ) . The minima in the first line are 2 α − α ′ 2\alpha-\alpha' 2 α − α ′ 2 α − α ′ 2\alpha-\alpha' 2 α − α ′ 1 / 8 1/8 1/8 1 / 30 1/30 1/30 α = 3 / 25 < 1 / 8 \alpha=3/25<1/8 α = 3/25 < 1/8 P C ( α 1 ) \mathcal{P}^{\mathrm{C}}(\alpha_1) P C ( α 1 ) α 1 = 8 / 25 \alpha_1=8/25 α 1 = 8/25
The critical scale
We fix α 1 = 8 / 25 \alpha_1=8/25 α 1 = 8/25 κ 1 = 49 / 400 \kappa_1=49/400 κ 1 = 49/400
1 4 < α 1 < 1 3 , 3 α 1 8 < κ 1 < 1 8 , 1 15 < κ 1 < 2 α 1 − 1 3 . (420) \frac{1}{4}<\alpha_1<\frac{1}{3},\qquad\frac{3\alpha_1}{8}<\kappa_1<\frac{1}{8},\qquad\frac{1}{15}<\kappa_1<2\alpha_1-\frac{1}{3}. \tag*{(420)} 4 1 < α 1 < 3 1 , 8 3 α 1 < κ 1 < 8 1 , 15 1 < κ 1 < 2 α 1 − 3 1 . (420) and we set q = N − 1 − κ 1 q=N^{-1-\kappa_1} q = N − 1 − κ 1 t α 1 = N ℓ α 1 2 = N − 1 − 2 α 1 t_{\alpha_1}=N\ell_{\alpha_1}^2=N^{-1-2\alpha_1} t α 1 = N ℓ α 1 2 = N − 1 − 2 α 1 κ 1 < α 1 \kappa_1<\alpha_1 κ 1 < α 1 ε > 0 \varepsilon>0 ε > 0 (420) ; for example, we need 5 ε < 8 κ 1 − 3 α 1 = 1 / 50 5\varepsilon<8\kappa_1-3\alpha_1=1/50 5 ε < 8 κ 1 − 3 α 1 = 1/50 I I I I ′ I' I ′ I I I I ′ ′ I'' I ′′ I ′ I' I ′ x 0 ≥ 1 x_0\geq1 x 0 ≥ 1
The matrix H t α 1 H_{t_{\alpha_1}} H t α 1 As in the induction step, conditionally on H H H H t α 1 H_{t_{\alpha_1}} H t α 1 D y + W D_y+W D y + W y i = e − t α 1 / 2 λ i y_i=e^{-t_{\alpha_1}/2}\lambda_i y i = e − t α 1 /2 λ i W W W H H H ( 1 − e − t α 1 ) 1 / 2 G (1-e^{-t_{\alpha_1}})^{1/2}G ( 1 − e − t α 1 ) 1/2 G q q q § 8.3 , and we work on the event { H ∈ G ε / 2 } ∩ { max i ≤ j ∣ W i j ∣ ≤ N ε ℓ α 1 } \{H\in\mathcal{G}_{\varepsilon/2}\}\cap\{\max_{i\leq j}|W_{ij}|\leq N^\varepsilon\ell_{\alpha_1}\} { H ∈ G ε /2 } ∩ { max i ≤ j ∣ W ij ∣ ≤ N ε ℓ α 1 } w 0 = N ε ℓ α 1 w_0=N^\varepsilon\ell_{\alpha_1} w 0 = N ε ℓ α 1 N 2 ε ℓ α 1 2 ( q − 1 + N log N ) ≤ C N 2 ε − α 1 + κ 1 → 0 N^{2\varepsilon}\ell_{\alpha_1}^2(q^{-1}+N\log N)\leq CN^{2\varepsilon-\alpha_1+\kappa_1}\to0 N 2 ε ℓ α 1 2 ( q − 1 + N log N ) ≤ C N 2 ε − α 1 + κ 1 → 0 H t α 1 H_{t_{\alpha_1}} H t α 1 D y + W D_y+W D y + W
C N 4 ε ℓ α 1 2 q − 1 = C N − 1 − 2 α 1 + κ 1 + 4 ε = o ( d 2 ) , CN^{4\varepsilon}\ell_{\alpha_1}^2q^{-1}=CN^{-1-2\alpha_1+\kappa_1+4\varepsilon}=o(d_2), C N 4 ε ℓ α 1 2 q − 1 = C N − 1 − 2 α 1 + κ 1 + 4 ε = o ( d 2 ) , since κ 1 < 2 α 1 − 1 / 3 \kappa_1<2\alpha_1-1/3 κ 1 < 2 α 1 − 1/3 q / 2 q/2 q /2 d 2 d_2 d 2
We next use Proposition 9.2 to control the sizes of the clusters. The smallest label i i i λ i + 2 − λ i < 2 e t α 1 / 2 q ≤ 4 q \lambda_{i+2}-\lambda_i<2e^{t_{\alpha_1}/2}q\leq4q λ i + 2 − λ i < 2 e t α 1 /2 q ≤ 4 q λ i + 3 − λ i < 3 e t α 1 / 2 q ≤ 6 q \lambda_{i+3}-\lambda_i<3e^{t_{\alpha_1}/2}q\leq6q λ i + 3 − λ i < 3 e t α 1 /2 q ≤ 6 q κ = κ 1 ∈ ( 1 / 15 , 1 / 8 ) \kappa=\kappa_1\in(1/15,1/8) κ = κ 1 ∈ ( 1/15 , 1/8 ) I ′ ′ I'' I ′′ I I I I ′ ′ I'' I ′′ N 1 − 8 κ 1 + ε N^{1-8\kappa_1+\varepsilon} N 1 − 8 κ 1 + ε I ′ I' I ′ I ′ ′ I'' I ′′ N ≥ N 0 N\geq N_0 N ≥ N 0 2 N ε q 2N^\varepsilon q 2 N ε q
Two-element clusters. Let Q ( y , E ) = ∑ j = 1 n θ j φ j ( E ) f p j ( y ) Q(y,E)=\sum_{j=1}^n\theta_j\varphi_j(E)f_{p_j}(y) Q ( y , E ) = ∑ j = 1 n θ j φ j ( E ) f p j ( y ) θ j ≥ 0 \theta_j\geq0 θ j ≥ 0 φ j ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi_j\in C^\infty_c((-2,2)) φ j ∈ C c ∞ (( − 2 , 2 )) I ′ I' I ′ p j ≥ p ∗ p_j\geq p_* p j ≥ p ∗ p 0 = min j p j p_0=\min_jp_j p 0 = min j p j E i = ( λ i + λ i + 1 ) / 2 E_i=(\lambda_i+\lambda_{i+1})/2 E i = ( λ i + λ i + 1 ) /2 K = { i , i + 1 } \mathcal{K}=\{i,i+1\} K = { i , i + 1 } g ^ i = g ^ K \hat{g}_i = \hat{g}_{\mathcal{K}} g ^ i = g ^ K α 1 \alpha_1 α 1 α \alpha α E ∈ R E \in\mathbb{R} E ∈ R ψ E ( y ) = 1 − e − Q ( y , E ) \psi_E(y) = 1 - e^{-Q(y,E)} ψ E ( y ) = 1 − e − Q ( y , E ) 0 ≤ ψ E ( y ) ≤ Q ( y , E ) ≤ C Q f p 0 ( y ) 0 \le\psi_E(y) \le Q(y,E) \le C_Q f_{p_0}(y) 0 ≤ ψ E ( y ) ≤ Q ( y , E ) ≤ C Q f p 0 ( y )
ψ ˉ i = E [ ψ E i ( g ^ i / d 2 ) ∣ H ] , m Q ( E ) = ∫ 0 ∞ y 2 ψ E ( y ) d y . (421) \bar{\psi}_i = \mathbb{E}\left[\psi_{E_i}\left(\hat{g}_i/d_2\right)\mid H\right], \qquad m_Q(E) = \int_0^\infty y^2\psi_E(y)\,dy. \tag*{(421)} ψ ˉ i = E [ ψ E i ( g ^ i / d 2 ) ∣ H ] , m Q ( E ) = ∫ 0 ∞ y 2 ψ E ( y ) d y . (421) By Lemma 3.10 as in (9.30), now applied to ψ E i / C Q \psi_{E_i}/C_Q ψ E i / C Q d 2 / ℓ α 1 d_2/\ell_{\alpha_1} d 2 / ℓ α 1
ψ ˉ i = ( d 2 ℓ α 1 ) 3 [ e − g ^ i 2 / ( 4 ℓ α 1 2 ) 2 π m Q ( E i ) + o ( 1 ) f p 0 ( g ^ i ℓ α 1 ) ] , \bar{\psi}_i = \left(\frac{d_2}{\ell_{\alpha_1}}\right)^3\left[\frac{e^{-\hat{g}_i^2/(4\ell_{\alpha_1}^2)}}{2\sqrt{\pi}}m_Q(E_i)+o(1)f_{p_0}\left(\frac{\hat{g}_i}{\ell_{\alpha_1}}\right)\right], ψ ˉ i = ( ℓ α 1 d 2 ) 3 [ 2 π e − g ^ i 2 / ( 4 ℓ α 1 2 ) m Q ( E i ) + o ( 1 ) f p 0 ( ℓ α 1 g ^ i ) ] , uniformly in i i i max i ψ ˉ i ≤ C ( d 2 / ℓ α 1 ) 3 → 0 \max_i\bar{\psi}_i \le C(d_2/\ell_{\alpha_1})^3 \to0 max i ψ ˉ i ≤ C ( d 2 / ℓ α 1 ) 3 → 0
We now sum the ψ ˉ i \bar{\psi}_i ψ ˉ i m Q m_Q m Q I ′ I^{\prime} I ′ P C ( α 1 ) \mathcal{P}^{\mathrm{C}}(\alpha_1) P C ( α 1 ) m Q m_Q m Q y ↦ e − y 2 / 4 y\mapsto e^{-y^2/4} y ↦ e − y 2 /4 ( d 2 / ℓ α 1 ) 3 M α 1 C = 1 (d_2/\ell_{\alpha_1})^3M_{\alpha_1}^{\mathrm{C}}=1 ( d 2 / ℓ α 1 ) 3 M α 1 C = 1 ∫ 0 ∞ r 2 e − r 2 / 4 d r = 2 π \int_0^\infty r^2e^{-r^2/4}\,dr=2\sqrt{\pi} ∫ 0 ∞ r 2 e − r 2 /4 d r = 2 π
∑ i ψ ˉ i ⟶ ∫ − 2 2 ∫ 0 ∞ c 2 ( E ) y 2 ( 1 − e − Q ( y , E ) ) d y d E (422) \sum_i\bar{\psi}_i \longrightarrow\int_{-2}^{2}\int_0^\infty c_2(E)y^2\left(1-e^{-Q(y,E)}\right)\,dy\,dE \tag*{(422)} i ∑ ψ ˉ i ⟶ ∫ − 2 2 ∫ 0 ∞ c 2 ( E ) y 2 ( 1 − e − Q ( y , E ) ) d y d E (422) in probability, where the sum is over the first labels of the two-element clusters. Here the pairs that do not form a two-element cluster were removed from the statistic of P C ( α 1 ) \mathcal{P}^{\mathrm{C}}(\alpha_1) P C ( α 1 ) 8 κ 1 > 3 α 1 8\kappa_1>3\alpha_1 8 κ 1 > 3 α 1 H H H exp ( − ∑ i Q ( g ^ i / d 2 , E i ) ) \exp(-\sum_i Q(\hat{g}_i/d_2,E_i)) exp ( − ∑ i Q ( g ^ i / d 2 , E i )) H H H ∏ i ( 1 − ψ ˉ i ) \prod_i(1-\bar{\psi}_i) ∏ i ( 1 − ψ ˉ i ) max i ψ ˉ i → 0 \max_i\bar{\psi}_i\to0 max i ψ ˉ i → 0 ∑ i ψ ˉ i = O P ( 1 ) \sum_i\bar{\psi}_i=O_{\mathbb{P}}(1) ∑ i ψ ˉ i = O P ( 1 )
Three-element clusters. Let K = { i , i + 1 , i + 2 } \mathcal{K}=\{i,i+1,i+2\} K = { i , i + 1 , i + 2 } I ′ ′ I^{\prime\prime} I ′′ B K = ( D y + W ) K K B_{\mathcal{K}}=(D_y+W)_{\mathcal{K}\mathcal{K}} B K = ( D y + W ) KK W K K W_{\mathcal{K}\mathcal{K}} W KK H H H σ 2 = ( 1 − e − t α 1 ) / N ∈ [ ℓ α 1 2 / 2 , ℓ α 1 2 ] \sigma^2=(1-e^{-t\alpha_1})/N\in[\ell_{\alpha_1}^2/2,\ell_{\alpha_1}^2] σ 2 = ( 1 − e − t α 1 ) / N ∈ [ ℓ α 1 2 /2 , ℓ α 1 2 ] R = N ε R=N^\varepsilon R = N ε δ ∈ ( 0 , σ ] \delta\in(0,\sigma] δ ∈ ( 0 , σ ]
P ( gap ( B K ) ≤ δ ∣ H ) ≤ C N 4 ε ( δ ℓ α 1 ) 3 + C e − N 2 ε / C . \mathbb{P}\left(\operatorname{gap}(B_{\mathcal{K}})\le\delta\mid H\right)\le CN^{4\varepsilon}\left(\frac{\delta}{\ell_{\alpha_1}}\right)^3+Ce^{-N^{2\varepsilon}/C}. P ( gap ( B K ) ≤ δ ∣ H ) ≤ C N 4 ε ( ℓ α 1 δ ) 3 + C e − N 2 ε / C . We now bound the contribution of B K B_{\mathcal{K}} B K B K B_{\mathcal{K}} B K gap ( B K ) \operatorname{gap}(B_{\mathcal{K}}) gap ( B K ) f p 0 f_{p_0} f p 0 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) k ≥ 0 k\ge0 k ≥ 0 gap ( B K ) ≤ 2 k d 2 \operatorname{gap}(B_{\mathcal{K}})\le2^kd_2 gap ( B K ) ≤ 2 k d 2 f p 0 ( gap ( B K ) / d 2 ) ≤ 4 p 0 2 − 2 p 0 k f_{p_0}(\operatorname{gap}(B_{\mathcal{K}})/d_2)\le4^{p_0}2^{-2p_0k} f p 0 ( gap ( B K ) / d 2 ) ≤ 4 p 0 2 − 2 p 0 k
E [ f p 0 ( gap ( B K ) d 2 ) | H ] ≤ C N 4 ε ( d 2 ℓ α 1 ) 3 ∑ k ≥ 0 2 ( 3 − 2 p 0 ) k + C ( d 2 σ ) 2 p 0 + C e − N 2 ε / C log N ≤ C N 4 ε ( d 2 ℓ α 1 ) 3 , \begin{aligned}
\mathbb{E}\left[f_{p_0}\left(\frac{\operatorname{gap}(B_{\mathcal{K}})}{d_2}\right)\middle|H\right]\le CN^{4\varepsilon}\left(\frac{d_2}{\ell_{\alpha_1}}\right)^3\sum_{k\ge0}2^{(3-2p_0)k}+C\left(\frac{d_2}{\sigma}\right)^{2p_0}+Ce^{-N^{2\varepsilon}/C}\log N \\
&\le CN^{4\varepsilon}\left(\frac{d_2}{\ell_{\alpha_1}}\right)^3,
\end{aligned} E [ f p 0 ( d 2 gap ( B K ) ) H ] ≤ C N 4 ε ( ℓ α 1 d 2 ) 3 k ≥ 0 ∑ 2 ( 3 − 2 p 0 ) k + C ( σ d 2 ) 2 p 0 + C e − N 2 ε / C log N ≤ C N 4 ε ( ℓ α 1 d 2 ) 3 , where the second term bounds the contribution of the k k k 2 k d 2 > σ 2^kd_2>\sigma 2 k d 2 > σ O ( log N ) O(\log N) O ( log N ) k k k 2 k d 2 ≤ σ 2^kd_2\le\sigma 2 k d 2 ≤ σ H t α 1 H_{t\alpha_1} H t α 1 K \mathcal{K} K B K B_{\mathcal{K}} B K o ( d 2 ) o(d_2) o ( d 2 ) ∑ i < i ′ Q ( ( λ i ′ − λ i ) / d 2 , ( λ i + λ i ′ ) / 2 ) \sum_{i<i'} Q((\lambda_{i'}-\lambda_i)/d_2,(\lambda_i+\lambda_{i'})/2) ∑ i < i ′ Q (( λ i ′ − λ i ) / d 2 , ( λ i + λ i ′ ) /2 ) C f p 0 ( gap ( B K ) / d 2 ) C f_{p_0}(\operatorname{gap}(B_{\mathcal{K}})/d_2) C f p 0 ( gap ( B K ) / d 2 ) N 1 − 8 κ 1 + ε N^{1-8\kappa_1+\varepsilon} N 1 − 8 κ 1 + ε x 0 d 2 x_0d_2 x 0 d 2
C N 1 − 8 κ 1 + 5 ε ( d 2 ℓ α 1 ) 3 = C N 1 − 8 κ 1 + 5 ε N − 1 + 3 α 1 = C N 3 α 1 − 8 κ 1 + 5 ε → 0 , (423) CN^{1-8\kappa_1+5\varepsilon}\left(\frac{d_2}{\ell_{\alpha_1}}\right)^3
= CN^{1-8\kappa_1+5\varepsilon}N^{-1+3\alpha_1}
= CN^{3\alpha_1-8\kappa_1+5\varepsilon}\to0, \tag*{(423)} C N 1 − 8 κ 1 + 5 ε ( ℓ α 1 d 2 ) 3 = C N 1 − 8 κ 1 + 5 ε N − 1 + 3 α 1 = C N 3 α 1 − 8 κ 1 + 5 ε → 0 , (423) since 5 ε < 8 κ 1 − 3 α 1 5\varepsilon<8\kappa_1-3\alpha_1 5 ε < 8 κ 1 − 3 α 1
The remaining pairs. With probability tending to one, every cluster containing an eigenvalue in I ′ I' I ′ H t α 1 H_{t\alpha_1} H t α 1 q / 2 q/2 q /2 p 0 ≥ p ∗ p_0\ge p_* p 0 ≥ p ∗
C N 2 ( 2 d 2 q ) 2 p 0 = C N 2 − 2 p 0 ( 1 / 3 − κ 1 ) = o ( 1 ) . CN^2\left(\frac{2d_2}{q}\right)^{2p_0}
= CN^{2-2p_0(1/3-\kappa_1)}
= o(1). C N 2 ( q 2 d 2 ) 2 p 0 = C N 2 − 2 p 0 ( 1/3 − κ 1 ) = o ( 1 ) . Pairs whose midpoint lies outside I ′ I' I ′ { i , i + 1 } \{i,i+1\} { i , i + 1 } H t α 1 H_{t\alpha_1} H t α 1 g ^ i \widehat g_i g i o ( d 2 ) o(d_2) o ( d 2 ) o ( 1 ) o(1) o ( 1 ) E i E_i E i C N ε ℓ α 1 CN^{\varepsilon}\ell^{\alpha_1} C N ε ℓ α 1 φ j \varphi_j φ j C N ε ℓ α 1 CN^{\varepsilon}\ell^{\alpha_1} C N ε ℓ α 1 ∑ i f p 0 ( g ^ i / d 2 ) \sum_i f_{p_0}(\widehat g_i/d_2) ∑ i f p 0 ( g i / d 2 ) I ′ I' I ′ O P ( 1 ) O_{\mathrm P}(1) O P ( 1 ) Lemma 3.10 and the upper bound (6.20) in Lemma 6.6, so these changes contribute o P ( 1 ) o_{\mathrm P}(1) o P ( 1 )
Combining these facts with the two preceding paragraphs and using the bounded convergence theorem, we conclude that, for every Q Q Q
E exp ( − ∑ i < i ′ Q ( λ i ′ ′ ( H t α 1 ) − λ i ( H t α 1 ) d 2 , λ i ( H t α 1 ) + λ i ′ ′ ( H t α 1 ) 2 ) ) \mathbb{E}\exp\left(-\sum_{i<i'}Q\left(\frac{\lambda_{i'}'(H_{t\alpha_1})-\lambda_i(H_{t\alpha_1})}{d_2},\frac{\lambda_i(H_{t\alpha_1})+\lambda_{i'}'(H_{t\alpha_1})}{2}\right)\right) E exp ( − i < i ′ ∑ Q ( d 2 λ i ′ ′ ( H t α 1 ) − λ i ( H t α 1 ) , 2 λ i ( H t α 1 ) + λ i ′ ′ ( H t α 1 ) ) ) ⟶ exp ( − ∫ − 2 2 ∫ 0 ∞ c 2 ( E ) y 2 ( 1 − e − Q ( y , E ) ) d y d E ) . (424) \longrightarrow\exp\left(-\int_{-2}^{2}\int_{0}^{\infty}c_2(E)y^2\left(1-e^{-Q(y,E)}\right)\,dy\,dE\right). \tag*{(424)} ⟶ exp ( − ∫ − 2 2 ∫ 0 ∞ c 2 ( E ) y 2 ( 1 − e − Q ( y , E ) ) d y d E ) . (424) The same argument shows that the statistic
