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 gaps between consecutive eigenvalues are among the central objects of random matrix theory. In the bulk of the spectrum, the limiting distribution of an individual gap is known for a wide range of models [17 , 55 ] . The smallest and largest gaps within a fixed interval in the bulk are understood for far fewer models. Unlike an individual gap, they are not determined by the local eigenvalue statistics on any fixed scale. The motivation for their study is discussed in the introduction of [4 ] , which includes a comparison with numerical data for the Riemann zeta function.
We first describe what is known about the largest gaps. For the Gaussian unitary ensemble (GUE) and the circular unitary ensemble (CUE) [14 , 37 ] , Ben Arous and Bourgade found the first-order asymptotics of the largest gaps in the bulk, which are of order log N / N \sqrt{\log N}/N log N / N [4 ] . Feng and Wei identified these fluctuations [21 ] . They proved that the largest gaps of the CUE and the GUE, after centering and rescaling, converge to a Poisson point process with an exponential intensity, so that the largest gap has a Gumbel limit. Charlier recently extended this result to unitary-invariant ensembles for a broad class of regular analytic potentials [6 ] . Bourgade proved universality of the smallest gaps in both symmetry classes, and of the largest gaps in the complex Hermitian class, for generalized Wigner matrices satisfying a smoothness assumption [5 ] . His proof combines new quantitative relaxation estimates for Dyson Brownian motion with reverse heat flow. Landon, Lopatto, and Marcinek proved comparison theorems for the largest gaps of real symmetric and complex Hermitian generalized Wigner matrices without a smoothness assumption [33 ] . In the complex Hermitian case, both works use the GUE result of Feng and Wei [21 ] to identify the limiting distribution. Both approaches also apply to the real symmetric largest-gap problem, but identifying its limiting distribution required the corresponding GOE result, which was not known.
The smallest gaps were studied earlier. Vinson’s dissertation [52 ] was the first rigorous work on extremal gaps. It determined the limiting distribution of the smallest gap of the CUE and the GUE. Ben Arous and Bourgade found the joint limit of the smallest gaps of these two ensembles [4 ] , building on Soshnikov’s work on determinantal point processes [46 ] . The limiting distribution of the smallest gaps is now also known for the Gaussian orthogonal ensemble (GOE) [23 ] , for the circular β \beta β [27 ] with β \beta β [20 ] , for the Gaussian symplectic ensemble [25 ] , and for complex Wishart ensembles and one-cut unitary-invariant ensembles with real-analytic confining potentials [42 ] . Bourgade’s universality theorem [5 ] covers the smallest gaps of real symmetric and complex Hermitian generalized Wigner matrices with smooth entries, and Figalli and Guionnet proved limiting laws for the smallest gaps and first-order asymptotics for the largest gaps in perturbative several-matrix models with β = 2 \beta= 2 β = 2 [26 ] . For Wigner matrices and related models, lower bounds for the smallest gaps were proved in [40 ] , and for sparse random matrices in [34 ] . Extremal gaps have also been studied for non-Hermitian matrices [35 , 36 , 38 ] and for the zeros of random functions [19 , 22 , 24 ] .
The results above that identify limiting laws or first-order asymptotics for the largest eigenvalue gaps rest on a determinantal structure [45 ] , either of the ensemble itself or of the Gaussian ensemble with which it is compared. The GOE does not have this structure, and the limiting distribution of its largest gaps has remained open; the problem was raised in [5 , 33 , 35 ] . For the circular orthogonal and symplectic ensembles (COE and CSE) [14 , 37 ] , Feng and Wei noted that even the order of the largest gaps is unknown [21 ] . Our main result, Theorem 1.1 , determines the limiting distribution of the largest gaps in the bulk of the spectrum of the GOE.
The analysis of the largest gaps rests on precise asymptotics for the probability that a long interval contains no eigenvalue. For the sine process, which is the limit of the GUE eigenvalues after rescaling in the bulk [2 ] , this probability is a Fredholm determinant of the sine kernel, and its asymptotics were predicted by des Cloizeaux and Mehta [12 ] and Dyson [13 ] . The leading term was proved by Widom [53 ] , and the constant term, known as the Widom–Dyson constant, by Krasovsky [31 ] , Ehrhardt [15 ] , and Deift, Its, Krasovsky, and Zhou [8 ] . For β = 1 \beta= 1 β = 1 β = 4 \beta= 4 β = 4 [16 ] . For general β > 0 \beta> 0 β > 0 β \beta β β _\beta β [50 ] , and the probability that it has no point in a long interval was determined up to a constant factor by Valkó and Virág [51 ] .