∑ i < i ′ χ ( λ i + λ i ′ ′ 2 ) f p ∗ ( λ i ′ ′ − λ i d 2 ) , λ i = λ i ( H t α 1 ) , \sum_{i<i'}\chi\left(\frac{\lambda_i+\lambda_{i'}'}{2}\right)f_{p_*}\left(\frac{\lambda_{i'}'-\lambda_i}{d_2}\right),\qquad\lambda_i=\lambda_i(H_{t\alpha_1}), i < i ′ ∑ χ ( 2 λ i + λ i ′ ′ ) f p ∗ ( d 2 λ i ′ ′ − λ i ) , λ i = λ i ( H t α 1 ) , is tight for every nonnegative χ ∈ C ∞ ( ( − 2 , 2 ) ) \chi\in C^\infty((-2,2)) χ ∈ C ∞ (( − 2 , 2 )) Lemma 6.2 at the scale d 2 = ℓ 1 / 3 d_2=\ell^{1/3} d 2 = ℓ 1/3 φ j \varphi_j φ j φ j , N \varphi_{j,N} φ j , N ∣ e − a − e − b ∣ ≤ ∣ a − b ∣ |e^{-a}-e^{-b}|\leq|a-b| ∣ e − a − e − b ∣ ≤ ∣ a − b ∣ a , b ≥ 0 a,b\geq0 a , b ≥ 0
Return to H H H The observable on the left side of (9.36), with smoothed weights, is of the form considered in Definition 5.4 at the scale d 2 = ℓ 1 / 3 d_2=\ell^{1/3} d 2 = ℓ 1/3 α = 1 / 3 \alpha=1/3 α = 1/3 t = t α 1 t=t_{\alpha_1} t = t α 1 C N 1 / 2 − 2 α 1 + 12 ε = C N − 7 / 50 + 12 ε = o ( 1 ) CN^{1/2-2\alpha_1+12\varepsilon}=CN^{-7/50+12\varepsilon}=o(1) C N 1/2 − 2 α 1 + 12 ε = C N − 7/50 + 12 ε = o ( 1 ) α 1 > 1 / 4 \alpha_1>1/4 α 1 > 1/4 H H H H t α 1 H_{t_{\alpha_1}} H t α 1
Next, we show that close triples are absent for H H H d 2 d_2 d 2 U d 2 ( 3 ) U_{d_2}^{(3)} U d 2 ( 3 ) d 2 d_2 d 2 q q q p = p ∗ p=p_* p = p ∗ χ ∈ C ∞ ( ( − 2 , 2 ) ) \chi\in C^\infty((-2,2)) χ ∈ C ∞ (( − 2 , 2 )) I ′ I' I ′ χ ≥ 2 \chi\geq2 χ ≥ 2 I I I η χ > 0 \eta_\chi>0 η χ > 0 2 η χ 2\eta_\chi 2 η χ supp χ \operatorname{supp}\chi supp χ I ′ I' I ′ U d 2 ( 3 ) ( H t α 1 ) → 0 U_{d_2}^{(3)}(H_{t_{\alpha_1}})\to0 U d 2 ( 3 ) ( H t α 1 ) → 0 H t α 1 H_{t_{\alpha_1}} H t α 1 C f p ∗ ( gap ( B R ) / d 2 ) C f_{p_*}(\operatorname{gap}(B_{\mathcal{R}})/d_2) C f p ∗ ( gap ( B R ) / d 2 ) q / 2 q/2 q /2 C N 3 ( 2 d 2 / q ) 2 p ∗ = o ( 1 ) CN^3(2d_2/q)^{2p_*}=o(1) C N 3 ( 2 d 2 / q ) 2 p ∗ = o ( 1 ) I ′ I' I ′ η χ \eta_\chi η χ supp χ \operatorname{supp}\chi supp χ C N 3 exp ( − η χ 2 N 1 / 5 / 2 ) = o ( 1 ) CN^3\exp(-\eta_\chi^2N^{1/5}/2)=o(1) C N 3 exp ( − η χ 2 N 1/5 /2 ) = o ( 1 ) U d 2 ( 3 ) ( H t α 1 ) → 0 U_{d_2}^{(3)}(H_{t_{\alpha_1}})\to0 U d 2 ( 3 ) ( H t α 1 ) → 0 E e − U d 2 ( 3 ) ( H t α 1 ) → 1 \mathbb{E}e^{-U_{d_2}^{(3)}(H_{t_{\alpha_1}})}\to1 E e − U d 2 ( 3 ) ( H t α 1 ) → 1
We now transfer this to H H H r = 3 r=3 r = 3 d 2 d_2 d 2 q q q κ = 1 / 3 \kappa=1/3 κ = 1/3 θ = M = 1 \theta=M=1 θ = M = 1 t = t α 1 t=t_{\alpha_1} t = t α 1 E e − U d 2 ( 3 ) ( H ) → 1 \mathbb{E}e^{-U_{d_2}^{(3)}(H)}\to1 E e − U d 2 ( 3 ) ( H ) → 1 U d 2 ( 3 ) ( H ) → 0 U_{d_2}^{(3)}(H)\to0 U d 2 ( 3 ) ( H ) → 0 λ i ∈ I \lambda_i\in I λ i ∈ I λ i + 2 − λ i ≤ x 0 d 2 \lambda_{i+2}-\lambda_i\leq x_0d_2 λ i + 2 − λ i ≤ x 0 d 2 ( i , i + 1 , i + 2 ) (i,i+1,i+2) ( i , i + 1 , i + 2 ) U d 2 ( 3 ) ( H ) U_{d_2}^{(3)}(H) U d 2 ( 3 ) ( H ) ( 1 + x 0 2 ) − 3 p ∗ (1+x_0^2)^{-3p_*} ( 1 + x 0 2 ) − 3 p ∗ N ≥ N 0 N\geq N_0 N ≥ N 0 i i i λ i ∈ I \lambda_i\in I λ i ∈ I λ i + 2 − λ i ≤ x 0 d 2 \lambda_{i+2}-\lambda_i\leq x_0d_2 λ i + 2 − λ i ≤ x 0 d 2 i < i ′ i<i' i < i ′ λ i ∈ I \lambda_i\in I λ i ∈ I λ i ′ − λ i ≤ x 0 d 2 \lambda_{i'}-\lambda_i\leq x_0d_2 λ i ′ − λ i ≤ x 0 d 2
Conclusion. We have shown that (9.36) holds for H H H Q = θ χ f p ∗ Q=\theta\chi f_{p_*} Q = θ χ f p ∗ θ → 0 \theta\to0 θ → 0
∑ i < i ′ χ N ( λ i + λ i ′ 2 ) f p ∗ ( λ i ′ − λ i d 2 ) \sum_{i<i'}\chi_N\left(\frac{\lambda_i+\lambda_{i'}}{2}\right)f_{p_*}\left(\frac{\lambda_{i'}-\lambda_i}{d_2}\right) i < i ′ ∑ χ N ( 2 λ i + λ i ′ ) f p ∗ ( d 2 λ i ′ − λ i ) are tight, and Lemma 5.1 then allows us to remove the smoothing of the weights as in the proof of Lemma 6.6. Since the Laplace transform determines the law of a random vector with nonnegative coordinates, it follows that for all nonnegative φ 1 , … , φ n ∈ C c ∞ ( ( − 2 , 2 ) ) \varphi_1,\ldots,\varphi_n\in C_c^\infty((-2,2)) φ 1 , … , φ n ∈ C c ∞ (( − 2 , 2 )) p 1 , … , p n ≥ p ∗ p_1,\ldots,p_n\geq p_* p 1 , … , p n ≥ p ∗
∑ i < i ′ φ j ( λ i + λ i ′ 2 ) f p j ( λ i ′ − λ i d 2 ) , j ∈ [ n ] , \sum_{i<i'}\varphi_j\left(\frac{\lambda_i+\lambda_{i'}}{2}\right)f_{p_j}\left(\frac{\lambda_{i'}-\lambda_i}{d_2}\right),\qquad j\in[n], i < i ′ ∑ φ j ( 2 λ i + λ i ′ ) f p j ( d 2 λ i ′ − λ i ) , j ∈ [ n ] , converges jointly in law to the vector of integrals of the functions ( y , E ) ↦ φ j ( E ) f p j ( y ) (y,E)\mapsto\varphi_j(E)f_{p_j}(y) ( y , E ) ↦ φ j ( E ) f p j ( y ) Π \Pi Π [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) c 2 ( E ) y 2 d y d E c_2(E)y^2\,dy\,dE c 2 ( E ) y 2 d y d E
We now pass to general test functions. Every continuous function f f f [ 0 , ∞ ) × ( − 2 , 2 ) [0,\infty)\times(-2,2) [ 0 , ∞ ) × ( − 2 , 2 ) ω f p ∗ ( y ) 1 I ′ ( E ) \omega f_{p_*}(y)\mathbf{1}_{I'}(E) ω f p ∗ ( y ) 1 I ′ ( E ) ( 1 + y 2 ) − 1 (1+y^2)^{-1} ( 1 + y 2 ) − 1 ∑ i < i ′ χ ( ( λ i + λ i ′ ) / 2 ) f p ∗ ( ( λ i ′ − λ i ) / d 2 ) \sum_{i<i'}\chi((\lambda_i+\lambda_{i'})/2)f_{p_*}((\lambda_{i'}-\lambda_i)/d_2) ∑ i < i ′ χ (( λ i + λ i ′ ) /2 ) f p ∗ (( λ i ′ − λ i ) / d 2 ) ∑ i < i ′ f ( ( λ i ′ − λ i ) / d 2 , ( λ i + λ i ′ ) / 2 ) \sum_{i<i'}f((\lambda_{i'}-\lambda_i)/d_2,(\lambda_i+\lambda_{i'})/2) ∑ i < i ′ f (( λ i ′ − λ i ) / d 2 , ( λ i + λ i ′ ) /2 ) ∫ f d Π \int f\,d\Pi ∫ f d Π ∑ i < i ′ δ ( ( λ i ′ − λ i ) / d 2 , ( λ i + λ i ′ ) / 2 ) \sum_{i<i'}\delta((\lambda_{i'}-\lambda_i)/d_2,(\lambda_i+\lambda_{i'})/2) ∑ i < i ′ δ (( λ i ′ − λ i ) / d 2 , ( λ i + λ i ′ ) /2 ) Π \Pi Π
Finally, we pass to consecutive eigenvalues. By the absence of close triples for H H H d 2 d_2 d 2 I I I E E E λ i ′ − λ i ≤ x 0 d 2 \lambda_{i'}-\lambda_i\leq x_0d_2 λ i ′ − λ i ≤ x 0 d 2 O ( d 2 ) O(d_2) O ( d 2 ) (8) . The limit (9) follows from Lemma 4.12, whose proof applies with Π \Pi Π c 2 ( E ) y 2 d y d E c_2(E)y^2\,dy\,dE c 2 ( E ) y 2 d y d E Ξ \Xi Ξ ν \nu ν ∫ 0 x c 2 ( E ) y 2 d y = π 2 ρ s c ( E ) 4 x 3 / 9 \int_0^x c_2(E)y^2\,dy=\pi^2\rho_{\mathrm{sc}}(E)^4x^3/9 ∫ 0 x c 2 ( E ) y 2 d y = π 2 ρ sc ( E ) 4 x 3 /9
Pair statistics of the Gaussian orthogonal ensemble
In this appendix, we prove Proposition 3.6 . Throughout, we fix α ∈ ( 0 , 1 / 4 ) \alpha\in(0,1/4) α ∈ ( 0 , 1/4 ) p > 1 p > 1 p > 1 f f f Proposition 3.6 , together with the bulk interval I I I C f C_f C f
We follow the approach of Feng, Li, and Yao [33 ] , who treat the Gaussian symplectic ensemble in detail and sketch the orthogonal case. The correlation functions of the GOE are Pfaffians of a 2 × 2 2 \times2 2 × 2 (51) is an integral of the two-point function, and the variance is controlled by the two-, three-, and four-point functions. However, the GOE analogues of the kernel bounds in [33 Lemma 3.2] are asserted in [33 Section 7.1] without proof, and the expansion of the two-point function that we need is not stated there, so we give complete proofs. They rest on elementary estimates for Hermite functions, proved in Appendix A.1, and on the density of states, which we deduce from eigenvalue rigidity in Lemma A.7 . The main step is the exact identity for the slope of the two-point function at the diagonal in Lemma A.8 ; together with the density of states, it determines the two-point function at separations below N − 1 N^{-1} N − 1 Lemma A.9 , which controls the variance.
In this appendix only, we abbreviate λ i = λ i ( G ) \lambda_i = \lambda_i(G) λ i = λ i ( G ) δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) I δ = [ − 2 + δ , 2 − δ ] I_\delta= [-2+\delta,2-\delta] I δ = [ − 2 + δ , 2 − δ ] C , c > 0 C,c > 0 C , c > 0 N 0 N_0 N 0 δ \delta δ α \alpha α p p p I I I f f f N N N
Hermite functions
Let He n \operatorname{He}_n He n e − w 2 / 2 e^{-w^2/2} e − w 2 /2
∑ n = 0 ∞ He n ( w ) ζ n n ! = e w ζ − ζ 2 / 2 , w , ζ ∈ R . (425) \sum_{n=0}^{\infty} \frac{\operatorname{He}_n(w)\zeta^n}{n!} = e^{w\zeta-\zeta^2/2}, \qquad w,\zeta\in\mathbb{R}. \tag*{(425)} n = 0 ∑ ∞ n ! He n ( w ) ζ n = e w ζ − ζ 2 /2 , w , ζ ∈ R . (425) For integers n ≥ 0 n \ge0 n ≥ 0
h n ( w ) = ( 2 π ) − 1 / 4 ( n ! ) − 1 / 2 He n ( w ) e − w 2 / 4 , w ∈ R , (426) h_n(w) = (2\pi)^{-1/4}(n!)^{-1/2}\operatorname{He}_n(w)e^{-w^2/4}, \qquad w \in\mathbb{R}, \tag*{(426)} h n ( w ) = ( 2 π ) − 1/4 ( n ! ) − 1/2 He n ( w ) e − w 2 /4 , w ∈ R , (426) and we set h − 1 = 0 h_{-1}=0 h − 1 = 0 h n h_n h n L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) [2 Section 3.2] ), and h n ( − w ) = ( − 1 ) n h n ( w ) h_n(-w)=(-1)^n h_n(w) h n ( − w ) = ( − 1 ) n h n ( w ) w w w ζ \zeta ζ H e n ′ = n H e n − 1 He_n' = nHe_{n-1} H e n ′ = n H e n − 1 w H e n = H e n + 1 + n H e n − 1 wHe_n = He_{n+1} + nHe_{n-1} w H e n = H e n + 1 + n H e n − 1 H e n ′ ′ − w H e n ′ + n H e n = 0 He_n'' - wHe_n' + nHe_n = 0 H e n ′′ − w H e n ′ + n H e n = 0
e n ( w ) = n + 1 2 − w 2 4 , n ≥ 0 , w ∈ R . (427) e_n(w) = n + \frac{1}{2} - \frac{w^2}{4}, \qquad n \ge0,\ w \in\mathbb{R}. \tag*{(427)} e n ( w ) = n + 2 1 − 4 w 2 , n ≥ 0 , w ∈ R . (427) In terms of the Hermite functions, the three relations above become
w h n = n + 1 h n + 1 + n h n − 1 , h n ′ = n h n − 1 − n + 1 h n + 1 2 , h n ′ ′ = − e n h n , (428) wh_n = \sqrt{n+1}\,h_{n+1} + \sqrt{n}\,h_{n-1}, \qquad h_n' = \frac{\sqrt{n}\,h_{n-1} - \sqrt{n+1}\,h_{n+1}}{2}, \qquad h_n'' = -e_nh_n, \tag*{(428)} w h n = n + 1 h n + 1 + n h n − 1 , h n ′ = 2 n h n − 1 − n + 1 h n + 1 , h n ′′ = − e n h n , (428) for all n ≥ 0 n \ge0 n ≥ 0 w , w ′ ∈ R w,w' \in\mathbb{R} w , w ′ ∈ R j ≥ 0 j \ge0 j ≥ 0
( w − w ′ ) h j ( w ) h j ( w ′ ) = j + 1 ( h j + 1 ( w ) h j ( w ′ ) − h j ( w ) h j + 1 ( w ′ ) ) − j ( h j ( w ) h j − 1 ( w ′ ) − h j − 1 ( w ) h j ( w ′ ) ) . (w-w')h_j(w)h_j(w') = \sqrt{j+1}\left(h_{j+1}(w)h_j(w') - h_j(w)h_{j+1}(w')\right) - \sqrt{j}\left(h_j(w)h_{j-1}(w') - h_{j-1}(w)h_j(w')\right). ( w − w ′ ) h j ( w ) h j ( w ′ ) = j + 1 ( h j + 1 ( w ) h j ( w ′ ) − h j ( w ) h j + 1 ( w ′ ) ) − j ( h j ( w ) h j − 1 ( w ′ ) − h j − 1 ( w ) h j ( w ′ ) ) . Summing over 0 ≤ j ≤ n − 1 0 \le j \le n-1 0 ≤ j ≤ n − 1
∑ j = 0 n − 1 h j ( w ) h j ( w ′ ) = n h n ( w ) h n − 1 ( w ′ ) − h n − 1 ( w ) h n ( w ′ ) w − w ′ , n ≥ 1 , w ≠ w ′ . (429) \sum_{j=0}^{n-1} h_j(w)h_j(w') = \sqrt{n}\,\frac{h_n(w)h_{n-1}(w') - h_{n-1}(w)h_n(w')}{w-w'}, \qquad n \ge1,\ w \ne w'. \tag*{(429)} j = 0 ∑ n − 1 h j ( w ) h j ( w ′ ) = n w − w ′ h n ( w ) h n − 1 ( w ′ ) − h n − 1 ( w ) h n ( w ′ ) , n ≥ 1 , w = w ′ . (429) Letting w ′ → w w' \to w w ′ → w
∑ j = 0 n − 1 h j ( w ) 2 = n ( h n ′ ( w ) h n − 1 ( w ) − h n − 1 ′ ( w ) h n ( w ) ) . (430) \sum_{j=0}^{n-1} h_j(w)^2 = \sqrt{n}\left(h_n'(w)h_{n-1}(w) - h_{n-1}'(w)h_n(w)\right). \tag*{(430)} j = 0 ∑ n − 1 h j ( w ) 2 = n ( h n ′ ( w ) h n − 1 ( w ) − h n − 1 ′ ( w ) h n ( w ) ) . (430) For integers n ≥ 0 n \ge0 n ≥ 0 w ∈ R w \in\mathbb{R} w ∈ R
F n ( w ) = 1 2 ∫ R sgn ( w − w ′ ) h n ( w ′ ) d w ′ , (431) F_n(w) = \frac{1}{2}\int_{\mathbb{R}} \operatorname{sgn}(w-w')h_n(w')\,\mathrm{d}w', \tag*{(431)} F n ( w ) = 2 1 ∫ R sgn ( w − w ′ ) h n ( w ′ ) d w ′ , (431) and we set F − 1 = 0 F_{-1}=0 F − 1 = 0 F n ′ = h n F_n'=h_n F n ′ = h n h n ′ h_n' h n ′ h n ( w ) → 0 h_n(w) \to0 h n ( w ) → 0 ∣ w ∣ → ∞ |w| \to\infty ∣ w ∣ → ∞
1 2 ∫ R sgn ( w − w ′ ) h n ′ ( w ′ ) d w ′ = h n ( w ) , w ∈ R . \frac{1}{2}\int_{\mathbb{R}} \operatorname{sgn}(w-w')h_n'(w')\,\mathrm{d}w' = h_n(w), \qquad w \in\mathbb{R}. 2 1 ∫ R sgn ( w − w ′ ) h n ′ ( w ′ ) d w ′ = h n ( w ) , w ∈ R . Inserting the second identity in (A.4) into the left side, we obtain
h n = n F n − 1 − n + 1 F n + 1 2 , n ≥ 0. (432) h_n = \frac{\sqrt{n}F_{n-1} - \sqrt{n+1}F_{n+1}}{2}, \qquad n \ge0. \tag*{(432)} h n = 2 n F n − 1 − n + 1 F n + 1 , n ≥ 0. (432) We will need the following estimates.
Lemma A.1. Let δ ∈ ( 0 , 1 ] \delta\in(0,1] δ ∈ ( 0 , 1 ] C > 0 C>0 C > 0 δ \delta δ n ≥ 1 n \ge1 n ≥ 1
(a) For ∣ w ∣ ≤ ( 2 − δ ) n |w| \le(2-\delta)\sqrt{n} ∣ w ∣ ≤ ( 2 − δ ) n k ∈ { 0 , 1 , 2 , 3 } k \in\{0,1,2,3\} k ∈ { 0 , 1 , 2 , 3 } ∣ h n ( k ) ( w ) ∣ ≤ C n k / 2 − 1 / 4 |h_n^{(k)}(w)| \le Cn^{k/2-1/4} ∣ h n ( k ) ( w ) ∣ ≤ C n k /2 − 1/4
(b) For ∣ w ∣ , ∣ w ′ ∣ ≤ ( 2 − δ ) n |w|,|w'|\le(2-\delta)\sqrt{n} ∣ w ∣ , ∣ w ′ ∣ ≤ ( 2 − δ ) n ∣ ∫ w w ′ h n ′ ∣ ≤ C n − 3 / 4 \left|\int_w^{w'}h_n'\right|\le Cn^{-3/4} ∫ w w ′ h n ′ ≤ C n − 3/4
(c) If n n n C − 1 n − 1 / 4 ≤ ∫ R h n ≤ C n − 1 / 4 C^{-1}n^{-1/4}\le\int_{\mathbb{R}}h_n\le Cn^{-1/4} C − 1 n − 1/4 ≤ ∫ R h n ≤ C n − 1/4 n n n ∫ 0 ∞ h n ≤ C n − 1 / 4 \int_0^\infty h_n\le Cn^{-1/4} ∫ 0 ∞ h n ≤ C n − 1/4
(d) For ∣ w ∣ ≤ ( 2 − δ ) n |w|\le(2-\delta)\sqrt{n} ∣ w ∣ ≤ ( 2 − δ ) n ∣ F n ( w ) ∣ ≤ C n − 1 / 4 |F_n(w)|\le Cn^{-1/4} ∣ F n ( w ) ∣ ≤ C n − 1/4
Proof. We begin with (a) for k ∈ { 0 , 1 } k\in\{0,1\} k ∈ { 0 , 1 } [68 Section 7.31] ). On the interval [ 0 , ( 4 n + 2 ) 1 / 2 ) [0,(4n+2)^{1/2}) [ 0 , ( 4 n + 2 ) 1/2 ) e n > 0 e_n>0 e n > 0
Θ ( w ) = h n ( w ) 2 + h n ′ ( w ) 2 e n ( w ) . \Theta(w)=h_n(w)^2+\frac{h_n'(w)^2}{e_n(w)}. Θ ( w ) = h n ( w ) 2 + e n ( w ) h n ′ ( w ) 2 . Using h n ′ ′ = − e n h n h_n''=-e_nh_n h n ′′ = − e n h n e n ′ = − w / 2 e_n'=-w/2 e n ′ = − w /2
Θ ′ = 2 h n h n ′ + 2 h n ′ h n ′ ′ e n − e n ′ ( h n ′ ) 2 e n 2 = w ( h n ′ ) 2 2 e n 2 ≥ 0. \Theta'=2h_nh_n'+\frac{2h_n'h_n''}{e_n}-\frac{e_n'(h_n')^2}{e_n^2}=\frac{w(h_n')^2}{2e_n^2}\ge0. Θ ′ = 2 h n h n ′ + e n 2 h n ′ h n ′′ − e n 2 e n ′ ( h n ′ ) 2 = 2 e n 2 w ( h n ′ ) 2 ≥ 0. Hence Θ \Theta Θ [ 0 , ( 4 n + 2 ) 1 / 2 ) [0,(4n+2)^{1/2}) [ 0 , ( 4 n + 2 ) 1/2 )
Set w 1 = ( 2 − δ ) n w_1=(2-\delta)\sqrt{n} w 1 = ( 2 − δ ) n w 2 = ( 2 − δ / 2 ) n w_2=(2-\delta/2)\sqrt{n} w 2 = ( 2 − δ /2 ) n w 2 < ( 4 n + 2 ) 1 / 2 w_2<(4n+2)^{1/2} w 2 < ( 4 n + 2 ) 1/2
e n ( w ) ≥ n ( 1 − ( 1 − δ / 4 ) 2 ) ≥ δ n 4 , w ∈ [ w 1 , w 2 ] . e_n(w)\ge n\left(1-(1-\delta/4)^2\right)\ge\frac{\delta n}{4},\qquad w\in[w_1,w_2]. e n ( w ) ≥ n ( 1 − ( 1 − δ /4 ) 2 ) ≥ 4 δ n , w ∈ [ w 1 , w 2 ] . Let 0 ≤ w ≤ w 1 0\le w\le w_1 0 ≤ w ≤ w 1 Θ \Theta Θ Θ ( w ) ≤ Θ ( w ′ ) \Theta(w)\le\Theta(w') Θ ( w ) ≤ Θ ( w ′ ) w ′ ∈ [ w 1 , w 2 ] w'\in[w_1,w_2] w ′ ∈ [ w 1 , w 2 ] w ′ w' w ′
Θ ( w ) ≤ 1 w 2 − w 1 ∫ w 1 w 2 Θ ( w ′ ) d w ′ ≤ 2 δ n ( ∫ R h n 2 + 4 δ n ∫ R ( h n ′ ) 2 ) = 2 δ n ( 1 + 2 n + 1 δ n ) ≤ 8 δ 2 n , \Theta(w)\le\frac{1}{w_2-w_1}\int_{w_1}^{w_2}\Theta(w')\,dw'\le\frac{2}{\delta\sqrt{n}}\left(\int_{\mathbb{R}}h_n^2+\frac{4}{\delta n}\int_{\mathbb{R}}(h_n')^2\right)
=\frac{2}{\delta\sqrt{n}}\left(1+\frac{2n+1}{\delta n}\right)\le\frac{8}{\delta^2\sqrt{n}}, Θ ( w ) ≤ w 2 − w 1 1 ∫ w 1 w 2 Θ ( w ′ ) d w ′ ≤ δ n 2 ( ∫ R h n 2 + δ n 4 ∫ R ( h n ′ ) 2 ) = δ n 2 ( 1 + δ n 2 n + 1 ) ≤ δ 2 n 8 , where in the equality we used the identity
∫ R ( h n ′ ) 2 = 2 n + 1 4 , \int_{\mathbb{R}}(h_n')^2=\frac{2n+1}{4}, ∫ R ( h n ′ ) 2 = 4 2 n + 1 , which follows from the second identity in (A.4) and orthonormality. Consequently,
h n ( w ) 2 ≤ Θ ( w ) ≤ 8 δ 2 n , h n ′ ( w ) 2 ≤ e n ( w ) Θ ( w ) ≤ 8 ( n + 1 ) δ 2 n . h_n(w)^2\le\Theta(w)\le\frac{8}{\delta^2\sqrt{n}},\qquad h_n'(w)^2\le e_n(w)\Theta(w)\le\frac{8(n+1)}{\delta^2\sqrt{n}}. h n ( w ) 2 ≤ Θ ( w ) ≤ δ 2 n 8 , h n ′ ( w ) 2 ≤ e n ( w ) Θ ( w ) ≤ δ 2 n 8 ( n + 1 ) . The same bounds hold for − w 1 ≤ w ≤ 0 -w_1\le w\le0 − w 1 ≤ w ≤ 0 h n h_n h n k ∈ { 0 , 1 } k\in\{0,1\} k ∈ { 0 , 1 }
The cases k ∈ { 2 , 3 } k\in\{2,3\} k ∈ { 2 , 3 } h n ′ ′ = − e n h n h_n''=-e_nh_n h n ′′ = − e n h n
h_n'''=(w/2)h_n-e_nh_n', \qquad{#eq:A.9}
together with ∣ e n ( w ) ∣ ≤ n + 1 |e_n(w)|\le n+1 ∣ e n ( w ) ∣ ≤ n + 1 ∣ w ∣ ≤ 2 n |w|\le2\sqrt{n} ∣ w ∣ ≤ 2 n
e n ( w ) ≥ n ( 1 − ( 1 − δ / 2 ) 2 ) ≥ δ n 2 , ∣ w ∣ ≤ ( 2 − δ ) n . e_n(w) \ge n\left(1-\left(1-\delta/2\right)^2\right) \ge\frac{\delta n}{2}, \qquad|w| \le(2-\delta)\sqrt{n}. e n ( w ) ≥ n ( 1 − ( 1 − δ /2 ) 2 ) ≥ 2 δ n , ∣ w ∣ ≤ ( 2 − δ ) n . On this interval, h n = − h n ′ ′ / e n h_n=-h_n''/e_n h n = − h n ′′ / e n
∫ w w ′ h n = − [ h n ′ e n − ( 1 e n ) ′ h n ] w w ′ − ∫ w w ′ ( 1 e n ) ′ ′ h n . \int_w^{w'} h_n=-\left[\frac{h_n'}{e_n}-\left(\frac{1}{e_n}\right)'h_n\right]_w^{w'}-\int_w^{w'}\left(\frac{1}{e_n}\right)''h_n. ∫ w w ′ h n = − [ e n h n ′ − ( e n 1 ) ′ h n ] w w ′ − ∫ w w ′ ( e n 1 ) ′′ h n . We have
( 1 e n ) ′ = w 2 e n 2 , ( 1 e n ) ′ ′ = 1 2 e n 2 + w 2 2 e n 3 . \left(\frac{1}{e_n}\right)'=\frac{w}{2e_n^2}, \qquad\left(\frac{1}{e_n}\right)''=\frac{1}{2e_n^2}+\frac{w^2}{2e_n^3}. ( e n 1 ) ′ = 2 e n 2 w , ( e n 1 ) ′′ = 2 e n 2 1 + 2 e n 3 w 2 . so 1 / e n 1/e_n 1/ e n ( 1 / e n ) ′ (1/e_n)' ( 1/ e n ) ′ ( 1 / e n ) ′ ′ (1/e_n)'' ( 1/ e n ) ′′ C n − 1 Cn^{-1} C n − 1 C n − 3 / 2 Cn^{-3/2} C n − 3/2 C n − 2 Cn^{-2} C n − 2 ∣ w ′ − w ∣ ≤ 4 n |w'-w|\le4\sqrt{n} ∣ w ′ − w ∣ ≤ 4 n
∣ ∫ w w ′ h n ∣ ≤ C ( n − 1 n 1 / 4 + n − 3 / 2 n − 1 / 4 + n n − 2 n − 1 / 4 ) ≤ C n − 3 / 4 . \left|\int_w^{w'}h_n\right|\le C\left(n^{-1}n^{1/4}+n^{-3/2}n^{-1/4}+\sqrt{n}\,n^{-2}n^{-1/4}\right)\le Cn^{-3/4}. ∫ w w ′ h n ≤ C ( n − 1 n 1/4 + n − 3/2 n − 1/4 + n n − 2 n − 1/4 ) ≤ C n − 3/4 . We now prove (c). In the computations below, the exchange of sums and integrals is justified by absolute convergence.
Suppose first that n = 2 l n=2l n = 2 l l ≥ 1 l\ge1 l ≥ 1 e − w 2 / 4 e^{-w^2/4} e − w 2 /4 w ∈ R w\in\mathbb{R} w ∈ R
∑ n ′ = 0 ∞ ζ n ′ n ′ ! ∫ R e − w 2 / 4 He n ′ ′ ( w ) d w = ∫ R e − ( w − 2 ζ ) 2 / 4 + ζ 2 / 2 d w = 2 π e ζ 2 / 2 . \sum_{n'=0}^{\infty}\frac{\zeta^{n'}}{n'!}\int_{\mathbb{R}}e^{-w^2/4}\operatorname{He}_{n'}'(w)\,dw
=\int_{\mathbb{R}}e^{-(w-2\zeta)^2/4+\zeta^2/2}\,dw
=2\sqrt{\pi}\,e^{\zeta^2/2}. n ′ = 0 ∑ ∞ n ′ ! ζ n ′ ∫ R e − w 2 /4 He n ′ ′ ( w ) d w = ∫ R e − ( w − 2 ζ ) 2 /4 + ζ 2 /2 d w = 2 π e ζ 2 /2 . Comparing the coefficients of ζ 2 l \zeta^{2l} ζ 2 l
∫ R e − w 2 / 4 He 2 l ′ ( w ) d w = 2 π ( 2 l ) ! 2 l l ! . \int_{\mathbb{R}}e^{-w^2/4}\operatorname{He}_{2l}'(w)\,dw=\frac{2\sqrt{\pi}(2l)!}{2^l l!}. ∫ R e − w 2 /4 He 2 l ′ ( w ) d w = 2 l l ! 2 π ( 2 l )! . and therefore
∫ R h 2 l = 2 π ( 2 π ) 1 / 4 ⋅ ( 2 l ) ! 2 l l ! = 2 π ( 2 π ) 1 / 4 ( 4 − l ( 2 l l ) ) 1 / 2 . \int_{\mathbb{R}}h_{2l}=\frac{2\sqrt{\pi}}{(2\pi)^{1/4}}\cdot\frac{\sqrt{(2l)!}}{2^l l!}
=\frac{2\sqrt{\pi}}{(2\pi)^{1/4}}\left(4^{-l}\binom{2l}{l}\right)^{1/2}. ∫ R h 2 l = ( 2 π ) 1/4 2 π ⋅ 2 l l ! ( 2 l )! = ( 2 π ) 1/4 2 π ( 4 − l ( l 2 l ) ) 1/2 . The claim for even n n n
( 4 l ) − 1 / 2 ≤ 4 − l ( 2 l l ) ≤ ( π l ) − 1 / 2 . (4l)^{-1/2}\le4^{-l}\binom{2l}{l}\le(\pi l)^{-1/2}. ( 4 l ) − 1/2 ≤ 4 − l ( l 2 l ) ≤ ( π l ) − 1/2 . Suppose next that n = 2 l + 1 n=2l+1 n = 2 l + 1 l ≥ 1 l\ge1 l ≥ 1 ζ ∈ R \zeta\in\mathbb{R} ζ ∈ R
∫ 0 ∞ e − w 2 / 4 e w ζ − ζ 2 / 2 d w = e ζ 2 / 2 ∫ − 2 ζ ∞ e − v 2 / 4 d v = π e ζ 2 / 2 + 2 ζ ∫ 0 1 e ζ 2 ( 1 / 2 − v 2 ) d v . \int_0^{\infty}e^{-w^2/4}e^{w\zeta-\zeta^2/2}\,dw
=e^{\zeta^2/2}\int_{-2\zeta}^{\infty}e^{-v^2/4}\,dv
=\sqrt{\pi}e^{\zeta^2/2}+2\zeta\int_0^1e^{\zeta^2(1/2-v^2)}\,dv. ∫ 0 ∞ e − w 2 /4 e w ζ − ζ 2 /2 d w = e ζ 2 /2 ∫ − 2 ζ ∞ e − v 2 /4 d v = π e ζ 2 /2 + 2 ζ ∫ 0 1 e ζ 2 ( 1/2 − v 2 ) d v . The first term on the right side is even in ζ \zeta ζ
j l = ∫ 0 1 ( 1 − 2 v 2 ) l d v . j_l=\int_{0}^{1}(1-2v^2)^l\,\mathrm{d}v. j l = ∫ 0 1 ( 1 − 2 v 2 ) l d v . Comparing the coefficients of ζ 2 l + 1 \zeta^{2l+1} ζ 2 l + 1
∫ 0 ∞ h 2 l + 1 = 2 1 − l ( 2 l + 1 ) ! ( 2 π ) 1 / 4 l ! j l = 2 2 l + 1 ( 2 π ) 1 / 4 ( 4 − l ( 2 l l ) ) 1 / 2 j l . \int_{0}^{\infty}h_{2l+1}=\frac{2^{1-l}\sqrt{(2l+1)!}}{(2\pi)^{1/4}l!}j_l=\frac{2\sqrt{2l+1}}{(2\pi)^{1/4}}\left(4^{-l}\binom{2l}{l}\right)^{1/2}j_l. ∫ 0 ∞ h 2 l + 1 = ( 2 π ) 1/4 l ! 2 1 − l ( 2 l + 1 )! j l = ( 2 π ) 1/4 2 2 l + 1 ( 4 − l ( l 2 l ) ) 1/2 j l . To bound j l j_l j l
0 ≤ 1 − 2 v 2 ≤ e − 2 v 2 , v ∈ [ 0 , 2 − 1 / 2 ] , 0\leq1-2v^2\leq e^{-2v^2},\qquad v\in[0,2^{-1/2}], 0 ≤ 1 − 2 v 2 ≤ e − 2 v 2 , v ∈ [ 0 , 2 − 1/2 ] , 0 ≤ 2 v 2 − 1 ≤ 1 − 2 ( 1 − v ) ≤ e − 2 ( 1 − v ) , v ∈ [ 2 − 1 / 2 , 1 ] . 0\leq2v^2-1\leq1-2(1-v)\leq e^{-2(1-v)},\qquad v\in[2^{-1/2},1]. 0 ≤ 2 v 2 − 1 ≤ 1 − 2 ( 1 − v ) ≤ e − 2 ( 1 − v ) , v ∈ [ 2 − 1/2 , 1 ] . They give
∣ j l ∣ ≤ ∫ 0 ∞ e − 2 l v 2 d v + ∫ 0 ∞ e − 2 l v d v ≤ 2 l − 1 / 2 , \lvert j_l\rvert\leq\int_{0}^{\infty}e^{-2lv^2}\,\mathrm{d}v+\int_{0}^{\infty}e^{-2lv}\,\mathrm{d}v\leq2l^{-1/2}, ∣ j l ∣ ≤ ∫ 0 ∞ e − 2 l v 2 d v + ∫ 0 ∞ e − 2 l v d v ≤ 2 l − 1/2 , and therefore
∣ ∫ 0 ∞ h 2 l + 1 ∣ ≤ C l 1 / 2 2 − l − 1 / 4 l − 1 / 2 ≤ C n − 1 / 4 . \left\lvert\int_{0}^{\infty}h_{2l+1}\right\rvert\leq Cl^{1/2}2^{-l-1/4}l^{-1/2}\leq Cn^{-1/4}. ∫ 0 ∞ h 2 l + 1 ≤ C l 1/2 2 − l − 1/4 l − 1/2 ≤ C n − 1/4 . The case n = 1 n=1 n = 1
Finally, we prove (d). When n n n h n h_n h n n n n h n h_n h n
F n ( w ) = ∫ 0 w h n ( n even ) , F n ( w ) = ∫ 0 w h n − ∫ 0 ∞ h n ( n odd ) , F_n(w)=\int_{0}^{w}h_n\quad(n\text{ even}),\qquad F_n(w)=\int_{0}^{w}h_n-\int_{0}^{\infty}h_n\quad(n\text{ odd}), F n ( w ) = ∫ 0 w h n ( n even ) , F n ( w ) = ∫ 0 w h n − ∫ 0 ∞ h n ( n odd ) , and (b) with w ′ = 0 w'=0 w ′ = 0
The Pfaffian formula
The entries G a b G_{ab} G ab a ≤ b a\leq b a ≤ b 2 / N 2/N 2/ N a = b a=b a = b 1 / N 1/N 1/ N a < b a<b a < b G G G exp ( − N Tr A 2 / 4 ) \exp(-N\operatorname{Tr} A^2/4) exp ( − N Tr A 2 /4 ) N × N N\times N N × N A A A G G G
℘ N ( x 1 , … , x N ) = 1 Z N ∏ 1 ≤ i < j ≤ N ∣ x i − x j ∣ ∏ i = 1 N e − N x i 2 / 4 (433) \wp_N(x_1,\ldots,x_N)=\frac{1}{Z_N}\prod_{1\leq i<j\leq N}\lvert x_i-x_j\rvert\prod_{i=1}^{N}e^{-Nx_i^2/4} \tag*{(433)} ℘ N ( x 1 , … , x N ) = Z N 1 1 ≤ i < j ≤ N ∏ ∣ x i − x j ∣ i = 1 ∏ N e − N x i 2 /4 (433) on R N \mathbb{R}^N R N Z N Z_N Z N [2 Section 2.5] ; this is [33 (1.1)] for the orthogonal ensemble). For n ∈ [ N ] n\in[N] n ∈ [ N ] n n n
R n , N ( x 1 , … , x n ) = N ! ( N − n ) ! ∫ R N − n ℘ N ( x 1 , … , x N ) d x n + 1 ⋯ d x N . (434) R_{n,N}(x_1,\ldots,x_n)=\frac{N!}{(N-n)!}\int_{\mathbb{R}^{N-n}}\wp_N(x_1,\ldots,x_N)\,\mathrm{d}x_{n+1}\cdots\mathrm{d}x_N. \tag*{(434)} R n , N ( x 1 , … , x n ) = ( N − n )! N ! ∫ R N − n ℘ N ( x 1 , … , x N ) d x n + 1 ⋯ d x N . (434) It is nonnegative and symmetric in its arguments. Since P N \mathcal{P}_N P N
E ∑ ( i 1 , … , i n ) F ( λ i 1 , … , λ i n ) = ∫ R n F ( x 1 , … , x n ) R n , N ( x 1 , … , x n ) d x 1 ⋯ d x n (435) \mathbb{E}\sum_{(i_1,\ldots,i_n)}\mathcal{F}(\lambda_{i_1},\ldots,\lambda_{i_n})=\int_{\mathbb{R}^n}\mathcal{F}(x_1,\ldots,x_n)R_{n,N}(x_1,\ldots,x_n)\,\mathrm{d}x_1\cdots\mathrm{d}x_n \tag*{(435)} E ( i 1 , … , i n ) ∑ F ( λ i 1 , … , λ i n ) = ∫ R n F ( x 1 , … , x n ) R n , N ( x 1 , … , x n ) d x 1 ⋯ d x n (435) for every measurable F : R n → [ 0 , ∞ ) \mathcal{F}\colon\mathbb{R}^n\to[0,\infty) F : R n → [ 0 , ∞ ) n n n [ N ] [N] [ N ] F \mathcal{F} F
We recall the definition of the Pfaffian. For an antisymmetric 2 n × 2 n 2n\times2n 2 n × 2 n W W W
Pf W = ∑ m sgn ( m ) ∏ { σ , σ ′ } ∈ m , σ < σ ′ W σ σ ′ , (436) \operatorname{Pf}W=\sum_{\mathfrak{m}}\operatorname{sgn}(\mathfrak{m})\prod_{\{\sigma,\sigma'\}\in\mathfrak{m},\,\sigma<\sigma'}W_{\sigma\sigma'}, \tag*{(436)} Pf W = m ∑ sgn ( m ) { σ , σ ′ } ∈ m , σ < σ ′ ∏ W σ σ ′ , (436) where the sum runs over the perfect matchings m \mathfrak{m} m [ 2 n ] [2n] [ 2 n ] sgn ( m ) \operatorname{sgn}(\mathfrak{m}) sgn ( m ) ( σ 1 , σ 1 ′ , … , σ n , σ n ′ ) (\sigma_1,\sigma_1',\ldots,\sigma_n,\sigma_n') ( σ 1 , σ 1 ′ , … , σ n , σ n ′ ) [ 2 n ] [2n] [ 2 n ] { σ r , σ r ′ } \{\sigma_r,\sigma_r'\} { σ r , σ r ′ } m \mathfrak{m} m σ r < σ r ′ \sigma_r<\sigma_r' σ r < σ r ′ Pf ( B T W B ) = det ( B ) Pf ( W ) \operatorname{Pf}(B^{\mathsf T}WB)=\det(B)\operatorname{Pf}(W) Pf ( B T W B ) = det ( B ) Pf ( W ) 2 n × 2 n 2n\times2n 2 n × 2 n B B B [33 Section 2.2] . Second, let 1 ≤ l < n 1\le l<n 1 ≤ l < n P 1 = [ 2 l ] P_1=[2l] P 1 = [ 2 l ] P 2 = [ 2 n ] ∖ [ 2 l ] P_2=[2n]\setminus[2l] P 2 = [ 2 n ] ∖ [ 2 l ] W P 1 P 1 W_{P_1P_1} W P 1 P 1 W P 2 P 2 W_{P_2P_2} W P 2 P 2 m \mathfrak{m} m m \mathfrak{m} m P 1 P_1 P 1 P 2 P_2 P 2 Pf ( W P 1 P 1 ) Pf ( W P 2 P 2 ) \operatorname{Pf}(W_{P_1P_1})\operatorname{Pf}(W_{P_2P_2}) Pf ( W P 1 P 1 ) Pf ( W P 2 P 2 ) [ 2 n ] [2n] [ 2 n ] ( 2 n − 1 ) ! ! (2n-1)!! ( 2 n − 1 )!!
∣ Pf W ∣ ≤ ( 2 n − 1 ) ! ! max σ , σ ′ ∣ W σ σ ′ ∣ n . |\operatorname{Pf}W|\le(2n-1)!!\max_{\sigma,\sigma'}|W_{\sigma\sigma'}|^n. ∣ Pf W ∣ ≤ ( 2 n − 1 )!! σ , σ ′ max ∣ W σ σ ′ ∣ n . We now define the matrix kernel of the GOE, following [33 (7.2)] up to the names of the functions.