The eigenvalues of the GOE form a Pfaffian point process rather than a determinantal one: the k k k 2 k × 2 k 2k \times2k 2 k × 2 k 2 × 2 2 \times2 2 × 2 sgn ( x − y ) / 2 \operatorname{sgn}(x-y)/2 sgn ( x − y ) /2 [37 ] . For even matrix dimensions, Tracy and Widom [48 ] expressed the square of the probability that the GOE has no eigenvalue in a union of intervals as the corresponding probability for the GUE, multiplied by the determinant of a matrix of fixed size; see also [1 , 27 , 28 , 44 , 49 , 54 ] for the correlation kernels of orthogonal and symplectic ensembles, and [9 , 10 , 11 ] for their use in proving universality. Our proof builds on the identity of Tracy and Widom. However, we derive it differently, for even and odd matrix dimensions, and we analyze it on intervals whose length in units of the local mean spacing grows with N N N § 2.4 .
Main results
Let H N H_N H N N × N N \times N N × N exp ( − N Tr H 2 / 4 ) \exp(-N\operatorname{Tr} H^2/4) exp ( − N Tr H 2 /4 ) H i j H_{ij} H ij i ≤ j i \le j i ≤ j 2 / N 2/N 2/ N i = j i=j i = j 1 / N 1/N 1/ N i < j i<j i < j λ 1 < λ 2 < ⋯ < λ N \lambda_1 < \lambda_2 < \cdots< \lambda_N λ 1 < λ 2 < ⋯ < λ N H N H_N H N Z N Z_N Z N
1 Z N ∏ 1 ≤ i < j ≤ N ∣ λ j − λ i ∣ ∏ i = 1 N e − N λ i 2 / 4 (1) \frac{1}{Z_N}\prod_{1\le i<j\le N}|\lambda_j-\lambda_i|\prod_{i=1}^{N}e^{-N\lambda_i^2/4} \tag*{(1)} Z N 1 1 ≤ i < j ≤ N ∏ ∣ λ j − λ i ∣ i = 1 ∏ N e − N λ i 2 /4 (1) on { λ 1 < ⋯ < λ N } \{\lambda_1 < \cdots< \lambda_N\} { λ 1 < ⋯ < λ N } [2 ] . We define
σ ( y ) = 4 − y 2 , ∣ y ∣ ≤ 2. (2) \sigma(y)=\sqrt{4-y^2},\qquad|y|\le2. \tag*{(2)} σ ( y ) = 4 − y 2 , ∣ y ∣ ≤ 2. (2) With this normalization, the empirical eigenvalue distribution converges to the semicircle law. This is the probability measure on [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] σ ( y ) / ( 2 π ) \sigma(y)/(2\pi) σ ( y ) / ( 2 π ) y ∈ ( − 2 , 2 ) y\in(-2,2) y ∈ ( − 2 , 2 ) 2 π / ( N σ ( y ) ) 2\pi/(N\sigma(y)) 2 π / ( N σ ( y ))
We define the constant κ 1 \kappa_1 κ 1 F F F
κ 1 = 2 13 / 24 exp ( 3 2 ζ ′ ( − 1 ) ) , F ( s ) = s 2 64 + s 8 , s ≥ 0 , (3) \kappa_1=2^{13/24}\exp\left(\frac{3}{2}\zeta'(-1)\right),\qquad F(s)=\frac{s^2}{64}+\frac{s}{8},\qquad s\ge0, \tag*{(3)} κ 1 = 2 13/24 exp ( 2 3 ζ ′ ( − 1 ) ) , F ( s ) = 64 s 2 + 8 s , s ≥ 0 , (3) where ζ \zeta ζ 1 _1 1 N ≥ 2 N\ge2 N ≥ 2 b N > 0 b_N>0 b N > 0 F ( b N ) = log N F(b_N)=\log N F ( b N ) = log N a N = F ′ ( b N ) a_N=F'(b_N) a N = F ′ ( b N ) F ( b N ) = log N F(b_N)=\log N F ( b N ) = log N