Definition A.2. Let N ≥ 2 N\ge2 N ≥ 2 x , y ∈ R x,y\in\mathbb{R} x , y ∈ R
K N ( x , y ) = N ∑ j = 0 N − 1 h j ( N x ) h j ( N y ) , P ( x , y ) = N 2 h N − 1 ( N x ) F N ( N y ) , K_N(x,y)=\sqrt{N}\sum_{j=0}^{N-1}h_j(\sqrt{N}x)h_j(\sqrt{N}y),\qquad P(x,y)=\frac{N}{2}h_{N-1}(\sqrt{N}x)F_N(\sqrt{N}y), K N ( x , y ) = N j = 0 ∑ N − 1 h j ( N x ) h j ( N y ) , P ( x , y ) = 2 N h N − 1 ( N x ) F N ( N y ) , and S = K N + P S=K_N+P S = K N + P q ( x ) = 0 q(x)=0 q ( x ) = 0 N N N
q ( x ) = N h N − 1 ( N x ) ( ∫ R h N − 1 ) − 1 q(x)=\sqrt{N}h_{N-1}(\sqrt{N}x)\left(\int_{\mathbb{R}}h_{N-1}\right)^{-1} q ( x ) = N h N − 1 ( N x ) ( ∫ R h N − 1 ) − 1 if N N N ∫ R h N − 1 > 0 \int_{\mathbb{R}}h_{N-1}>0 ∫ R h N − 1 > 0 Lemma A.1 (c). We further set Q ( x ) = 1 2 ∫ R sgn ( x − x ′ ) q ( x ′ ) d x ′ Q(x)=\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-x')q(x')\,\mathrm{d}x' Q ( x ) = 2 1 ∫ R sgn ( x − x ′ ) q ( x ′ ) d x ′
T ( x , y ) = S ( x , y ) + q ( x ) , V ( x , y ) = − ∂ y S ( x , y ) , T(x,y)=S(x,y)+q(x),\qquad V(x,y)=-\partial_yS(x,y), T ( x , y ) = S ( x , y ) + q ( x ) , V ( x , y ) = − ∂ y S ( x , y ) , I ( x , y ) = 1 2 ∫ R sgn ( x − x ′ ) S ( x ′ , y ) d x ′ − 1 2 sgn ( x − y ) + Q ( x ) − Q ( y ) . I(x,y)=\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-x')S(x',y)\,\mathrm{d}x'-\frac{1}{2}\operatorname{sgn}(x-y)+Q(x)-Q(y). I ( x , y ) = 2 1 ∫ R sgn ( x − x ′ ) S ( x ′ , y ) d x ′ − 2 1 sgn ( x − y ) + Q ( x ) − Q ( y ) . Finally, we define the 2 × 2 2\times2 2 × 2
A ( x , y ) = ( I ( x , y ) T ( y , x ) − T ( x , y ) − V ( x , y ) ) . A(x,y)=\begin{pmatrix} I(x,y)&T(y,x)\\-T(x,y)&-V(x,y)\end{pmatrix}. A ( x , y ) = ( I ( x , y ) − T ( x , y ) T ( y , x ) − V ( x , y ) ) . We suppress the dependence on N N N P P P S S S q q q Q Q Q T T T V V V I I I A A A K N K_N K N ∣ G a b ∣ 2 = 1 / N \lvert G_{ab}\rvert^2 = 1/N ∣ G ab ∣ 2 = 1/ N a , b a,b a , b [2 Section 3.2] , and P P P q q q K N K_N K N P P P S S S q q q T T T V V V F N ′ = h N F_N' = h_N F N ′ = h N S ( ⋅ , y ) S(\mathord\cdot,y) S ( ⋅ , y ) q q q I I I Q Q Q Q ′ = q Q' = q Q ′ = q
Lemma A.3. Let N ≥ 2 N \ge2 N ≥ 2 n ∈ [ N ] n \in[N] n ∈ [ N ] x 1 , … , x n ∈ R x_1,\ldots,x_n \in\mathbb{R} x 1 , … , x n ∈ R
R n , N ( x 1 , … , x n ) = Pf [ A ( x i , x j ) ] i , j = 1 n (437) R_{n,N}(x_1,\ldots,x_n)=\operatorname{Pf}\left[A(x_i,x_j)\right]_{i,j=1}^{n} \tag*{(437)} R n , N ( x 1 , … , x n ) = Pf [ A ( x i , x j ) ] i , j = 1 n (437) where the 2 n × 2 n 2n \times2n 2 n × 2 n 2 × 2 2 \times2 2 × 2 A ( x i , x j ) A(x_i,x_j) A ( x i , x j ) ( i , j ) (i,j) ( i , j )
This is the classical Pfaffian, or quaternion determinant, formula of Mehta for the correlation functions of the GOE [56 ] . In the present normalization, it is [33 (7.1)–(7.2)] . In the notation of [33 ] , the matrix size is n n n N N N T T T V V V I I I q q q Q Q Q S n ( 1 ) + α n S_n^{(1)}+\alpha_n S n ( 1 ) + α n V n ( 1 ) V_n^{(1)} V n ( 1 ) J n ( 1 ) J_n^{(1)} J n ( 1 ) α n \alpha_n α n γ n \gamma_n γ n A ( x , y ) A(x,y) A ( x , y ) J K n ( 1 ) ( x , y ) JK_n^{(1)}(x,y) J K n ( 1 ) ( x , y ) N N N [72 ] ((389) and Section 10), and for odd N N N [1 ] ((20) and (76) ). These references use the weight e − x 2 / 2 e^{-x^2/2} e − x 2 /2 G G G ( N / 2 ) 1 / 2 (N/2)^{1/2} ( N /2 ) 1/2 Lemma A.4 (a) below, the matrix on the right side of (A.14) is antisymmetric.
We will use the following identities for the kernel.
Lemma A.4. Let N ≥ 2 N \ge2 N ≥ 2 x , y ∈ R x,y \in\mathbb{R} x , y ∈ R
(a) We have V ( x , y ) = − V ( y , x ) V(x,y)=-V(y,x) V ( x , y ) = − V ( y , x ) I ( x , y ) = − I ( y , x ) I(x,y)=-I(y,x) I ( x , y ) = − I ( y , x ) A ( x , y ) T = − A ( y , x ) A(x,y)^{T}=-A(y,x) A ( x , y ) T = − A ( y , x )
(b) We have
I ( x , y ) = ∫ y x T ( x ′ , y ) d x ′ − 1 2 sgn ( x − y ) . (438) I(x,y)=\int_y^x T(x',y)\,\mathrm{d}x'-\frac{1}{2}\operatorname{sgn}(x-y). \tag*{(438)} I ( x , y ) = ∫ y x T ( x ′ , y ) d x ′ − 2 1 sgn ( x − y ) . (438) Consequently, ∂ x I ( x , y ) = T ( x , y ) \partial_x I(x,y)=T(x,y) ∂ x I ( x , y ) = T ( x , y ) ∂ y I ( x , y ) = − T ( y , x ) \partial_y I(x,y)=-T(y,x) ∂ y I ( x , y ) = − T ( y , x ) x ≠ y x\ne y x = y
(c) We have R 1 , N ( x ) = T ( x , x ) R_{1,N}(x)=T(x,x) R 1 , N ( x ) = T ( x , x ) x ≠ y x\ne y x = y
R 2 , N ( x , y ) = T ( x , x ) T ( y , y ) − T ( x , y ) T ( y , x ) + I ( x , y ) V ( x , y ) . (439) R_{2,N}(x,y)=T(x,x)T(y,y)-T(x,y)T(y,x)+I(x,y)V(x,y). \tag*{(439)} R 2 , N ( x , y ) = T ( x , x ) T ( y , y ) − T ( x , y ) T ( y , x ) + I ( x , y ) V ( x , y ) . (439) Proof. Throughout the proof, we write w = N x w=\sqrt{N}x w = N x w ′ = N y w'=\sqrt{N}y w ′ = N y V V V
∑ j = 0 N − 1 h j ( w ) h j ′ ( w ′ ) = 1 2 ∑ j = 1 N − 1 j ( h j ( w ) h j − 1 ( w ′ ) − h j − 1 ( w ) h j ( w ′ ) ) − N 2 h N − 1 ( w ) h N ( w ′ ) . \sum_{j=0}^{N-1}h_j(w)h_j'(w')=\frac{1}{2}\sum_{j=1}^{N-1}\sqrt{j}\left(h_j(w)h_{j-1}(w')-h_{j-1}(w)h_j(w')\right)-\frac{\sqrt{N}}{2}h_{N-1}(w)h_N(w'). j = 0 ∑ N − 1 h j ( w ) h j ′ ( w ′ ) = 2 1 j = 1 ∑ N − 1 j ( h j ( w ) h j − 1 ( w ′ ) − h j − 1 ( w ) h j ( w ′ ) ) − 2 N h N − 1 ( w ) h N ( w ′ ) . Using F N ′ = h N F_N'=h_N F N ′ = h N
∂ y P ( x , y ) = N 3 / 2 2 h N − 1 ( w ) h N ( w ′ ) . (440) \partial_yP(x,y)=\frac{N^{3/2}}{2}h_{N-1}(w)h_N(w'). \tag*{(440)} ∂ y P ( x , y ) = 2 N 3/2 h N − 1 ( w ) h N ( w ′ ) . (440) and therefore
V ( x , y ) = − N ( ∑ j = 0 N − 1 h j ( w ) h j ′ ( w ′ ) + N 2 h N − 1 ( w ) h N ( w ′ ) ) = − N 2 ∑ j = 1 N − 1 j ( h j ( w ) h j − 1 ( w ′ ) − h j − 1 ( w ) h j ( w ′ ) ) . (441) \begin{aligned}
V(x,y)=-N\left(\sum_{j=0}^{N-1}h_j(w)h'_j(w')+\frac{\sqrt{N}}{2}h_{N-1}(w)h_N(w')\right) \\
&=-\frac{N}{2}\sum_{j=1}^{N-1}\sqrt{j}\left(h_j(w)h_{j-1}(w')-h_{j-1}(w)h_j(w')\right). \tag*{(441)}
\end{aligned} V ( x , y ) = − N ( j = 0 ∑ N − 1 h j ( w ) h j ′ ( w ′ ) + 2 N h N − 1 ( w ) h N ( w ′ ) ) = − 2 N j = 1 ∑ N − 1 j ( h j ( w ) h j − 1 ( w ′ ) − h j − 1 ( w ) h j ( w ′ ) ) . (441) which is antisymmetric in ( x , y ) (x,y) ( x , y )
Next, we consider I. Substituting N x ′ \sqrt{N}x' N x ′
1 2 ∫ R sgn ( x − x ′ ) N h j ( N x ′ ) d x ′ = F j ( w ) , \frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-x')\sqrt{N}h_j(\sqrt{N}x')\,\mathrm{d}x'=F_j(w), 2 1 ∫ R sgn ( x − x ′ ) N h j ( N x ′ ) d x ′ = F j ( w ) , and therefore
1 2 ∫ R sgn ( x − x ′ ) S ( x ′ , y ) d x ′ = ∑ j = 0 N − 1 F j ( w ) h j ( w ′ ) + N 2 F N − 1 ( w ) F N ( w ′ ) . \frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-x')S(x',y)\,\mathrm{d}x'=\sum_{j=0}^{N-1}F_j(w)h_j(w')+\frac{\sqrt{N}}{2}F_{N-1}(w)F_N(w'). 2 1 ∫ R sgn ( x − x ′ ) S ( x ′ , y ) d x ′ = j = 0 ∑ N − 1 F j ( w ) h j ( w ′ ) + 2 N F N − 1 ( w ) F N ( w ′ ) . Inserting (A.8) for h j ( w ′ ) h_j(w') h j ( w ′ )
1 2 ∫ R sgn ( x − x ′ ) S ( x ′ , y ) d x ′ = 1 2 ∑ j = 1 N − 1 j ( F j ( w ) F j − 1 ( w ′ ) − F j − 1 ( w ) F j ( w ′ ) ) . (442) \frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-x')S(x',y)\,\mathrm{d}x'=\frac{1}{2}\sum_{j=1}^{N-1}\sqrt{j}\left(F_j(w)F_{j-1}(w')-F_{j-1}(w)F_j(w')\right). \tag*{(442)} 2 1 ∫ R sgn ( x − x ′ ) S ( x ′ , y ) d x ′ = 2 1 j = 1 ∑ N − 1 j ( F j ( w ) F j − 1 ( w ′ ) − F j − 1 ( w ) F j ( w ′ ) ) . (442) Let I 0 ( x , y ) I_0(x,y) I 0 ( x , y ) Q ( x ) − Q ( y ) Q(x)-Q(y) Q ( x ) − Q ( y )
I ( x , y ) = I 0 ( x , y ) − 1 2 sgn ( x − y ) . I(x,y)=I_0(x,y)-\frac{1}{2}\operatorname{sgn}(x-y). I ( x , y ) = I 0 ( x , y ) − 2 1 sgn ( x − y ) . The function I 0 I_0 I 0 ( x , y ) ↦ sgn ( x − y ) (x,y)\mapsto\operatorname{sgn}(x-y) ( x , y ) ↦ sgn ( x − y ) I I I A ( x , y ) T = − A ( y , x ) A(x,y)^{T}=-A(y,x) A ( x , y ) T = − A ( y , x ) I I I V V V
We now prove (b). Since I 0 I_0 I 0 I 0 ( y , y ) = 0 I_0(y,y)=0 I 0 ( y , y ) = 0 Q ′ = q Q'=q Q ′ = q
∂ x I 0 ( x , y ) = S ( x , y ) + q ( x ) = T ( x , y ) , x ∈ R . \partial_x I_0(x,y)=S(x,y)+q(x)=T(x,y),\qquad x\in\mathbb{R}. ∂ x I 0 ( x , y ) = S ( x , y ) + q ( x ) = T ( x , y ) , x ∈ R . Hence
I 0 ( x , y ) = ∫ y x T ( x ′ , y ) d x ′ , I_0(x,y)=\int_y^x T(x',y)\,\mathrm{d}x', I 0 ( x , y ) = ∫ y x T ( x ′ , y ) d x ′ , which is (A.15). The formula for ∂ x I \partial_x I ∂ x I ∂ y I \partial_y I ∂ y I I ( x , y ) = − I ( y , x ) I(x,y)=-I(y,x) I ( x , y ) = − I ( y , x )
It remains to prove (c). By (a), we have I ( x , x ) = 0 I(x,x)=0 I ( x , x ) = 0 V ( x , x ) = 0 V(x,x)=0 V ( x , x ) = 0 A ( x , x ) A(x,x) A ( x , x ) T ( x , x ) T(x,x) T ( x , x ) − T ( x , x ) -T(x,x) − T ( x , x ) n = 1 n=1 n = 1 R 1 , N ( x ) = T ( x , x ) R_{1,N}(x)=T(x,x) R 1 , N ( x ) = T ( x , x ) n = 2 n=2 n = 2
W = ( 0 T ( x , x ) I ( x , y ) T ( y , x ) − T ( x , x ) 0 − T ( x , y ) − V ( x , y ) − I ( x , y ) T ( x , y ) 0 T ( y , y ) − T ( y , x ) V ( x , y ) − T ( y , y ) 0 ) . W=\begin{pmatrix}
0 & T(x,x) & I(x,y) & T(y,x) \\
-T(x,x) & 0 & -T(x,y) & -V(x,y) \\
-I(x,y) & T(x,y) & 0 & T(y,y) \\
-T(y,x) & V(x,y) & -T(y,y) & 0
\end{pmatrix}. W = 0 − T ( x , x ) − I ( x , y ) − T ( y , x ) T ( x , x ) 0 T ( x , y ) V ( x , y ) I ( x , y ) − T ( x , y ) 0 − T ( y , y ) T ( y , x ) − V ( x , y ) T ( y , y ) 0 . Therefore
R 2 , N ( x , y ) = W 12 W 34 − W 13 W 24 + W 14 W 23 , R_{2,N}(x,y)=W_{12}W_{34}-W_{13}W_{24}+W_{14}W_{23}, R 2 , N ( x , y ) = W 12 W 34 − W 13 W 24 + W 14 W 23 , which is (A.16).
Kernel estimates
For functions of two real variables, we write ∂ 1 \partial_1 ∂ 1 ∂ 2 \partial_2 ∂ 2 K N K_N K N
Lemma A.5. Let δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) C > 0 C>0 C > 0 N 0 N_0 N 0 N ≥ N 0 N\ge N_0 N ≥ N 0 x , y ∈ I δ x,y\in I_\delta x , y ∈ I δ
(a) For all integers k 1 , k 2 ≥ 0 k_1,k_2\ge0 k 1 , k 2 ≥ 0 k 1 + k 2 ≤ 3 k_1+k_2\le3 k 1 + k 2 ≤ 3 ∣ ∂ x k 1 ∂ y k 2 K N ( x , y ) ∣ ≤ C N 1 + k 1 + k 2 \left|\partial_x^{k_1}\partial_y^{k_2}K_N(x,y)\right|\le CN^{1+k_1+k_2} ∂ x k 1 ∂ y k 2 K N ( x , y ) ≤ C N 1 + k 1 + k 2
(b) If x ≠ y x\ne y x = y ∣ K N ( x , y ) ∣ ≤ C ∣ x − y ∣ − 1 \left|K_N(x,y)\right|\le C\lvert x-y\rvert^{-1} ∣ K N ( x , y ) ∣ ≤ C ∣ x − y ∣ − 1 ∣ ∂ y K N ( x , y ) ∣ ≤ C ( N ∣ x − y ∣ − 1 + ∣ x − y ∣ − 2 ) \left|\partial_yK_N(x,y)\right|\le C\left(N\lvert x-y\rvert^{-1}+\lvert x-y\rvert^{-2}\right) ∣ ∂ y K N ( x , y ) ∣ ≤ C ( N ∣ x − y ∣ − 1 + ∣ x − y ∣ − 2 )
(c) The derivative of the function x ↦ K N ( x , x ) x\mapsto K_N(x,x) x ↦ K N ( x , x ) C N CN C N I δ I_\delta I δ
(d) We have
∂ y 2 K N ( x , y ) ∣ y = x = − N 3 ( N ( 1 − x 2 4 ) + 1 2 ) K N ( x , x ) − N 2 3 h N ( N x ) h N − 1 ′ ( N x ) . (443) \left.\partial_y^2K_N(x,y)\right|_{y=x}=-\frac{N}{3}\left(N\left(1-\frac{x^2}{4}\right)+\frac{1}{2}\right)K_N(x,x)-\frac{N^2}{3}h_N(\sqrt{N}x)h_{N-1}'(\sqrt{N}x). \tag*{(443)} ∂ y 2 K N ( x , y ) y = x = − 3 N ( N ( 1 − 4 x 2 ) + 2 1 ) K N ( x , x ) − 3 N 2 h N ( N x ) h N − 1 ′ ( N x ) . (443) Proof. Let w = N x w=\sqrt{N}x w = N x w ′ = N y w'=\sqrt{N}y w ′ = N y
K ∘ ( w , w ′ ) = ∑ j = 0 N − 1 h j ( w ) h j ( w ′ ) , K^\circ(w,w')=\sum_{j=0}^{N-1}h_j(w)h_j(w'), K ∘ ( w , w ′ ) = j = 0 ∑ N − 1 h j ( w ) h j ( w ′ ) , so that K N ( x , y ) = N K ∘ ( w , w ′ ) K_N(x,y)=\sqrt{N}K^\circ(w,w') K N ( x , y ) = N K ∘ ( w , w ′ )
We have ∣ w ∣ , ∣ w ′ ∣ ≤ ( 2 − δ ) N \lvert w\rvert,\lvert w'\rvert\le(2-\delta)\sqrt{N} ∣ w ∣ , ∣ w ′ ∣ ≤ ( 2 − δ ) N N 0 N_0 N 0 ( 2 − δ ) N ≤ ( 2 − δ / 2 ) n (2-\delta)\sqrt{N}\le(2-\delta/2)\sqrt{n} ( 2 − δ ) N ≤ ( 2 − δ /2 ) n N ≥ N 0 N\ge N_0 N ≥ N 0 n ∈ { N − 1 , N } n\in\{N-1,N\} n ∈ { N − 1 , N } δ / 2 \delta/2 δ /2 δ \delta δ
∣ h n ( k ) ( w ) ∣ + ∣ h n ( k ) ( w ′ ) ∣ ≤ C N k / 2 − 1 / 4 , n ∈ { N − 1 , N } , k ∈ { 0 , 1 , 2 , 3 } . (444) \left|h_n^{(k)}(w)\right|+\left|h_n^{(k)}(w')\right|\le CN^{k/2-1/4},\qquad n\in\{N-1,N\},\ k\in\{0,1,2,3\}. \tag*{(444)} h n ( k ) ( w ) + h n ( k ) ( w ′ ) ≤ C N k /2 − 1/4 , n ∈ { N − 1 , N } , k ∈ { 0 , 1 , 2 , 3 } . (444) We begin with (a). Using (A.6) with n = N n=N n = N
∑ j = 0 N − 1 h j ( w ) 2 = N ( h N ′ ( w ) h N − 1 ( w ) − h N − 1 ′ ( w ) h N ( w ) ) ≤ C N . \sum_{j=0}^{N-1}h_j(w)^2=\sqrt{N}\left(h_N'(w)h_{N-1}(w)-h_{N-1}'(w)h_N(w)\right)\le C\sqrt{N}. j = 0 ∑ N − 1 h j ( w ) 2 = N ( h N ′ ( w ) h N − 1 ( w ) − h N − 1 ′ ( w ) h N ( w ) ) ≤ C N . Squaring the second identity in (A.4) and using the Cauchy–Schwarz inequality, we find
∑ j = 0 N − 1 h j ′ ( ω ) 2 ≤ 1 2 ∑ j = 0 N − 1 ( j h j − 1 ( ω ) 2 + ( j + 1 ) h j + 1 ( ω ) 2 ) ≤ N ∑ j = 0 N h j ( ω ) 2 ≤ C N 3 / 2 . \sum_{j=0}^{N-1} h_j'(\omega)^2 \le\frac{1}{2}\sum_{j=0}^{N-1}\left(jh_{j-1}(\omega)^2+(j+1)h_{j+1}(\omega)^2\right)\le N\sum_{j=0}^{N}h_j(\omega)^2\le CN^{3/2}. j = 0 ∑ N − 1 h j ′ ( ω ) 2 ≤ 2 1 j = 0 ∑ N − 1 ( j h j − 1 ( ω ) 2 + ( j + 1 ) h j + 1 ( ω ) 2 ) ≤ N j = 0 ∑ N h j ( ω ) 2 ≤ C N 3/2 . where in the last step we used (A.20) for h N ( ω ) h_N(\omega) h N ( ω ) j ≤ N − 1 j\le N-1 j ≤ N − 1
∣ h j ′ ′ ( ω ) ∣ = ∣ e j ( ω ) ∣ ∣ h j ( ω ) ∣ ≤ 2 N ∣ h j ( ω ) ∣ , |h_j''(\omega)|=|e_j(\omega)||h_j(\omega)|\le2N|h_j(\omega)|, ∣ h j ′′ ( ω ) ∣ = ∣ e j ( ω ) ∣∣ h j ( ω ) ∣ ≤ 2 N ∣ h j ( ω ) ∣ , ∣ h j ′ ′ ′ ( ω ) ∣ ≤ ∣ ω ∣ 2 ∣ h j ( ω ) ∣ + 2 N ∣ h j ′ ( ω ) ∣ ≤ N ∣ h j ( ω ) ∣ + 2 N ∣ h j ′ ( ω ) ∣ . |h_j'''(\omega)|\le\frac{|\omega|}{2}|h_j(\omega)|+2N|h_j'(\omega)|\le\sqrt{N}|h_j(\omega)|+2N|h_j'(\omega)|. ∣ h j ′′′ ( ω ) ∣ ≤ 2 ∣ ω ∣ ∣ h j ( ω ) ∣ + 2 N ∣ h j ′ ( ω ) ∣ ≤ N ∣ h j ( ω ) ∣ + 2 N ∣ h j ′ ( ω ) ∣. Combining the last three bounds, we find
∑ j = 0 N − 1 h j ( k ) ( ω ) 2 ≤ C N k + 1 / 2 , k ∈ { 0 , 1 , 2 , 3 } , (445) \sum_{j=0}^{N-1}h_j^{(k)}(\omega)^2\le CN^{k+1/2},\qquad k\in\{0,1,2,3\}, \tag*{(445)} j = 0 ∑ N − 1 h j ( k ) ( ω ) 2 ≤ C N k + 1/2 , k ∈ { 0 , 1 , 2 , 3 } , (445) and the same bound holds at ω ′ \omega' ω ′
∣ ∂ x k 1 ∂ y k 2 K N ( x , y ) ∣ = N ( 1 + k 1 + k 2 ) / 2 ∣ ∑ j = 0 N − 1 h j ( k 1 ) ( ω ) h j ( k 2 ) ( ω ′ ) ∣ ≤ N ( 1 + k 1 + k 2 ) / 2 ( C N k 1 + 1 / 2 ) 1 / 2 ( C N k 2 + 1 / 2 ) 1 / 2 = C N 1 + k 1 + k 2 . \begin{aligned}
|\partial_x^{k_1}\partial_y^{k_2}K_N(x,y)|=N^{(1+k_1+k_2)/2}\left|\sum_{j=0}^{N-1}h_j^{(k_1)}(\omega)h_j^{(k_2)}(\omega')\right| \\
&\le N^{(1+k_1+k_2)/2}\left(CN^{k_1+1/2}\right)^{1/2}\left(CN^{k_2+1/2}\right)^{1/2}=CN^{1+k_1+k_2}.
\end{aligned} ∣ ∂ x k 1 ∂ y k 2 K N ( x , y ) ∣ = N ( 1 + k 1 + k 2 ) /2 j = 0 ∑ N − 1 h j ( k 1 ) ( ω ) h j ( k 2 ) ( ω ′ ) ≤ N ( 1 + k 1 + k 2 ) /2 ( C N k 1 + 1/2 ) 1/2 ( C N k 2 + 1/2 ) 1/2 = C N 1 + k 1 + k 2 . To prove (b), we write (A.5) in the form
K N ( x , y ) = N Z ( ω , ω ′ ) x − y , Z ( ω , ω ′ ) = h N ( ω ) h N − 1 ( ω ′ ) − h N − 1 ( ω ) h N ( ω ′ ) . K_N(x,y)=\sqrt{N}\frac{Z(\omega,\omega')}{x-y},\qquad Z(\omega,\omega')=h_N(\omega)h_{N-1}(\omega')-h_{N-1}(\omega)h_N(\omega'). K N ( x , y ) = N x − y Z ( ω , ω ′ ) , Z ( ω , ω ′ ) = h N ( ω ) h N − 1 ( ω ′ ) − h N − 1 ( ω ) h N ( ω ′ ) . By (A.20), we have
∣ Z ( ω , ω ′ ) ∣ ≤ C N − 1 / 2 , ∣ ∂ ω ′ Z ( ω , ω ′ ) ∣ ≤ C . |Z(\omega,\omega')|\le CN^{-1/2},\qquad|\partial_{\omega'}Z(\omega,\omega')|\le C. ∣ Z ( ω , ω ′ ) ∣ ≤ C N − 1/2 , ∣ ∂ ω ′ Z ( ω , ω ′ ) ∣ ≤ C . The bounds in (b) then follow from the last two displays and the identity
∂ y K N ( x , y ) = N ∂ ω ′ Z ( ω , ω ′ ) x − y + N Z ( ω , ω ′ ) ( x − y ) 2 . \partial_yK_N(x,y)=\frac{N\partial_{\omega'}Z(\omega,\omega')}{x-y}+\frac{\sqrt{N}Z(\omega,\omega')}{(x-y)^2}. ∂ y K N ( x , y ) = x − y N ∂ ω ′ Z ( ω , ω ′ ) + ( x − y ) 2 N Z ( ω , ω ′ ) . Next, we prove (c). The second identity in (A.4) and a telescoping sum give
d d x K N ( x , x ) = 2 N ∑ j = 0 N − 1 h j ( ω ) h j ′ ( ω ) = − N 3 / 2 h N − 1 ( ω ) h N ( ω ) , \frac{d}{dx}K_N(x,x)=2N\sum_{j=0}^{N-1}h_j(\omega)h_j'(\omega)=-N^{3/2}h_{N-1}(\omega)h_N(\omega), d x d K N ( x , x ) = 2 N j = 0 ∑ N − 1 h j ( ω ) h j ′ ( ω ) = − N 3/2 h N − 1 ( ω ) h N ( ω ) , and the right side is bounded by C N CN C N
It remains to prove (d), which holds for all x ∈ R x \in\mathbb{R} x ∈ R w w w z ( w ′ ) = Z ( w , w ′ ) z(w') = Z(w,w') z ( w ′ ) = Z ( w , w ′ ) z z z z ( w ) = 0 z(w) = 0 z ( w ) = 0 z z z w w w
K ∘ ( w , w ′ ) = − N z ( w ′ ) w ′ − w = − N ∑ l = 1 ∞ z ( l ) ( w ) l ! ( w ′ − w ) l − 1 , w ′ ≠ w . K^\circ(w,w')=-\sqrt{N}\frac{z(w')}{w'-w}=-\sqrt{N}\sum_{l=1}^{\infty}\frac{z^{(l)}(w)}{l!}(w'-w)^{l-1},\qquad w'\ne w. K ∘ ( w , w ′ ) = − N w ′ − w z ( w ′ ) = − N l = 1 ∑ ∞ l ! z ( l ) ( w ) ( w ′ − w ) l − 1 , w ′ = w . Hence
K ∘ ( w , w ) = − N z ′ ( w ) , ∂ w ′ 2 K ∘ ( w , w ′ ) ∣ w ′ = w = − N 3 z ′ ′ ′ ( w ) . K^\circ(w,w)=-\sqrt{N}z'(w),\qquad\left.\partial_{w'}^2K^\circ(w,w')\right|_{w'=w}=-\frac{\sqrt{N}}{3}z'''(w). K ∘ ( w , w ) = − N z ′ ( w ) , ∂ w ′ 2 K ∘ ( w , w ′ ) w ′ = w = − 3 N z ′′′ ( w ) . To compute z ′ ′ ′ z''' z ′′′ n ∈ { N − 1 , N } n\in\{N-1,N\} n ∈ { N − 1 , N } e N − 1 = e N − 1 e_{N-1}=e_N-1 e N − 1 = e N − 1
z ′ ′ ′ ( w ) = − e N − 1 ( w ) h N ( w ) h N − 1 ′ ( w ) + e N ( w ) h N − 1 ( w ) h N ′ ( w ) = − e N ( w ) z ′ ( w ) + h N ( w ) h N − 1 ′ ( w ) . \begin{aligned}
z'''(w)&=-e_{N-1}(w)h_N(w)h'_{N-1}(w)+e_N(w)h_{N-1}(w)h'_N(w)\\
&=-e_N(w)z'(w)+h_N(w)h'_{N-1}(w).
\end{aligned} z ′′′ ( w ) = − e N − 1 ( w ) h N ( w ) h N − 1 ′ ( w ) + e N ( w ) h N − 1 ( w ) h N ′ ( w ) = − e N ( w ) z ′ ( w ) + h N ( w ) h N − 1 ′ ( w ) . Combining the last two displays, we obtain
∂ w ′ 2 K ∘ ( w , w ′ ) ∣ w ′ = w = − e N ( w ) 3 K ∘ ( w , w ) − N 3 h N ( w ) h N − 1 ′ ( w ) . \left.\partial_{w'}^2K^\circ(w,w')\right|_{w'=w}=-\frac{e_N(w)}{3}K^\circ(w,w)-\frac{\sqrt{N}}{3}h_N(w)h'_{N-1}(w). ∂ w ′ 2 K ∘ ( w , w ′ ) w ′ = w = − 3 e N ( w ) K ∘ ( w , w ) − 3 N h N ( w ) h N − 1 ′ ( w ) . Finally, inserting the relations
∂ y 2 K N ( x , y ) = N 3 / 2 ∂ w ′ 2 K ∘ ( w , w ′ ) , K ∘ ( w , w ) = K N ( x , x ) N , e N ( N x ) = N ( 1 − x 2 4 ) + 1 2 \partial_y^2K_N(x,y)=N^{3/2}\partial_{w'}^2K^\circ(w,w'),\qquad K^\circ(w,w)=\frac{K_N(x,x)}{\sqrt{N}},\qquad e_N(\sqrt{N}x)=N\left(1-\frac{x^2}{4}\right)+\frac{1}{2} ∂ y 2 K N ( x , y ) = N 3/2 ∂ w ′ 2 K ∘ ( w , w ′ ) , K ∘ ( w , w ) = N K N ( x , x ) , e N ( N x ) = N ( 1 − 4 x 2 ) + 2 1 into the preceding formula for ∂ w ′ 2 K ∘ ( w , w ′ ) ∣ w ′ = w \left.\partial_{w'}^2K^\circ(w,w')\right|_{w'=w} ∂ w ′ 2 K ∘ ( w , w ′ ) w ′ = w
We now turn to the functions T T T V V V I I I
Lemma A.6. Let δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) C > 0 C>0 C > 0 N 0 N_0 N 0 N ≥ N 0 N\ge N_0 N ≥ N 0 x , y ∈ I δ x,y\in I_\delta x , y ∈ I δ
(a) For all integers k 1 , k 2 ≥ 0 k_1,k_2\ge0 k 1 , k 2 ≥ 0 k 1 + k 2 ≤ 2 k_1+k_2\le2 k 1 + k 2 ≤ 2 ∣ ∂ x k 1 ∂ y k 2 T ( x , y ) ∣ ≤ C N 1 + k 1 + k 2 |\partial_x^{k_1}\partial_y^{k_2}T(x,y)|\le CN^{1+k_1+k_2} ∣ ∂ x k 1 ∂ y k 2 T ( x , y ) ∣ ≤ C N 1 + k 1 + k 2 ∣ ∂ x k 1 ∂ y k 2 V ( x , y ) ∣ ≤ C N 2 + k 1 + k 2 |\partial_x^{k_1}\partial_y^{k_2}V(x,y)|\le CN^{2+k_1+k_2} ∣ ∂ x k 1 ∂ y k 2 V ( x , y ) ∣ ≤ C N 2 + k 1 + k 2
(b) If x ≠ y x\ne y x = y ∣ T ( x , y ) ∣ ≤ C ( ∣ x − y ∣ − 1 + N 1 / 2 ) |T(x,y)|\le C(|x-y|^{-1}+N^{1/2}) ∣ T ( x , y ) ∣ ≤ C ( ∣ x − y ∣ − 1 + N 1/2 ) ∣ V ( x , y ) ∣ ≤ C ( N ∣ x − y ∣ − 1 + ∣ x − y ∣ − 2 + N ) |V(x,y)|\le C(N|x-y|^{-1}+|x-y|^{-2}+N) ∣ V ( x , y ) ∣ ≤ C ( N ∣ x − y ∣ − 1 + ∣ x − y ∣ − 2 + N )
(c) We have ∣ I ( x , y ) ∣ ≤ C ( 1 + log ( 1 + N ∣ x − y ∣ ) ) |I(x,y)|\le C(1+\log(1+N|x-y|)) ∣ I ( x , y ) ∣ ≤ C ( 1 + log ( 1 + N ∣ x − y ∣ ))
(d) We have ∣ T ( x , x ) − K N ( x , x ) ∣ ≤ C N 1 / 2 |T(x,x)-K_N(x,x)|\le CN^{1/2} ∣ T ( x , x ) − K N ( x , x ) ∣ ≤ C N 1/2 x ↦ T ( x , x ) x\mapsto T(x,x) x ↦ T ( x , x ) C N 3 / 2 CN^{3/2} C N 3/2 I δ I_\delta I δ
(e) We have
∣ ∂ 2 V ( x , x ) − N 3 ( N ( 1 − x 2 4 ) + 1 2 ) K N ( x , x ) ∣ ≤ C N 2 . (446) \left|\partial_2V(x,x)-\frac{N}{3}\left(N\left(1-\frac{x^2}{4}\right)+\frac{1}{2}\right)K_N(x,x)\right|\le CN^2. \tag*{(446)} ∂ 2 V ( x , x ) − 3 N ( N ( 1 − 4 x 2 ) + 2 1 ) K N ( x , x ) ≤ C N 2 . (446) Proof. We write w = N x w = \sqrt{N}x w = N x w ′ = N y w' = \sqrt{N}y w ′ = N y N 0 N_0 N 0 Lemma A.5 , so that (A.20) holds. Since ∣ w ′ ∣ ≤ ( 2 − δ / 2 ) N |w'| \le(2-\delta/2)\sqrt{N} ∣ w ′ ∣ ≤ ( 2 − δ /2 ) N Lemma A.1 (d) and (c), with δ / 2 \delta/2 δ /2 δ \delta δ
∣ F N ( w ′ ) ∣ ≤ C N − 1 / 4 , ∫ R h N − 1 ≥ c N − 1 / 4 if N is odd . |F_N(w')| \le CN^{-1/4}, \qquad\int_{\mathbb{R}} h_{N-1} \ge cN^{-1/4}\quad\text{if } N \text{ is odd}. ∣ F N ( w ′ ) ∣ ≤ C N − 1/4 , ∫ R h N − 1 ≥ c N − 1/4 if N is odd . Recall that T = K N + P + q T=K_N+P+q T = K N + P + q V = − ∂ y K N − ∂ y P V=-\partial_yK_N-\partial_yP V = − ∂ y K N − ∂ y P P P P q q q k 1 ≥ 0 k_1\ge0 k 1 ≥ 0 k 2 ≥ 1 k_2\ge1 k 2 ≥ 1 k 1 + k 2 ≤ 3 k_1+k_2\le3 k 1 + k 2 ≤ 3
∂ x k 1 P ( x , y ) = N 1 + k 1 / 2 2 h N − 1 ( k 1 ) ( w ) F N ( w ′ ) , \partial_x^{k_1}P(x,y)=\frac{N^{1+k_1/2}}{2}h_{N-1}^{(k_1)}(w)F_N(w'), ∂ x k 1 P ( x , y ) = 2 N 1 + k 1 /2 h N − 1 ( k 1 ) ( w ) F N ( w ′ ) , ∂ x k 1 ∂ y k 2 P ( x , y ) = N 1 + ( k 1 + k 2 ) / 2 2 h N − 1 ( k 1 ) ( w ) h N ( k 2 − 1 ) ( w ′ ) , \partial_x^{k_1}\partial_y^{k_2}P(x,y)=\frac{N^{1+(k_1+k_2)/2}}{2}h_{N-1}^{(k_1)}(w)h_N^{(k_2-1)}(w'), ∂ x k 1 ∂ y k 2 P ( x , y ) = 2 N 1 + ( k 1 + k 2 ) /2 h N − 1 ( k 1 ) ( w ) h N ( k 2 − 1 ) ( w ′ ) , and, for odd N N N
q ( k 1 ) ( x ) = N ( 1 + k 1 ) / 2 h N − 1 ( k 1 ) ( w ) ( ∫ R h N − 1 ) − 1 . q^{(k_1)}(x)=N^{(1+k_1)/2}h_{N-1}^{(k_1)}(w)\left(\int_{\mathbb{R}}h_{N-1}\right)^{-1}. q ( k 1 ) ( x ) = N ( 1 + k 1 ) /2 h N − 1 ( k 1 ) ( w ) ( ∫ R h N − 1 ) − 1 . Therefore
∣ ∂ x k 1 P ( x , y ) ∣ + ∣ q ( k 1 ) ( x ) ∣ ≤ C N k 1 + 1 / 2 , ∣ ∂ x k 1 ∂ y k 2 P ( x , y ) ∣ ≤ C N k 1 + k 2 , (447) \left|\partial_x^{k_1}P(x,y)\right|+\left|q^{(k_1)}(x)\right|\le CN^{k_1+1/2},\qquad\left|\partial_x^{k_1}\partial_y^{k_2}P(x,y)\right|\le CN^{k_1+k_2}, \tag*{(447)} ∂ x k 1 P ( x , y ) + q ( k 1 ) ( x ) ≤ C N k 1 + 1/2 , ∂ x k 1 ∂ y k 2 P ( x , y ) ≤ C N k 1 + k 2 , (447) where we used (A.20) and the bounds on F N ( w ′ ) F_N(w') F N ( w ′ ) ∫ R h N − 1 \int_{\mathbb{R}}h_{N-1} ∫ R h N − 1 Lemma A.5 (a), and the bounds in (b) follow from (A.23) with k 1 + k 2 ≤ 1 k_1+k_2\le1 k 1 + k 2 ≤ 1 Lemma A.5 (b).