b N = 8 log N + 1 4 − 4 , a N = b N + 4 32 , a N b N = 2 log N − b N 8 . (4) b_N=8\sqrt{\log N+\frac{1}{4}}-4,\qquad a_N=\frac{b_N+4}{32},\qquad a_Nb_N=2\log N-\frac{b_N}{8}. \tag*{(4)} b N = 8 log N + 4 1 − 4 , a N = 32 b N + 4 , a N b N = 2 log N − 8 b N . (4) We fix an interval I = [ y − , y + ] I=[y_-,y_+] I = [ y − , y + ] − 2 < y − < y + < 2 -2<y_-<y_+<2 − 2 < y − < y + < 2 λ i + 1 − λ i \lambda_{i+1}-\lambda_i λ i + 1 − λ i I I I
σ I = min y ∈ I σ ( y ) , ∂ ∗ I = { ξ ∈ { y − , y + } : σ ( ξ ) = σ I } . (5) \sigma_I=\min_{y\in I}\sigma(y),\qquad\partial_*I=\{\xi\in\{y_-,y_+\}:\sigma(\xi)=\sigma_I\}. \tag*{(5)} σ I = y ∈ I min σ ( y ) , ∂ ∗ I = { ξ ∈ { y − , y + } : σ ( ξ ) = σ I } . (5) Observe that ∂ ∗ I \partial_*I ∂ ∗ I I I I y − ≠ − y + y_-\ne-y_+ y − = − y + y − = − y + y_-=-y_+ y − = − y + ξ ≠ 0 \xi\ne0 ξ = 0 ξ ∈ ∂ ∗ I \xi\in\partial_*I ξ ∈ ∂ ∗ I
Λ I = κ 1 ∑ ξ ∈ ∂ ∗ I σ I 3 ∣ ξ ∣ . (6) \Lambda_I=\kappa_1\sum_{\xi\in\partial_*I}\frac{\sigma_I^3}{|\xi|}. \tag*{(6)} Λ I = κ 1 ξ ∈ ∂ ∗ I ∑ ∣ ξ ∣ σ I 3 . (6) We let G N ( I ) \mathcal{G}_N(I) G N ( I ) i ∈ { 1 , … , N − 1 } i \in\{1,\ldots,N-1\} i ∈ { 1 , … , N − 1 } λ i , λ i + 1 ∈ I \lambda_i,\lambda_{i+1} \in I λ i , λ i + 1 ∈ I i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I ) rescaled gap
τ i , N = a N ( N σ I ( λ i + 1 − λ i ) − b N ) + 9 8 log b N . (7) \tau_{i,N}=a_N\left(N\sigma_I(\lambda_{i+1}-\lambda_i)-b_N\right)+\frac{9}{8}\log b_N. \tag*{(7)} τ i , N = a N ( N σ I ( λ i + 1 − λ i ) − b N ) + 8 9 log b N . (7) Finally, we let X N \mathcal{X}_N X N R \mathbb{R} R τ i , N \tau_{i,N} τ i , N i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I ) A ⊂ R A \subset\mathbb{R} A ⊂ R X N ( A ) \mathcal{X}_N(A) X N ( A ) X N \mathcal{X}_N X N A A A i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I ) τ i , N ∈ A \tau_{i,N} \in A τ i , N ∈ A
Theorem 1.1. Let m ≥ 1 m \ge1 m ≥ 1 A 1 , … , A m ⊂ R A_1,\ldots,A_m \subset\mathbb{R} A 1 , … , A m ⊂ R ( a , b ] (a,b] ( a , b ] a < b a < b a < b ( a , ∞ ) (a,\infty) ( a , ∞ ) N → ∞ N \to\infty N → ∞ ( X N ( A 1 ) , … , X N ( A m ) ) (\mathcal{X}_N(A_1),\ldots,\mathcal{X}_N(A_m)) ( X N ( A 1 ) , … , X N ( A m )) Λ I ∫ A l e − x d x \Lambda_I \int_{A_l} e^{-x}\,\mathrm{d}x Λ I ∫ A l e − x d x 1 ≤ l ≤ m 1 \le l \le m 1 ≤ l ≤ m X N \mathcal{X}_N X N R \mathbb{R} R Λ I e − x d x \Lambda_I e^{-x}\,\mathrm{d}x Λ I e − x d x