To prove (c), we use (A.15) and bound the contributions of K N K_N K N P P P q q q Lemma A.5 (a) and (b), we have ∣ K N ( x ′ , y ) ∣ ≤ C min ( N , ∣ x ′ − y ∣ − 1 ) |K_N(x',y)|\le C\min(N,|x'-y|^{-1}) ∣ K N ( x ′ , y ) ∣ ≤ C min ( N , ∣ x ′ − y ∣ − 1 ) x ′ ∈ I δ x'\in I_\delta x ′ ∈ I δ
∣ ∫ y x K N ( x ′ , y ) d x ′ ∣ ≤ C ∫ 0 ∣ x − y ∣ min ( N , v − 1 ) d v ≤ C ( 1 + log ( 1 + N ∣ x − y ∣ ) ) . \left|\int_y^x K_N(x',y)\,\mathrm{d}x'\right|\le C\int_0^{|x-y|}\min(N,v^{-1})\,\mathrm{d}v\le C(1+\log(1+N|x-y|)). ∫ y x K N ( x ′ , y ) d x ′ ≤ C ∫ 0 ∣ x − y ∣ min ( N , v − 1 ) d v ≤ C ( 1 + log ( 1 + N ∣ x − y ∣ )) . Next, substituting N x ′ \sqrt{N}x' N x ′
∫ y x P ( x ′ , y ) d x ′ = N 2 F N ( w ′ ) ∫ w ′ w h N − 1 , ∫ y x q ( x ′ ) d x ′ = ( ∫ R h N − 1 ) − 1 ∫ w ′ w h N − 1 , \int_y^x P(x',y)\,\mathrm{d}x'=\frac{\sqrt{N}}{2}F_N(w')\int_{w'}^w h_{N-1},
\qquad
\int_y^x q(x')\,\mathrm{d}x'=\left(\int_{\mathbb{R}}h_{N-1}\right)^{-1}\int_{w'}^w h_{N-1}, ∫ y x P ( x ′ , y ) d x ′ = 2 N F N ( w ′ ) ∫ w ′ w h N − 1 , ∫ y x q ( x ′ ) d x ′ = ( ∫ R h N − 1 ) − 1 ∫ w ′ w h N − 1 , where the second identity is for odd N N N ∣ w ∣ , ∣ w ′ ∣ ≤ ( 2 − δ / 2 ) N − 1 |w|,|w'|\le(2-\delta/2)\sqrt{N-1} ∣ w ∣ , ∣ w ′ ∣ ≤ ( 2 − δ /2 ) N − 1 Lemma A.1 (b) with δ / 2 \delta/2 δ /2 δ \delta δ
∣ ∫ w ′ w h N − 1 ∣ ≤ C N − 3 / 4 , \left|\int_{w'}^w h_{N-1}\right|\le CN^{-3/4}, ∫ w ′ w h N − 1 ≤ C N − 3/4 , and therefore the integrals of P ( ⋅ , y ) P(\mathord{\cdot},y) P ( ⋅ , y ) q q q y y y x x x C N − 1 / 2 CN^{-1/2} C N − 1/2
We now prove (d). We have
T ( x , x ) − K N ( x , x ) = P ( x , x ) + q ( x ) , T(x,x)-K_N(x,x)=P(x,x)+q(x), T ( x , x ) − K N ( x , x ) = P ( x , x ) + q ( x ) , d d x T ( x , x ) = d d x K N ( x , x ) + ∂ 1 P ( x , x ) + ∂ 2 P ( x , x ) + q ′ ( x ) . \frac{\mathrm{d}}{\mathrm{d}x}T(x,x)=\frac{\mathrm{d}}{\mathrm{d}x}K_N(x,x)+\partial_1P(x,x)+\partial_2P(x,x)+q'(x). d x d T ( x , x ) = d x d K N ( x , x ) + ∂ 1 P ( x , x ) + ∂ 2 P ( x , x ) + q ′ ( x ) . and (A.23) and Lemma A.5(c) bound the right sides by C N 1 / 2 CN^{1/2} C N 1/2 C N 3 / 2 CN^{3/2} C N 3/2
∂ 2 2 V ( x , x ) = − ∂ y 2 K N ( x , y ) ∣ y = x − ∂ y 2 P ( x , y ) ∣ y = x ′ . \partial_2^2V(x,x)=-\left.\partial_y^2K_N(x,y)\right|_{y=x}-\left.\partial_y^2P(x,y)\right|_{y=x'}. ∂ 2 2 V ( x , x ) = − ∂ y 2 K N ( x , y ) y = x − ∂ y 2 P ( x , y ) y = x ′ . and the second term is bounded by C N 2 CN^2 C N 2 N 2 ∣ h N ( w ) h N − 1 ′ ( w ) ∣ / 3 N^2\lvert h_N(w)h'_{N-1}(w)\rvert/3 N 2 ∣ h N ( w ) h N − 1 ′ ( w )∣ /3 C N 2 CN^2 C N 2
We now deduce the density of states from eigenvalue rigidity, using the bound on the derivative of x ↦ T ( x , x ) x\mapsto T(x,x) x ↦ T ( x , x ) c ( E ) = ( π 2 / 6 ) ρ s c ( E ) 3 c(E)=(\pi^2/6)\rho_{\mathrm{sc}}(E)^3 c ( E ) = ( π 2 /6 ) ρ sc ( E ) 3 1 − x 2 / 4 = π 2 ρ s c ( x ) 2 1-x^2/4=\pi^2\rho_{\mathrm{sc}}(x)^2 1 − x 2 /4 = π 2 ρ sc ( x ) 2 ∣ x ∣ ≤ 2 \lvert x\rvert\leq2 ∣ x ∣ ≤ 2
Lemma A.7. Let δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) C > 0 C>0 C > 0 N 0 N_0 N 0 N ≥ N 0 N\geq N_0 N ≥ N 0 x ∈ I δ x\in I_\delta x ∈ I δ
∣ R 1 , N ( x ) − N ρ s c ( x ) ∣ + ∣ K N ( x , x ) − N ρ s c ( x ) ∣ ≤ C N 7 / 8 , \left\lvert R_{1,N}(x)-N\rho_{\mathrm{sc}}(x)\right\rvert+\left\lvert K_N(x,x)-N\rho_{\mathrm{sc}}(x)\right\rvert\leq CN^{7/8}, ∣ R 1 , N ( x ) − N ρ sc ( x ) ∣ + ∣ K N ( x , x ) − N ρ sc ( x ) ∣ ≤ C N 7/8 , ∣ 1 2 ∂ 2 V ( x , x ) − N 3 c ( x ) ∣ ≤ C N 23 / 8 . (448) \left\lvert\frac{1}{2}\partial_2V(x,x)-N^3c(x)\right\rvert\leq CN^{23/8}. \tag*{(448)} 2 1 ∂ 2 V ( x , x ) − N 3 c ( x ) ≤ C N 23/8 . (448) Proof. Set ε = 1 / 8 \varepsilon=1/8 ε = 1/8 c ∗ ∈ ( 0 , 1 / 4 ) c_*\in(0,1/4) c ∗ ∈ ( 0 , 1/4 )
∫ − 2 − 2 + δ / 4 ρ s c > 2 c ∗ . \int_{-2}^{-2+\delta/4}\rho_{\mathrm{sc}}>2c_*. ∫ − 2 − 2 + δ /4 ρ sc > 2 c ∗ . For N ≥ c ∗ − 1 N\geq c_*^{-1} N ≥ c ∗ − 1 γ N − i = − γ i \gamma_{N-i}=-\gamma_i γ N − i = − γ i
γ i < − 2 + δ 4 ( i ≤ c ∗ N + 1 ) , γ i > 2 − δ 4 ( i ≥ ( 1 − c ∗ ) N − 1 ) . \gamma_i<-2+\frac{\delta}{4}\quad(i\leq c_*N+1),\qquad\gamma_i>2-\frac{\delta}{4}\quad(i\geq(1-c_*)N-1). γ i < − 2 + 4 δ ( i ≤ c ∗ N + 1 ) , γ i > 2 − 4 δ ( i ≥ ( 1 − c ∗ ) N − 1 ) . Let E N \mathcal{E}_N E N ∣ λ i − γ i ∣ ≤ N − 1 + ε \lvert\lambda_i-\gamma_i\rvert\leq N^{-1+\varepsilon} ∣ λ i − γ i ∣ ≤ N − 1 + ε i i i c ∗ N ≤ i ≤ ( 1 − c ∗ ) N c_*N\leq i\leq(1-c_*)N c ∗ N ≤ i ≤ ( 1 − c ∗ ) N G G G P ( E N c ) ≤ N − 2 \mathbb{P}(\mathcal{E}_N^c)\leq N^{-2} P ( E N c ) ≤ N − 2 N ≥ N 0 N\geq N_0 N ≥ N 0
We first show that for all y 1 < y 2 y_1<y_2 y 1 < y 2 [ y 1 , y 2 ] ⊂ I δ / 2 [y_1,y_2]\subset I_{\delta/2} [ y 1 , y 2 ] ⊂ I δ /2
∣ E # { i : λ i ∈ [ y 1 , y 2 ] } − N ∫ y 1 y 2 ρ s c ∣ ≤ C N ε . (449) \left\lvert\mathbb{E}\#\{i:\lambda_i\in[y_1,y_2]\}-N\int_{y_1}^{y_2}\rho_{\mathrm{sc}}\right\rvert\leq CN^\varepsilon. \tag*{(449)} E # { i : λ i ∈ [ y 1 , y 2 ]} − N ∫ y 1 y 2 ρ sc ≤ C N ε . (449) Suppose that E N \mathcal{E}_N E N i < c ∗ N i<c_*N i < c ∗ N
λ i ≤ λ ⌈ c ∗ N ⌉ ≤ γ ⌈ c ∗ N ⌉ + N − 1 + ε < y 1 , \lambda_i\leq\lambda_{\lceil c_*N\rceil}\leq\gamma_{\lceil c_*N\rceil}+N^{-1+\varepsilon}<y_1, λ i ≤ λ ⌈ c ∗ N ⌉ ≤ γ ⌈ c ∗ N ⌉ + N − 1 + ε < y 1 , provided N ≥ N 0 N \ge N_0 N ≥ N 0 λ i > y 2 \lambda_i > y_2 λ i > y 2 i > ( 1 − c ∗ ) N i > (1-c_*)N i > ( 1 − c ∗ ) N
λ i ∈ [ y 1 , y 2 ] ⟹ γ i ∈ [ y 1 − N − 1 + ε , y 2 + N − 1 + ε ] , \lambda_i \in[y_1,y_2] \Longrightarrow\gamma_i \in[y_1-N^{-1+\varepsilon},y_2+N^{-1+\varepsilon}], λ i ∈ [ y 1 , y 2 ] ⟹ γ i ∈ [ y 1 − N − 1 + ε , y 2 + N − 1 + ε ] , γ i ∈ [ y 1 + N − 1 + ε , y 2 − N − 1 + ε ] ⟹ λ i ∈ [ y 1 , y 2 ] . \gamma_i \in[y_1+N^{-1+\varepsilon},y_2-N^{-1+\varepsilon}] \Longrightarrow\lambda_i \in[y_1,y_2]. γ i ∈ [ y 1 + N − 1 + ε , y 2 − N − 1 + ε ] ⟹ λ i ∈ [ y 1 , y 2 ] . Since ρ s c ≤ 1 / π \rho_{\mathrm{sc}} \le1/\pi ρ sc ≤ 1/ π π / N \pi/N π / N [ y 1 − N − 1 + ε , y 1 + N − 1 + ε ] [y_1-N^{-1+\varepsilon},y_1+N^{-1+\varepsilon}] [ y 1 − N − 1 + ε , y 1 + N − 1 + ε ] [ y 2 − N − 1 + ε , y 2 + N − 1 + ε ] [y_2-N^{-1+\varepsilon},y_2+N^{-1+\varepsilon}] [ y 2 − N − 1 + ε , y 2 + N − 1 + ε ] C N ε CN^\varepsilon C N ε [ y 1 , y 2 ] [y_1,y_2] [ y 1 , y 2 ] N ∫ y 1 y 2 ρ s c + O ( 1 ) N\int_{y_1}^{y_2}\rho_{\mathrm{sc}}+O(1) N ∫ y 1 y 2 ρ sc + O ( 1 ) E N \mathcal{E}_N E N
# { i : λ i ∈ [ y 1 , y 2 ] } = N ∫ y 1 y 2 ρ s c + O ( N ε ) . \#\{i:\lambda_i\in[y_1,y_2]\}=N\int_{y_1}^{y_2}\rho_{\mathrm{sc}}+O(N^\varepsilon). # { i : λ i ∈ [ y 1 , y 2 ]} = N ∫ y 1 y 2 ρ sc + O ( N ε ) . On E N c \mathcal{E}_N^c E N c N N N N P ( E N c ) ≤ N − 1 N\mathbb{P}(\mathcal{E}_N^c)\le N^{-1} N P ( E N c ) ≤ N − 1
Now let x ∈ I δ x\in I_\delta x ∈ I δ [ x , x + N − 3 / 4 ] ⊂ I δ / 2 [x,x+N^{-3/4}]\subset I_{\delta/2} [ x , x + N − 3/4 ] ⊂ I δ /2 n = 1 n=1 n = 1
E # { i : λ i ∈ [ x , x + N − 3 / 4 ] } = ∫ x x + N − 3 / 4 R 1 , N ( x ′ ) d x ′ . \mathbb{E}\#\{i:\lambda_i\in[x,x+N^{-3/4}]\}=\int_x^{x+N^{-3/4}}R_{1,N}(x')\,\mathrm{d}x'. E # { i : λ i ∈ [ x , x + N − 3/4 ]} = ∫ x x + N − 3/4 R 1 , N ( x ′ ) d x ′ . Note that R 1 , N ( x ′ ) = T ( x ′ , x ′ ) R_{1,N}(x')=T(x',x') R 1 , N ( x ′ ) = T ( x ′ , x ′ ) I δ / 2 I_{\delta/2} I δ /2
∣ R 1 , N ( x ′ ) − R 1 , N ( x ) ∣ ≤ C N 3 / 2 N − 3 / 4 = C N 3 / 4 , x ′ ∈ [ x , x + N − 3 / 4 ] . |R_{1,N}(x')-R_{1,N}(x)|\le CN^{3/2}N^{-3/4}=CN^{3/4},\qquad x'\in[x,x+N^{-3/4}]. ∣ R 1 , N ( x ′ ) − R 1 , N ( x ) ∣ ≤ C N 3/2 N − 3/4 = C N 3/4 , x ′ ∈ [ x , x + N − 3/4 ] . Using also (A.25) and the fact that ρ s c \rho_{\mathrm{sc}} ρ sc I δ / 2 I_{\delta/2} I δ /2
∣ R 1 , N ( x ) − N ρ s c ( x ) ∣ ≤ N 3 / 4 ∣ ∫ x x + N − 3 / 4 R 1 , N − N ∫ x x + N − 3 / 4 ρ s c ∣ + C N 3 / 4 + C N 1 / 4 ≤ C ( N ε + 3 / 4 + N 3 / 4 ) ≤ C N 7 / 8 . \begin{aligned}
|R_{1,N}(x)-N\rho_{\mathrm{sc}}(x)|\le N^{3/4}\left|\int_x^{x+N^{-3/4}}R_{1,N}-N\int_x^{x+N^{-3/4}}\rho_{\mathrm{sc}}\right|+CN^{3/4}+CN^{1/4} \\
&\le C(N^{\varepsilon+3/4}+N^{3/4})\le CN^{7/8}.
\end{aligned} ∣ R 1 , N ( x ) − N ρ sc ( x ) ∣ ≤ N 3/4 ∫ x x + N − 3/4 R 1 , N − N ∫ x x + N − 3/4 ρ sc + C N 3/4 + C N 1/4 ≤ C ( N ε + 3/4 + N 3/4 ) ≤ C N 7/8 . By Lemma A.6(d), we have ∣ K N ( x , x ) − R 1 , N ( x ) ∣ ≤ C N 1 / 2 |K_N(x,x)-R_{1,N}(x)|\le CN^{1/2} ∣ K N ( x , x ) − R 1 , N ( x ) ∣ ≤ C N 1/2 1 − x 2 / 4 = π 2 ρ s c ( x ) 2 1-x^2/4=\pi^2\rho_{\mathrm{sc}}(x)^2 1 − x 2 /4 = π 2 ρ sc ( x ) 2
1 2 ∂ 2 V ( x , x ) = ( π 2 6 N 2 ρ s c ( x ) 2 + N 12 ) ( N ρ s c ( x ) + O ( N 7 / 8 ) ) + O ( N 2 ) = N 3 c ( x ) + O ( N 23 / 8 ) , \frac{1}{2}\partial_2V(x,x)=\left(\frac{\pi^2}{6}N^2\rho_{\mathrm{sc}}(x)^2+\frac{N}{12}\right)\left(N\rho_{\mathrm{sc}}(x)+O(N^{7/8})\right)+O(N^2)=N^3c(x)+O(N^{23/8}), 2 1 ∂ 2 V ( x , x ) = ( 6 π 2 N 2 ρ sc ( x ) 2 + 12 N ) ( N ρ sc ( x ) + O ( N 7/8 ) ) + O ( N 2 ) = N 3 c ( x ) + O ( N 23/8 ) , which is the second bound in (A.24).
Correlation functions
We now estimate the correlation functions. In the next lemma, part (a) determines the mean of the statistic in (51) , and the cruder bound in part (b) will be used for the tails.
Lemma A.8. Let δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) C > 0 C>0 C > 0 N 0 N_0 N 0 N ≥ N 0 N\ge N_0 N ≥ N 0
(a) For all x , y ∈ I δ x,y \in I_\delta x , y ∈ I δ 0 < y − x ≤ N − 1 0 < y-x \le N^{-1} 0 < y − x ≤ N − 1
∣ R 2 , N ( x , y ) − N 3 c ( x ) ( y − x ) ∣ ≤ C ( N 23 / 8 ( y − x ) + N 4 ( y − x ) 2 ) , \left|R_{2,N}(x,y)-N^3c(x)(y-x)\right| \le C\left(N^{23/8}(y-x)+N^4(y-x)^2\right), R 2 , N ( x , y ) − N 3 c ( x ) ( y − x ) ≤ C ( N 23/8 ( y − x ) + N 4 ( y − x ) 2 ) , ∣ R 2 , N ( x , y ) ∣ ≤ C N 3 ( y − x ) . (450) \left|R_{2,N}(x,y)\right| \le CN^3(y-x). \tag*{(450)} ∣ R 2 , N ( x , y ) ∣ ≤ C N 3 ( y − x ) . (450) (b) For all x , y ∈ I δ x,y \in I_\delta x , y ∈ I δ ∣ x − y ∣ ≥ N − 1 \lvert x-y\rvert\ge N^{-1} ∣ x − y ∣ ≥ N − 1 ∣ R 2 , N ( x , y ) ∣ ≤ C N 2 log N \lvert R_{2,N}(x,y)\rvert\le CN^2\log N ∣ R 2 , N ( x , y )∣ ≤ C N 2 log N
Proof. We begin with (a). We set y v = x + v y_v=x+v y v = x + v v ∈ ( 0 , y − x ] v \in(0,y-x] v ∈ ( 0 , y − x ]
r ( v ) = R 2 , N ( x , y v ) = T ( x , x ) T ( y v , y v ) − T ( x , y v ) T ( y v , x ) + I ( x , y v ) V ( x , y v ) , r(v)=R_{2,N}(x,y_v)=T(x,x)T(y_v,y_v)-T(x,y_v)T(y_v,x)+I(x,y_v)V(x,y_v), r ( v ) = R 2 , N ( x , y v ) = T ( x , x ) T ( y v , y v ) − T ( x , y v ) T ( y v , x ) + I ( x , y v ) V ( x , y v ) , where the second equality is (A.16). By Lemma A.4 (b) and the antisymmetry of I I I
I ( x , y ′ ) = 1 2 − ∫ x y ′ T ( x ′ , x ) d x ′ , ∂ 2 I ( x , y ′ ) = − T ( y ′ , x ) , y ′ > x , I(x,y')=\frac{1}{2}-\int_x^{y'}T(x',x)\,\mathrm{d}x',\qquad\partial_2 I(x,y')=-T(y',x),\qquad y'>x, I ( x , y ′ ) = 2 1 − ∫ x y ′ T ( x ′ , x ) d x ′ , ∂ 2 I ( x , y ′ ) = − T ( y ′ , x ) , y ′ > x , so r r r ( 0 , y − x ] (0,y-x] ( 0 , y − x ]
r ′ ( v ) = T ( x , x ) ( ∂ 1 T + ∂ 2 T ) ( y v , y v ) − ∂ 2 T ( x , y v ) T ( y v , x ) − T ( x , y v ) ∂ 1 T ( y v , x ) − T ( y v , x ) V ( x , y v ) + I ( x , y v ) ∂ 2 V ( x , y v ) . r'(v)=T(x,x)(\partial_1T+\partial_2T)(y_v,y_v)-\partial_2T(x,y_v)T(y_v,x)-T(x,y_v)\partial_1T(y_v,x)
-T(y_v,x)V(x,y_v)+I(x,y_v)\partial_2V(x,y_v). r ′ ( v ) = T ( x , x ) ( ∂ 1 T + ∂ 2 T ) ( y v , y v ) − ∂ 2 T ( x , y v ) T ( y v , x ) − T ( x , y v ) ∂ 1 T ( y v , x ) − T ( y v , x ) V ( x , y v ) + I ( x , y v ) ∂ 2 V ( x , y v ) . We now let v ↓ 0 v \downarrow0 v ↓ 0 I ( x , y v ) → 1 / 2 I(x,y_v)\to1/2 I ( x , y v ) → 1/2 V ( x , x ) = 0 V(x,x)=0 V ( x , x ) = 0 Lemma A.4 (a). Since T T T V V V r ′ ( v ) r'(v) r ′ ( v ) − T ( x , x ) V ( x , x ) = 0 -T(x,x)V(x,x)=0 − T ( x , x ) V ( x , x ) = 0 r ( v ) → 0 r(v)\to0 r ( v ) → 0
lim v ↓ 0 r ′ ( v ) = 1 2 ∂ 2 V ( x , x ) . (451) \lim_{v\downarrow0}r'(v)=\frac{1}{2}\partial_2V(x,x). \tag*{(451)} v ↓ 0 lim r ′ ( v ) = 2 1 ∂ 2 V ( x , x ) . (451) Next, we bound r ′ r' r ′ r ′ ′ r'' r ′′ v ≤ N − 1 v\le N^{-1} v ≤ N − 1 Lemma A.6 (c) gives ∣ I ( x , y v ) ∣ ≤ C \lvert I(x,y_v)\rvert\le C ∣ I ( x , y v )∣ ≤ C T T T V V V I δ 2 I_\delta^2 I δ 2 Lemma A.6 (a), and ∂ 2 2 I ( x , y ′ ) = − ∂ 1 T ( y ′ , x ) \partial_2^2I(x,y')=-\partial_1T(y',x) ∂ 2 2 I ( x , y ′ ) = − ∂ 1 T ( y ′ , x ) y ′ > x y'>x y ′ > x r ′ r' r ′
∣ r ′ ( v ) ∣ ≤ C N 3 , ∣ r ′ ′ ( v ) ∣ ≤ C N 4 , v ∈ ( 0 , y − x ] . \lvert r'(v)\rvert\le CN^3,\qquad\lvert r''(v)\rvert\le CN^4,\qquad v\in(0,y-x]. ∣ r ′ ( v )∣ ≤ C N 3 , ∣ r ′′ ( v )∣ ≤ C N 4 , v ∈ ( 0 , y − x ] . For example, the last term of r ′ r' r ′ ∣ I ∂ 2 2 V ∣ + ∣ ∂ 2 I ∂ 2 V ∣ ≤ C N 4 \lvert I\partial_2^2V\rvert+\lvert\partial_2I\,\partial_2V\rvert\le CN^4 ∣ I ∂ 2 2 V ∣ + ∣ ∂ 2 I ∂ 2 V ∣ ≤ C N 4 r ′ ′ r'' r ′′ r ( v ) → 0 r(v)\to0 r ( v ) → 0 v ↓ 0 v\downarrow0 v ↓ 0
R 2 , N ( x , y ) = ∫ 0 y − x r ′ ( v ) d v = y − x 2 ∂ 2 V ( x , x ) + ∫ 0 y − x ( r ′ ( v ) − 1 2 ∂ 2 V ( x , x ) ) d v , R_{2,N}(x,y)=\int_0^{y-x}r'(v)\,\mathrm{d}v
=\frac{y-x}{2}\partial_2V(x,x)+\int_0^{y-x}\left(r'(v)-\frac{1}{2}\partial_2V(x,x)\right)\,\mathrm{d}v, R 2 , N ( x , y ) = ∫ 0 y − x r ′ ( v ) d v = 2 y − x ∂ 2 V ( x , x ) + ∫ 0 y − x ( r ′ ( v ) − 2 1 ∂ 2 V ( x , x ) ) d v , where the last integral is bounded by C N 4 ( y − x ) 2 CN^4(y-x)^2 C N 4 ( y − x ) 2 r ′ ′ r'' r ′′ Lemma A.7 , we obtain the first bound in (A.26). The second bound follows from the first identity in the last display and ∣ r ′ ∣ ≤ C N 3 \lvert r'\rvert\le CN^3 ∣ r ′ ∣ ≤ C N 3
We now prove (b), using (A.16) and Lemma A.6 . Since ∣ x − y ∣ ≥ N − 1 |x-y| \ge N^{-1} ∣ x − y ∣ ≥ N − 1 ∣ T ∣ ≤ C N |T| \le CN ∣ T ∣ ≤ C N I δ 2 I_{\delta}^{2} I δ 2
∣ V ( x , y ) ∣ ≤ C ( N 2 + N 2 + N ) ≤ C N 2 , ∣ I ( x , y ) ∣ ≤ C ( 1 + log ( 1 + 4 N ) ) ≤ C log N , |V(x,y)| \le C(N^{2}+N^{2}+N) \le CN^{2}, \qquad|I(x,y)| \le C(1+\log(1+4N)) \le C\log N, ∣ V ( x , y ) ∣ ≤ C ( N 2 + N 2 + N ) ≤ C N 2 , ∣ I ( x , y ) ∣ ≤ C ( 1 + log ( 1 + 4 N )) ≤ C log N , where we also used ∣ x − y ∣ ≤ 4 |x-y| \le4 ∣ x − y ∣ ≤ 4 ∣ R 2 , N ( x , y ) ∣ ≤ C N 2 log N |R_{2,N}(x,y)| \le CN^{2}\log N ∣ R 2 , N ( x , y ) ∣ ≤ C N 2 log N
To bound the higher correlation functions, we normalize the entries of the matrix in (A.14). Let D N = diag ( 1 , N − 1 ) D_{N}=\operatorname{diag}(1,N^{-1}) D N = diag ( 1 , N − 1 ) B n B_{n} B n n n n D N D_{N} D N det B n = N − n \det B_{n}=N^{-n} det B n = N − n
R n , N ( x 1 , … , x n ) = N n Pf [ D N A ( x i , x j ) D N ] i , j = 1 n . (452) R_{n,N}(x_{1},\ldots,x_{n})=N^{n}\operatorname{Pf}\left[D_{N}A(x_{i},x_{j})D_{N}\right]_{i,j=1}^{n}. \tag*{(452)} R n , N ( x 1 , … , x n ) = N n Pf [ D N A ( x i , x j ) D N ] i , j = 1 n . (452) where
D N A ( x , y ) D N = ( I ( x , y ) N − 1 T ( y , x ) − N − 1 T ( x , y ) − N − 2 V ( x , y ) ) . D_{N}A(x,y)D_{N}=
\begin{pmatrix}
I(x,y) & N^{-1}T(y,x)\\
-N^{-1}T(x,y) & -N^{-2}V(x,y)
\end{pmatrix}. D N A ( x , y ) D N = ( I ( x , y ) − N − 1 T ( x , y ) N − 1 T ( y , x ) − N − 2 V ( x , y ) ) . For the matrix in (A.28), we call row 2 i − 1 2i-1 2 i − 1 x i x_{i} x i 2 i 2i 2 i Lemma A.6 (a), all entries of D N A ( x , y ) D N D_{N}A(x,y)D_{N} D N A ( x , y ) D N I ( x , y ) I(x,y) I ( x , y ) x , y ∈ I δ x,y \in I_{\delta} x , y ∈ I δ ϱ > 0 \varrho>0 ϱ > 0
Δ 0 ( ϱ ) = ( N ϱ ) − 1 + N − 1 / 2 . (453) \Delta_{0}(\varrho)=(N\varrho)^{-1}+N^{-1/2}. \tag*{(453)} Δ 0 ( ϱ ) = ( N ϱ ) − 1 + N − 1/2 . (453) Lemma A.9. Let δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) C > 0 C>0 C > 0 N 0 N_{0} N 0 N ≥ N 0 N \ge N_{0} N ≥ N 0
(a) For all pairwise distinct x 1 , x 2 , x 3 ∈ I δ x_{1},x_{2},x_{3}\in I_{\delta} x 1 , x 2 , x 3 ∈ I δ ∣ x i − x j ∣ ≤ N − 1 |x_{i}-x_{j}| \le N^{-1} ∣ x i − x j ∣ ≤ N − 1 i , j i,j i , j ∣ R 3 , N ( x 1 , x 2 , x 3 ) ∣ ≤ C N 3 |R_{3,N}(x_{1},x_{2},x_{3})| \le CN^{3} ∣ R 3 , N ( x 1 , x 2 , x 3 ) ∣ ≤ C N 3
(b) For all pairwise distinct x 1 , x 2 , x 3 , x 4 ∈ I δ x_{1},x_{2},x_{3},x_{4}\in I_{\delta} x 1 , x 2 , x 3 , x 4 ∈ I δ ∣ R 4 , N ( x 1 , x 2 , x 3 , x 4 ) ∣ ≤ C N 4 ( log N ) 2 |R_{4,N}(x_{1},x_{2},x_{3},x_{4})| \le CN^{4}(\log N)^{2} ∣ R 4 , N ( x 1 , x 2 , x 3 , x 4 ) ∣ ≤ C N 4 ( log N ) 2
(c) Let x 1 , x 2 , x 3 , x 4 ∈ I δ x_{1},x_{2},x_{3},x_{4}\in I_{\delta} x 1 , x 2 , x 3 , x 4 ∈ I δ x 1 < x 2 ≤ x 1 + N − 1 x_{1}<x_{2}\le x_{1}+N^{-1} x 1 < x 2 ≤ x 1 + N − 1 x 3 < x 4 ≤ x 3 + N − 1 x_{3}<x_{4}\le x_{3}+N^{-1} x 3 < x 4 ≤ x 3 + N − 1 ϱ = ∣ x 3 − x 1 ∣ \varrho=|x_{3}-x_{1}| ϱ = ∣ x 3 − x 1 ∣ ϱ ≥ max ( 4 ( x 2 − x 1 + x 4 − x 3 ) , N − 1 ) \varrho\ge\max(4(x_{2}-x_{1}+x_{4}-x_{3}),N^{-1}) ϱ ≥ max ( 4 ( x 2 − x 1 + x 4 − x 3 ) , N − 1 )
∣ R 4 , N ( x 1 , x 2 , x 3 , x 4 ) − R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) ∣ ≤ C N 4 ( log N ) Δ 0 ( ϱ ) . (454) |R_{4,N}(x_{1},x_{2},x_{3},x_{4})-R_{2,N}(x_{1},x_{2})R_{2,N}(x_{3},x_{4})| \le CN^{4}(\log N)\Delta_{0}(\varrho). \tag*{(454)} ∣ R 4 , N ( x 1 , x 2 , x 3 , x 4 ) − R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) ∣ ≤ C N 4 ( log N ) Δ 0 ( ϱ ) . (454) Proof. We begin with (a). The points are pairwise within distance N − 1 N^{-1} N − 1 Lemma A.6 (c) shows that the entries I ( x i , x j ) I(x_{i},x_{j}) I ( x i , x j ) 6 × 6 6 \times6 6 × 6 ∣ R 3 , N ∣ ≤ 15 C 3 N 3 |R_{3,N}| \le15C^{3}N^{3} ∣ R 3 , N ∣ ≤ 15 C 3 N 3
Next, we prove (b). The entries I ( x i , x j ) I(x_{i},x_{j}) I ( x i , x j ) 8 × 8 8 \times8 8 × 8 C log N C\log N C log N Lemma A.6 (c), and all other entries are bounded. The entries I ( x i , x j ) I(x_{i},x_{j}) I ( x i , x j ) [ 8 ] [8] [ 8 ] { σ , σ ′ } \{\sigma,\sigma'\} { σ , σ ′ } σ \sigma σ σ ′ \sigma' σ ′ C ( log N ) 2 C(\log N)^{2} C ( log N ) 2 ∣ R 4 , N ∣ ≤ 105 C N 4 ( log N ) 2 |R_{4,N}| \le105CN^{4}(\log N)^{2} ∣ R 4 , N ∣ ≤ 105 C N 4 ( log N ) 2 [33 Section 5.2] . Let W W W 8 × 8 8 \times8 8 × 8 x 1 , x 2 , x 3 , x 4 x_1,x_2,x_3,x_4 x 1 , x 2 , x 3 , x 4 W ′ W' W ′ W W W W ′ W' W ′ x 2 x_2 x 2 x 1 x_1 x 1 x 4 x_4 x 4 x 3 x_3 x 3 W ′ = B T W B W' = B^{T}WB W ′ = B T W B B B B det B = 1 \det B = 1 det B = 1
Pf W ′ = Pf W = N − 4 R 4 , N ( x 1 , x 2 , x 3 , x 4 ) . \operatorname{Pf} W' = \operatorname{Pf} W = N^{-4}R_{4,N}(x_1,x_2,x_3,x_4). Pf W ′ = Pf W = N − 4 R 4 , N ( x 1 , x 2 , x 3 , x 4 ) . Let P 1 = { 1 , 2 , 3 , 4 } P_1 = \{1,2,3,4\} P 1 = { 1 , 2 , 3 , 4 } P 2 = { 5 , 6 , 7 , 8 } P_2 = \{5,6,7,8\} P 2 = { 5 , 6 , 7 , 8 } P 1 P_1 P 1 P 2 P_2 P 2 W P 1 P 1 ′ W'_{P_1P_1} W P 1 P 1 ′ W P 1 P 1 W_{P_1P_1} W P 1 P 1 n = 2 n = 2 n = 2
Pf W P 1 P 1 ′ = Pf W P 1 P 1 = N − 2 R 2 , N ( x 1 , x 2 ) , Pf W P 2 P 2 ′ = N − 2 R 2 , N ( x 3 , x 4 ) . \operatorname{Pf} W'_{P_1P_1} = \operatorname{Pf} W_{P_1P_1} = N^{-2}R_{2,N}(x_1,x_2), \qquad\operatorname{Pf} W'_{P_2P_2} = N^{-2}R_{2,N}(x_3,x_4). Pf W P 1 P 1 ′ = Pf W P 1 P 1 = N − 2 R 2 , N ( x 1 , x 2 ) , Pf W P 2 P 2 ′ = N − 2 R 2 , N ( x 3 , x 4 ) . We next bound the entries of W ′ W' W ′ x ′ ∈ [ x 1 , x 2 ] x' \in[x_1,x_2] x ′ ∈ [ x 1 , x 2 ] j ∈ { 3 , 4 } j \in\{3,4\} j ∈ { 3 , 4 }
∣ x ′ − x j ∣ ≥ ϱ − ( x 2 − x 1 ) − ( x 4 − x 3 ) ≥ ϱ 2 . |x' - x_j| \geq\varrho- (x_2-x_1) - (x_4-x_3) \geq\frac{\varrho}{2}. ∣ x ′ − x j ∣ ≥ ϱ − ( x 2 − x 1 ) − ( x 4 − x 3 ) ≥ 2 ϱ . and the same holds for x ′ ∈ [ x 3 , x 4 ] x' \in[x_3,x_4] x ′ ∈ [ x 3 , x 4 ] j ∈ { 1 , 2 } j \in\{1,2\} j ∈ { 1 , 2 } W P 1 P 1 ′ W'_{P_1P_1} W P 1 P 1 ′ W P 1 P 1 W_{P_1P_1} W P 1 P 1 ∣ x 1 − x 2 ∣ ≤ N − 1 |x_1-x_2| \leq N^{-1} ∣ x 1 − x 2 ∣ ≤ N − 1 W P 1 P 1 ′ W'_{P_1P_1} W P 1 P 1 ′ Lemma A.6 (a) and (c). The same holds for W P 2 P 2 ′ W'_{P_2P_2} W P 2 P 2 ′
Now consider an entry W σ σ ′ ′ W'_{\sigma\sigma'} W σ σ ′ ′ σ ∈ P 1 \sigma\in P_1 σ ∈ P 1 σ ′ ∈ P 2 \sigma' \in P_2 σ ′ ∈ P 2 σ \sigma σ σ ′ \sigma' σ ′ W σ σ ′ ′ = − N − 2 V ( x i , x j ) W'_{\sigma\sigma'} = -N^{-2}V(x_i,x_j) W σ σ ′ ′ = − N − 2 V ( x i , x j ) i ∈ { 1 , 2 } i \in\{1,2\} i ∈ { 1 , 2 } j ∈ { 3 , 4 } j \in\{3,4\} j ∈ { 3 , 4 } σ \sigma σ σ ′ \sigma' σ ′ W σ σ ′ ′ W'_{\sigma\sigma'} W σ σ ′ ′ ± N − 1 T ( x i , x j ) \pm N^{-1}T(x_i,x_j) ± N − 1 T ( x i , x j ) ± N − 1 T ( x j , x i ) \pm N^{-1}T(x_j,x_i) ± N − 1 T ( x j , x i ) i ∈ { 1 , 2 } i \in\{1,2\} i ∈ { 1 , 2 } j ∈ { 3 , 4 } j \in\{3,4\} j ∈ { 3 , 4 } Lemma A.6 (b) and N ϱ ≥ 1 N\varrho\geq1 N ϱ ≥ 1
N − 1 ∣ T ( x i , x j ) ∣ + N − 1 ∣ T ( x j , x i ) ∣ ≤ C ( ( N ϱ ) − 1 + N − 1 / 2 ) ≤ C Δ 0 ( ϱ ) , N^{-1}|T(x_i,x_j)| + N^{-1}|T(x_j,x_i)| \leq C\bigl((N\varrho)^{-1} + N^{-1/2}\bigr) \leq C\Delta_0(\varrho), N − 1 ∣ T ( x i , x j ) ∣ + N − 1 ∣ T ( x j , x i ) ∣ ≤ C ( ( N ϱ ) − 1 + N − 1/2 ) ≤ C Δ 0 ( ϱ ) , N − 2 ∣ V ( x i , x j ) ∣ ≤ C ( ( N ϱ ) − 1 + ( N ϱ ) − 2 + N − 1 ) ≤ C Δ 0 ( ϱ ) N^{-2}|V(x_i,x_j)| \leq C\bigl((N\varrho)^{-1} + (N\varrho)^{-2} + N^{-1}\bigr) \leq C\Delta_0(\varrho) N − 2 ∣ V ( x i , x j ) ∣ ≤ C ( ( N ϱ ) − 1 + ( N ϱ ) − 2 + N − 1 ) ≤ C Δ 0 ( ϱ ) for i ∈ { 1 , 2 } i \in\{1,2\} i ∈ { 1 , 2 } j ∈ { 3 , 4 } j \in\{3,4\} j ∈ { 3 , 4 } ∣ W σ σ ′ ′ ∣ ≤ C Δ 0 ( ϱ ) |W'_{\sigma\sigma'}| \leq C\Delta_0(\varrho) ∣ W σ σ ′ ′ ∣ ≤ C Δ 0 ( ϱ )
It remains to consider the first rows σ ∈ { 1 , 3 } \sigma\in\{1,3\} σ ∈ { 1 , 3 } σ ′ ∈ { 5 , 7 } \sigma' \in\{5,7\} σ ′ ∈ { 5 , 7 } W 15 ′ = I ( x 1 , x 3 ) W'_{15} = I(x_1,x_3) W 15 ′ = I ( x 1 , x 3 ) C log N C\log N C log N Lemma A.6 (c). Second, since x 3 ∉ [ x 1 , x 2 ] x_3 \notin[x_1,x_2] x 3 ∈ / [ x 1 , x 2 ]
W 35 ′ = I ( x 2 , x 3 ) − I ( x 1 , x 3 ) = ∫ x 1 x 2 T ( x ′ , x 3 ) d x ′ , W'_{35} = I(x_2,x_3) - I(x_1,x_3) = \int_{x_1}^{x_2} T(x',x_3)\,\mathrm{d}x', W 35 ′ = I ( x 2 , x 3 ) − I ( x 1 , x 3 ) = ∫ x 1 x 2 T ( x ′ , x 3 ) d x ′ , so Lemma A.6 (b) and the assumption that x 2 − x 1 ≤ N − 1 x_2-x_1 \leq N^{-1} x 2 − x 1 ≤ N − 1
∣ W 35 ′ ∣ ≤ C ( x 2 − x 1 ) ( ϱ − 1 + N 1 / 2 ) ≤ C Δ 0 ( ϱ ) . |W'_{35}| \leq C(x_2-x_1)(\varrho^{-1} + N^{1/2}) \leq C\Delta_0(\varrho). ∣ W 35 ′ ∣ ≤ C ( x 2 − x 1 ) ( ϱ − 1 + N 1/2 ) ≤ C Δ 0 ( ϱ ) . Third, by the antisymmetry of I I I
W 17 ′ = I ( x 1 , x 4 ) − I ( x 1 , x 3 ) = − ∫ x 3 x 4 T ( x ′ , x 1 ) d x ′ W'_{17}=I(x_1,x_4)-I(x_1,x_3)=-\int_{x_3}^{x_4}T(x',x_1)\,\mathrm{d}x' W 17 ′ = I ( x 1 , x 4 ) − I ( x 1 , x 3 ) = − ∫ x 3 x 4 T ( x ′ , x 1 ) d x ′ which is bounded in the same way. Fourth, the same argument gives
W 37 ′ = ∫ x 1 x 2 ( T ( x ′ , x 4 ) − T ( x ′ , x 3 ) ) d x ′ W'_{37}=\int_{x_1}^{x_2}\left(T(x',x_4)-T(x',x_3)\right)\,\mathrm{d}x' W 37 ′ = ∫ x 1 x 2 ( T ( x ′ , x 4 ) − T ( x ′ , x 3 ) ) d x ′ which is also bounded by C Δ 0 ( ϱ ) C\Delta_0(\varrho) C Δ 0 ( ϱ )
Finally, we expand Pf W ′ \operatorname{Pf}W' Pf W ′ m m m m m m P 1 P_1 P 1 P 2 P_2 P 2 Pf W P 1 P 1 ′ Pf W P 2 P 2 ′ \operatorname{Pf}W'_{P_1P_1}\operatorname{Pf}W'_{P_2P_2} Pf W P 1 P 1 ′ Pf W P 2 P 2 ′ ∣ P 1 ∣ = 4 \lvert P_1\rvert=4 ∣ P 1 ∣ = 4 P 1 P_1 P 1 P 2 P_2 P 2 { 1 , 5 } \{1,5\} { 1 , 5 } C Δ 0 ( ϱ ) C\Delta_0(\varrho) C Δ 0 ( ϱ ) C Δ 0 ( ϱ ) C\Delta_0(\varrho) C Δ 0 ( ϱ ) C log N C\log N C log N C C C Δ 0 ( ϱ ) ≤ 2 \Delta_0(\varrho)\leq2 Δ 0 ( ϱ ) ≤ 2
∣ N − 4 R 4 , N ( x 1 , x 2 , x 3 , x 4 ) − N − 4 R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) ∣ ≤ 105 C ( log N ) Δ 0 ( ϱ ) (455) \left|N^{-4}R_{4,N}(x_1,x_2,x_3,x_4)-N^{-4}R_{2,N}(x_1,x_2)R_{2,N}(x_3,x_4)\right|\leq105C(\log N)\Delta_0(\varrho) \tag*{(455)} N − 4 R 4 , N ( x 1 , x 2 , x 3 , x 4 ) − N − 4 R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) ≤ 105 C ( log N ) Δ 0 ( ϱ ) (455) which is (A.30).
Mean, variance, and tails
We now prove Proposition 3.6. Recall from (3.15) that ℓ α = N − 1 − α \ell_\alpha=N^{-1-\alpha} ℓ α = N − 1 − α M α = N 1 − 2 α = N 3 ℓ α 2 M_\alpha=N^{1-2\alpha}=N^3\ell_\alpha^2 M α = N 1 − 2 α = N 3 ℓ α 2 δ ∈ ( 0 , 1 / 4 ) \delta\in(0,1/4) δ ∈ ( 0 , 1/4 ) I ⊂ I 4 δ I\subset I_{4\delta} I ⊂ I 4 δ g : [ 0 , ∞ ) × R → R g:[0,\infty)\times\mathbb{R}\to\mathbb{R} g : [ 0 , ∞ ) × R → R
X α ( G ; g ) = ∑ 1 ≤ i < j ≤ N g ( λ j − λ i ℓ α , λ i + λ j 2 ) , L ( g ) = ∫ − 2 2 ∫ 0 ∞ c ( E ) r g ( r , E ) d r d E (456) \mathcal{X}_\alpha(G;g)=\sum_{1\leq i<j\leq N}g\left(\frac{\lambda_j-\lambda_i}{\ell_\alpha},\frac{\lambda_i+\lambda_j}{2}\right),\qquad\mathcal{L}(g)=\int_{-2}^{2}\int_0^\infty c(E)rg(r,E)\,\mathrm{d}r\,\mathrm{d}E \tag*{(456)} X α ( G ; g ) = 1 ≤ i < j ≤ N ∑ g ( ℓ α λ j − λ i , 2 λ i + λ j ) , L ( g ) = ∫ − 2 2 ∫ 0 ∞ c ( E ) r g ( r , E ) d r d E (456) where L ( g ) \mathcal{L}(g) L ( g ) X α ( G ; g ) \mathcal{X}_\alpha(G;g) X α ( G ; g ) (125) of G G G g g g E E E r r r
g ♯ ( x , y ) = g ( ∣ y − x ∣ ℓ α , x + y 2 ) , x , y ∈ R , g^\sharp(x,y)=g\left(\frac{\lvert y-x\rvert}{\ell_\alpha},\frac{x+y}{2}\right),\qquad x,y\in\mathbb{R}, g ♯ ( x , y ) = g ( ℓ α ∣ y − x ∣ , 2 x + y ) , x , y ∈ R , so that
X α ( G ; g ) = 1 2 ∑ i ≠ j g ♯ ( λ i , λ j ) . \mathcal{X}_\alpha(G;g)=\frac{1}{2}\sum_{i\ne j}g^\sharp(\lambda_i,\lambda_j). X α ( G ; g ) = 2 1 i = j ∑ g ♯ ( λ i , λ j ) . Throughout, we consider functions g g g E ∉ I E \notin I E ∈ / I g g g r r r n = 2 n=2 n = 2 y = x + v y=x+v y = x + v
E X α ( G ; g ) = ∫ x < y g ( y − x ℓ α , x + y 2 ) R 2 , N ( x , y ) d x d y = ∫ 0 ∞ ∫ R g ( v ℓ α , x + v 2 ) R 2 , N ( x , x + v ) d x d v . (457) \begin{aligned}
\mathbb{E}\mathcal{X}_{\alpha}(G;g)
&=\int_{x<y}g\left(\frac{y-x}{\ell_{\alpha}},\frac{x+y}{2}\right)R_{2,N}(x,y)\,dx\,dy\\
&=\int_{0}^{\infty}\int_{\mathbb{R}}g\left(\frac{v}{\ell_{\alpha}},x+\frac{v}{2}\right)R_{2,N}(x,x+v)\,dx\,dv. \tag*{(457)}
\end{aligned} E X α ( G ; g ) = ∫ x < y g ( ℓ α y − x , 2 x + y ) R 2 , N ( x , y ) d x d y = ∫ 0 ∞ ∫ R g ( ℓ α v , x + 2 v ) R 2 , N ( x , x + v ) d x d v . (457) The integrand in (A.32) vanishes unless x + v / 2 ∈ I ⊂ I 4 δ x+v/2\in I\subset I_{4\delta} x + v /2 ∈ I ⊂ I 4 δ x , x + v ∈ I 3 δ x,x+v\in I_{3\delta} x , x + v ∈ I 3 δ v ≤ δ v\leq\delta v ≤ δ
The mean. Let r 0 ≥ 1 r_0\geq1 r 0 ≥ 1 g g g g ( r , E ) = 0 g(r,E)=0 g ( r , E ) = 0 r > r 0 r>r_0 r > r 0 E ∉ I E\notin I E ∈ / I
lim N → ∞ E X α ( G ; g ) M α = L ( g ) . (458) \lim_{N\to\infty}\frac{\mathbb{E}\mathcal{X}_{\alpha}(G;g)}{M_{\alpha}}=\mathcal{L}(g). \tag*{(458)} N → ∞ lim M α E X α ( G ; g ) = L ( g ) . (458) Let N N N r 0 ℓ α ≤ N − 1 ≤ δ r_0\ell_{\alpha}\leq N^{-1}\leq\delta r 0 ℓ α ≤ N − 1 ≤ δ v ≤ r 0 ℓ α v\leq r_0\ell_{\alpha} v ≤ r 0 ℓ α x , x + v ∈ I 3 δ x,x+v\in I_{3\delta} x , x + v ∈ I 3 δ 3 δ 3\delta 3 δ δ \delta δ
E X α ( G ; g ) = N 3 ∫ 0 r 0 ℓ α ∫ R g ( v ℓ α , x + v 2 ) c ( x ) v d x d v + O ( ∫ 0 r 0 ℓ α ( N 23 / 8 v + N 4 v 2 ) d v ) = N 3 ℓ α 2 ∫ 0 r 0 ∫ I g ( r , E ) c ( E − ℓ α r 2 ) r d E d r + O ( N 23 / 8 ℓ α 2 + N 4 ℓ α 3 ) . \begin{aligned}
\mathbb{E}\mathcal{X}_{\alpha}(G;g)
&=N^3\int_{0}^{r_0\ell_{\alpha}}\int_{\mathbb{R}}g\left(\frac{v}{\ell_{\alpha}},x+\frac{v}{2}\right)c(x)v\,dx\,dv
+O\left(\int_{0}^{r_0\ell_{\alpha}}\left(N^{23/8}v+N^4v^2\right)\,dv\right)\\
&=N^3\ell_{\alpha}^2\int_{0}^{r_0}\int_I g(r,E)c\left(E-\frac{\ell_{\alpha}r}{2}\right)r\,dE\,dr
+O\left(N^{23/8}\ell_{\alpha}^2+N^4\ell_{\alpha}^3\right).
\end{aligned} E X α ( G ; g ) = N 3 ∫ 0 r 0 ℓ α ∫ R g ( ℓ α v , x + 2 v ) c ( x ) v d x d v + O ( ∫ 0 r 0 ℓ α ( N 23/8 v + N 4 v 2 ) d v ) = N 3 ℓ α 2 ∫ 0 r 0 ∫ I g ( r , E ) c ( E − 2 ℓ α r ) r d E d r + O ( N 23/8 ℓ α 2 + N 4 ℓ α 3 ) . where the implicit constants depend on r 0 r_0 r 0 ∥ g ∥ ∞ \lVert g\rVert_\infty ∥ g ∥ ∞ v = ℓ α r v=\ell_{\alpha}r v = ℓ α r x = E − ℓ α r / 2 x=E-\ell_{\alpha}r/2 x = E − ℓ α r /2 E ↦ c ( E ) E\mapsto c(E) E ↦ c ( E ) [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] c ( E − ℓ α r / 2 ) − c ( E ) → 0 c(E-\ell_{\alpha}r/2)-c(E)\to0 c ( E − ℓ α r /2 ) − c ( E ) → 0 E ∈ I E\in I E ∈ I r ∈ [ 0 , r 0 ] r\in[0,r_0] r ∈ [ 0 , r 0 ] M α = N 3 ℓ α 2 M_{\alpha}=N^3\ell_{\alpha}^2 M α = N 3 ℓ α 2 N 23 / 8 ℓ α 2 = N − 1 / 8 M α N^{23/8}\ell_{\alpha}^2=N^{-1/8}M_{\alpha} N 23/8 ℓ α 2 = N − 1/8 M α N 4 ℓ α 3 = N − α M α N^4\ell_{\alpha}^3=N^{-\alpha}M_{\alpha} N 4 ℓ α 3 = N − α M α
The variance. Let r 0 r_0 r 0 g g g
lim N → ∞ Var X α ( G ; g ) M α 2 = 0. (459) \lim_{N\to\infty}\frac{\operatorname{Var}\mathcal{X}_{\alpha}(G;g)}{M_{\alpha}^2}=0. \tag*{(459)} N → ∞ lim M α 2 Var X α ( G ; g ) = 0. (459) Expanding the square of ∑ i ≠ j g ♯ ( λ i , λ j ) \sum_{i\ne j}g^\sharp(\lambda_i,\lambda_j) ∑ i = j g ♯ ( λ i , λ j ) n ∈ { 2 , 3 , 4 } n\in\{2,3,4\} n ∈ { 2 , 3 , 4 }
Var X α ( G ; g ) = 1 2 ∫ g ♯ ( x 1 , x 2 ) 2 R 2 , N ( x 1 , x 2 ) + ∫ g ♯ ( x 1 , x 2 ) g ♯ ( x 1 , x 3 ) R 3 , N ( x 1 , x 2 , x 3 ) + 1 4 ∫ g ♯ ( x 1 , x 2 ) g ♯ ( x 3 , x 4 ) ( R 4 , N ( x 1 , x 2 , x 3 , x 4 ) − R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) ) . (460) \begin{aligned}
\operatorname{Var}\mathcal{X}_{\alpha}(G;g)
={}&\frac{1}{2}\int g^\sharp(x_1,x_2)^2R_{2,N}(x_1,x_2)
+\int g^\sharp(x_1,x_2)g^\sharp(x_1,x_3)R_{3,N}(x_1,x_2,x_3)\\
&+\frac{1}{4}\int g^\sharp(x_1,x_2)g^\sharp(x_3,x_4)
\left(R_{4,N}(x_1,x_2,x_3,x_4)-R_{2,N}(x_1,x_2)R_{2,N}(x_3,x_4)\right). \tag*{(460)}
\end{aligned} Var X α ( G ; g ) = 2 1 ∫ g ♯ ( x 1 , x 2 ) 2 R 2 , N ( x 1 , x 2 ) + ∫ g ♯ ( x 1 , x 2 ) g ♯ ( x 1 , x 3 ) R 3 , N ( x 1 , x 2 , x 3 ) + 4 1 ∫ g ♯ ( x 1 , x 2 ) g ♯ ( x 3 , x 4 ) ( R 4 , N ( x 1 , x 2 , x 3 , x 4 ) − R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) ) . (460) where the integrals are over R 2 \mathbb{R}^{2} R 2 R 3 \mathbb{R}^{3} R 3 R 4 \mathbb{R}^{4} R 4
( E X α ( G ; g ) ) 2 = 1 4 ∫ g ♯ ( x 1 , x 2 ) g ♯ ( x 3 , x 4 ) R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) d x 1 ⋯ d x 4 . \left(\mathbb{E}\mathcal{X}_{\alpha}(G;g)\right)^{2}=\frac{1}{4}\int g^{\sharp}(x_{1},x_{2})g^{\sharp}(x_{3},x_{4})R_{2,N}(x_{1},x_{2})R_{2,N}(x_{3},x_{4})\,\mathrm{d}x_{1}\cdots\mathrm{d}x_{4}. ( E X α ( G ; g ) ) 2 = 4 1 ∫ g ♯ ( x 1 , x 2 ) g ♯ ( x 3 , x 4 ) R 2 , N ( x 1 , x 2 ) R 2 , N ( x 3 , x 4 ) d x 1 ⋯ d x 4 . Let N N N 8 r 0 ℓ α ≤ N − 1 ≤ δ 8r_{0}\ell_{\alpha}\leq N^{-1}\leq\delta 8 r 0 ℓ α ≤ N − 1 ≤ δ r 0 r_{0} r 0 ∥ g ∥ ∞ \lVert g\rVert_{\infty} ∥ g ∥ ∞
Since R 2 , N ≥ 0 R_{2,N}\geq0 R 2 , N ≥ 0 ∥ g ∥ ∞ E X α ( G ; ∣ g ∣ ) \lVert g\rVert_{\infty}\mathbb{E}\mathcal{X}_{\alpha}(G;|g|) ∥ g ∥ ∞ E X α ( G ; ∣ g ∣ ) O ( M α ) O(M_{\alpha}) O ( M α ) ∣ g ∣ |g| ∣ g ∣ ∣ x 2 − x 1 ∣ , ∣ x 3 − x 1 ∣ ≤ r 0 ℓ α |x_{2}-x_{1}|,|x_{3}-x_{1}|\leq r_{0}\ell_{\alpha} ∣ x 2 − x 1 ∣ , ∣ x 3 − x 1 ∣ ≤ r 0 ℓ α x 1 , x 2 , x 3 ∈ I 3 δ x_{1},x_{2},x_{3}\in I_{3\delta} x 1 , x 2 , x 3 ∈ I 3 δ 2 r 0 ℓ α ≤ N − 1 2r_{0}\ell_{\alpha}\leq N^{-1} 2 r 0 ℓ α ≤ N − 1
C N 3 ⋅ ∣ I 3 δ ∣ ( 2 r 0 ℓ α ) 2 ≤ C M α . CN^{3}\cdot|I_{3\delta}|(2r_{0}\ell_{\alpha})^{2}\leq CM_{\alpha}. C N 3 ⋅ ∣ I 3 δ ∣ ( 2 r 0 ℓ α ) 2 ≤ C M α . We now turn to the third term. The integrand is invariant under exchanging x 1 x_{1} x 1 x 2 x_{2} x 2 x 3 x_{3} x 3 x 4 x_{4} x 4 1 / 4 1/4 1/4 1 1 1 x 1 < x 2 x_{1}<x_{2} x 1 < x 2 x 3 < x 4 x_{3}<x_{4} x 3 < x 4 x 2 − x 1 ≤ r 0 ℓ α x_{2}-x_{1}\leq r_{0}\ell_{\alpha} x 2 − x 1 ≤ r 0 ℓ α x 4 − x 3 ≤ r 0 ℓ α x_{4}-x_{3}\leq r_{0}\ell_{\alpha} x 4 − x 3 ≤ r 0 ℓ α I 3 δ I_{3\delta} I 3 δ ϱ 0 = N − 1 / 2 \varrho_{0}=N^{-1/2} ϱ 0 = N − 1/2 ϱ = ∣ x 3 − x 1 ∣ \varrho=|x_{3}-x_{1}| ϱ = ∣ x 3 − x 1 ∣ ϱ ≤ ϱ 0 \varrho\leq\varrho_{0} ϱ ≤ ϱ 0 ∣ R 2 , N ( x 1 , x 2 ) ∣ ≤ C N 3 ( x 2 − x 1 ) ≤ C N 2 |R_{2,N}(x_{1},x_{2})|\leq CN^{3}(x_{2}-x_{1})\leq CN^{2} ∣ R 2 , N ( x 1 , x 2 ) ∣ ≤ C N 3 ( x 2 − x 1 ) ≤ C N 2 ∣ I 3 δ ∣ ( r 0 ℓ α ) 2 ⋅ 2 ϱ 0 |I_{3\delta}|(r_{0}\ell_{\alpha})^{2}\cdot2\varrho_{0} ∣ I 3 δ ∣ ( r 0 ℓ α ) 2 ⋅ 2 ϱ 0
C N 4 ( log N ) 2 ℓ α 2 ϱ 0 . CN^{4}(\log N)^{2}\ell_{\alpha}^{2}\varrho_{0}. C N 4 ( log N ) 2 ℓ α 2 ϱ 0 . In the remaining part, where ϱ > ϱ 0 \varrho>\varrho_{0} ϱ > ϱ 0
ϱ ≥ N − 1 ≥ 8 r 0 ℓ α ≥ 4 ( x 2 − x 1 + x 4 − x 3 ) , \varrho\geq N^{-1}\geq8r_{0}\ell_{\alpha}\geq4(x_{2}-x_{1}+x_{4}-x_{3}), ϱ ≥ N − 1 ≥ 8 r 0 ℓ α ≥ 4 ( x 2 − x 1 + x 4 − x 3 ) , so Lemma A.9(c) applies. The contribution of this part is bounded by
C N 4 ( log N ) ( r 0 ℓ α ) 2 ∫ I 3 δ ∫ { x 3 : ϱ 0 < ∣ x 3 − x 1 ∣ ≤ 4 } Δ 0 ( ∣ x 3 − x 1 ∣ ) d x 3 d x 1 ≤ C N 4 ℓ α 2 ( log N ) ( N − 1 log N + N − 1 / 2 ) . CN^{4}(\log N)(r_{0}\ell_{\alpha})^{2}\int_{I_{3\delta}}\int_{\{x_{3}:\varrho_{0}<|x_{3}-x_{1}|\leq4\}}\Delta_{0}(|x_{3}-x_{1}|)\,\mathrm{d}x_{3}\,\mathrm{d}x_{1}
\leq CN^{4}\ell_{\alpha}^{2}(\log N)\left(N^{-1}\log N+N^{-1/2}\right). C N 4 ( log N ) ( r 0 ℓ α ) 2 ∫ I 3 δ ∫ { x 3 : ϱ 0 < ∣ x 3 − x 1 ∣ ≤ 4 } Δ 0 ( ∣ x 3 − x 1 ∣ ) d x 3 d x 1 ≤ C N 4 ℓ α 2 ( log N ) ( N − 1 log N + N − 1/2 ) . Collecting the bounds and using N 4 ℓ α 2 / M α 2 = N − 2 ℓ α − 2 = N 2 α N^{4}\ell_{\alpha}^{2}/M_{\alpha}^{2}=N^{-2}\ell_{\alpha}^{-2}=N^{2\alpha} N 4 ℓ α 2 / M α 2 = N − 2 ℓ α − 2 = N 2 α
Var X α ( G ; g ) M α 2 ≤ C ( M α − 1 + N 2 α ( log N ) 2 N − 1 / 2 ) = C ( N 2 α − 1 + ( log N ) 2 N 2 α − 1 / 2 ) . \frac{\operatorname{Var}\mathcal{X}_{\alpha}(G;g)}{M_{\alpha}^{2}}\leq C\left(M_{\alpha}^{-1}+N^{2\alpha}(\log N)^{2}N^{-1/2}\right)=C\left(N^{2\alpha-1}+(\log N)^{2}N^{2\alpha-1/2}\right). M α 2 Var X α ( G ; g ) ≤ C ( M α − 1 + N 2 α ( log N ) 2 N − 1/2 ) = C ( N 2 α − 1 + ( log N ) 2 N 2 α − 1/2 ) . The right side tends to zero because α < 1 / 4 \alpha<1/4 α < 1/4 α < 1 / 4 \alpha<1/4 α < 1/4
The tails. Let r 0 ≥ 1 r_0 \ge1 r 0 ≥ 1 g g g ∣ g ( r , E ) ∣ ≤ C f ( 1 + r 2 ) − p |g(r,E)| \le C_f(1+r^2)^{-p} ∣ g ( r , E ) ∣ ≤ C f ( 1 + r 2 ) − p ( r , E ) (r,E) ( r , E ) g ( r , E ) = 0 g(r,E)=0 g ( r , E ) = 0 r < r 0 r<r_0 r < r 0 E ∉ I E\notin I E ∈ / I
lim sup N → ∞ E X α ( G ; ∣ g ∣ ) M α ≤ C C f ∫ r 0 ∞ r ( 1 + r 2 ) − p d r . (461) \limsup_{N\to\infty}\frac{\mathbb{E}\mathcal{X}_{\alpha}(G;|g|)}{M_{\alpha}}\le CC_f\int_{r_0}^{\infty}r(1+r^2)^{-p}\,dr. \tag*{(461)} N → ∞ lim sup M α E X α ( G ; ∣ g ∣ ) ≤ C C f ∫ r 0 ∞ r ( 1 + r 2 ) − p d r . (461) To prove this, we let N ≥ δ − 1 N\ge\delta^{-1} N ≥ δ − 1 v v v ∣ g ∣ |g| ∣ g ∣ g g g r 0 ℓ α ≤ v ≤ N − 1 r_0\ell_\alpha\le v\le N^{-1} r 0 ℓ α ≤ v ≤ N − 1 ∣ R 2 , N ( x , x + v ) ∣ ≤ C N 3 v |R_{2,N}(x,x+v)|\le CN^3v ∣ R 2 , N ( x , x + v ) ∣ ≤ C N 3 v v = ℓ α r v=\ell_\alpha r v = ℓ α r
C C f ∣ I ∣ ∫ r 0 ℓ α N − 1 ( 1 + v 2 ℓ α 2 ) − p N 3 v d v ≤ C C f M α ∫ r 0 ∞ r ( 1 + r 2 ) − p d r . CC_f|I|\int_{r_0\ell_\alpha}^{N^{-1}}\left(1+\frac{v^2}{\ell_\alpha^2}\right)^{-p}N^3v\,dv\le CC_fM_\alpha\int_{r_0}^{\infty}r(1+r^2)^{-p}\,dr. C C f ∣ I ∣ ∫ r 0 ℓ α N − 1 ( 1 + ℓ α 2 v 2 ) − p N 3 v d v ≤ C C f M α ∫ r 0 ∞ r ( 1 + r 2 ) − p d r . In the range N − 1 ≤ v ≤ δ N^{-1}\le v\le\delta N − 1 ≤ v ≤ δ ( 1 + v 2 / ℓ α 2 ) − p ≤ ( ℓ α / v ) 2 p \left(1+v^2/\ell_\alpha^2\right)^{-p}\le(\ell_\alpha/v)^{2p} ( 1 + v 2 / ℓ α 2 ) − p ≤ ( ℓ α / v ) 2 p I 3 δ I_{3\delta} I 3 δ
C C f N 2 ( log N ) ∫ N − 1 δ ( ℓ α v ) 2 p d v ≤ C N 2 p + 1 ℓ α 2 p log N = C N 1 − 2 α p log N = C M α N − 2 α ( p − 1 ) log N . \begin{aligned}
CC_fN^2(\log N)\int_{N^{-1}}^\delta\left(\frac{\ell_\alpha}{v}\right)^{2p}\,dv\le CN^{2p+1}\ell_\alpha^{2p}\log N \\
&=CN^{1-2\alpha p}\log N=CM_\alpha N^{-2\alpha(p-1)}\log N.
\end{aligned} C C f N 2 ( log N ) ∫ N − 1 δ ( v ℓ α ) 2 p d v ≤ C N 2 p + 1 ℓ α 2 p log N = C N 1 − 2 α p log N = C M α N − 2 α ( p − 1 ) log N . The range v > δ v>\delta v > δ ∣ g ( ( λ j − λ i ) / ℓ α , ( λ i + λ j ) / 2 ) ∣ |g((\lambda_j-\lambda_i)/\ell_\alpha,(\lambda_i+\lambda_j)/2)| ∣ g (( λ j − λ i ) / ℓ α , ( λ i + λ j ) /2 ) ∣ i < j i<j i < j λ j − λ i > δ \lambda_j-\lambda_i>\delta λ j − λ i > δ N 2 N^2 N 2 C f ( ℓ α / δ ) 2 p C_f(\ell_\alpha/\delta)^{2p} C f ( ℓ α / δ ) 2 p C N 2 ℓ α 2 p = C N 2 − 2 p − 2 α p CN^2\ell_\alpha^{2p}=CN^{2-2p-2\alpha p} C N 2 ℓ α 2 p = C N 2 − 2 p − 2 α p p > 1 p>1 p > 1 o ( M α ) o(M_\alpha) o ( M α )
Conclusion. Let r 0 ≥ 1 r_0\ge1 r 0 ≥ 1 g 1 ( r , E ) = 1 r ≤ r 0 f ( r , E ) g_1(r,E)=\mathbf{1}_{r\le r_0}f(r,E) g 1 ( r , E ) = 1 r ≤ r 0 f ( r , E ) g 2 = f − g 1 g_2=f-g_1 g 2 = f − g 1 g 1 g_1 g 1 g 2 g_2 g 2
E ∣ X α ( G ; g 1 ) M α − L ( g 1 ) ∣ ≤ ( Var X α ( G ; g 1 ) M α 2 + ( E X α ( G ; g 1 ) M α − L ( g 1 ) ) 2 ) 1 / 2 ⟶ 0. \mathbb{E}\left|\frac{\mathcal{X}_{\alpha}(G;g_1)}{M_\alpha}-\mathcal{L}(g_1)\right|\le\left(\frac{\operatorname{Var}\mathcal{X}_{\alpha}(G;g_1)}{M_\alpha^2}+\left(\frac{\mathbb{E}\mathcal{X}_{\alpha}(G;g_1)}{M_\alpha}-\mathcal{L}(g_1)\right)^2\right)^{1/2}\longrightarrow0. E M α X α ( G ; g 1 ) − L ( g 1 ) ≤ ( M α 2 Var X α ( G ; g 1 ) + ( M α E X α ( G ; g 1 ) − L ( g 1 ) ) 2 ) 1/2 ⟶ 0. Since X α ( G ; f ) = X α ( G ; g 1 ) + X α ( G ; g 2 ) \mathcal{X}_{\alpha}(G;f)=\mathcal{X}_{\alpha}(G;g_1)+\mathcal{X}_{\alpha}(G;g_2) X α ( G ; f ) = X α ( G ; g 1 ) + X α ( G ; g 2 ) L ( f ) = L ( g 1 ) + L ( g 2 ) \mathcal{L}(f)=\mathcal{L}(g_1)+\mathcal{L}(g_2) L ( f ) = L ( g 1 ) + L ( g 2 )
lim sup N → ∞ E ∣ X α ( G ; f ) M α − L ( f ) ∣ ≤ lim sup N → ∞ E X α ( G ; ∣ g 2 ∣ ) M α + ∣ L ( g 2 ) ∣ ≤ C C f ∫ r 0 ∞ r ( 1 + r 2 ) − p d r , \limsup_{N\to\infty}\mathbb{E}\left|\frac{\mathcal{X}_{\alpha}(G;f)}{M_\alpha}-\mathcal{L}(f)\right|\le\limsup_{N\to\infty}\frac{\mathbb{E}\mathcal{X}_{\alpha}(G;|g_2|)}{M_\alpha}+|\mathcal{L}(g_2)|\le CC_f\int_{r_0}^{\infty}r(1+r^2)^{-p}\,dr, N → ∞ lim sup E M α X α ( G ; f ) − L ( f ) ≤ N → ∞ lim sup M α E X α ( G ; ∣ g 2 ∣ ) + ∣ L ( g 2 ) ∣ ≤ C C f ∫ r 0 ∞ r ( 1 + r 2 ) − p d r , where in the last inequality we also used
∣ L ( g 2 ) ∣ ≤ C f ( sup E ∈ [ − 2 , 2 ] c ( E ) ) ∣ I ∣ ∫ r 0 ∞ r ( 1 + r 2 ) − p d r . |\mathcal{L}(g_2)|\le C_f\left(\sup_{E\in[-2,2]}c(E)\right)|I|\int_{r_0}^{\infty}r(1+r^2)^{-p}\,dr. ∣ L ( g 2 ) ∣ ≤ C f ( E ∈ [ − 2 , 2 ] sup c ( E ) ) ∣ I ∣ ∫ r 0 ∞ r ( 1 + r 2 ) − p d r . Since p > 1 p>1 p > 1 r 0 → ∞ r_0\to\infty r 0 → ∞ X α ( G ; f ) / M α → L ( f ) \mathcal{X}_{\alpha}(G;f)/M_\alpha\to\mathcal{L}(f) X α ( G ; f ) / M α → L ( f ) L 1 L^1 L 1 X α ( G ; f ) / M α \mathcal{X}_{\alpha}(G;f)/M_\alpha X α ( G ; f ) / M α L ( f ) \mathcal{L}(f) L ( f ) (51) , this completes the proof of Proposition 3.6.
References 