We let τ 1 , N ↓ ≥ τ 2 , N ↓ ≥ ⋯ \tau_{1,N}^{\downarrow} \ge\tau_{2,N}^{\downarrow} \ge\cdots τ 1 , N ↓ ≥ τ 2 , N ↓ ≥ ⋯ τ i , N \tau_{i,N} τ i , N i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I ) τ k , N ↓ = − ∞ \tau_{k,N}^{\downarrow}=-\infty τ k , N ↓ = − ∞ G N ( I ) \mathcal{G}_N(I) G N ( I ) k k k m 1 , N m_{1,N} m 1 , N λ i + 1 − λ i \lambda_{i+1}-\lambda_i λ i + 1 − λ i i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I ) m 1 , N = 0 m_{1,N}=0 m 1 , N = 0 G N ( I ) \mathcal{G}_N(I) G N ( I ) Y N Y_N Y N θ N \theta_N θ N Y N = O P ( θ N ) Y_N=O_{\mathrm{P}}(\theta_N) Y N = O P ( θ N ) ( Y N / θ N ) N (Y_N/\theta_N)_N ( Y N / θ N ) N
Corollary 1.2. For every k ≥ 1 k \ge1 k ≥ 1 x ∈ R x \in\mathbb{R} x ∈ R
lim N → ∞ P ( τ k , N ↓ ≤ x ) = exp ( − Λ I e − x ) ∑ j = 0 k − 1 ( Λ I e − x ) j j ! . (8) \lim_{N\to\infty}\mathbb{P}\left(\tau_{k,N}^{\downarrow}\le x\right)=\exp\left(-\Lambda_I e^{-x}\right)\sum_{j=0}^{k-1}\frac{\left(\Lambda_I e^{-x}\right)^j}{j!}. \tag*{(8)} N → ∞ lim P ( τ k , N ↓ ≤ x ) = exp ( − Λ I e − x ) j = 0 ∑ k − 1 j ! ( Λ I e − x ) j . (8) In particular, τ 1 , N ↓ − log Λ I \tau_{1,N}^{\downarrow}-\log\Lambda_I τ 1 , N ↓ − log Λ I x ↦ exp ( − e − x ) x\mapsto\exp(-e^{-x}) x ↦ exp ( − e − x )
lim N → ∞ P ( τ 1 , N ↓ − log Λ I ≤ x ) = exp ( − e − x ) , x ∈ R . (9) \lim_{N\to\infty}\mathbb{P}\left(\tau_{1,N}^{\downarrow}-\log\Lambda_I\le x\right)=\exp\left(-e^{-x}\right),\qquad x\in\mathbb{R}. \tag*{(9)} N → ∞ lim P ( τ 1 , N ↓ − log Λ I ≤ x ) = exp ( − e − x ) , x ∈ R . (9) Moreover,
N σ I m 1 , N = 8 log N − 4 − 9 4 log log N log N + O P ( ( log N ) − 1 / 2 ) . (10) N\sigma_I m_{1,N}=8\sqrt{\log N}-4-\frac{9}{4}\frac{\log\log N}{\sqrt{\log N}}+O_{\mathrm{P}}\left((\log N)^{-1/2}\right). \tag*{(10)} N σ I m 1 , N = 8 log N − 4 − 4 9 log N log log N + O P ( ( log N ) − 1/2 ) . (10) We now explain where the constant κ 1 \kappa_1 κ 1 F F F A ⊂ R A \subset\mathbb{R} A ⊂ R hole probability
E 1 , N ( A ) = P ( λ i ∉ A for all i ∈ { 1 , … , N } ) . (11) E_{1,N}(A)=\mathbb{P}\left(\lambda_i\notin A\text{ for all }i\in\{1,\ldots,N\}\right). \tag*{(11)} E 1 , N ( A ) = P ( λ i ∈ / A for all i ∈ { 1 , … , N } ) . (11) The limit of the hole probability of a short interval is expressed through the sine kernel. We define the sine kernel S S S
S ( u ) = sin u π u , u ≠ 0 , S ( 0 ) = 1 π . (12) S(u)=\frac{\sin u}{\pi u},\qquad u\ne0,\qquad S(0)=\frac{1}{\pi}. \tag*{(12)} S ( u ) = π u sin u , u = 0 , S ( 0 ) = π 1 . (12) It is the kernel of the sine process, rescaled to have mean spacing π \pi π t > 0 t > 0 t > 0 S + , t S_{+,t} S + , t S − , t S_{-,t} S − , t L 2 ( 0 , t ) L^2(0,t) L 2 ( 0 , t ) S ( u − v ) + S ( u + v ) S(u-v)+S(u+v) S ( u − v ) + S ( u + v ) S ( u − v ) − S ( u + v ) S(u-v)-S(u+v) S ( u − v ) − S ( u + v )
D + ( t ) = det ( Id − S + , t ) , D − ( t ) = det ( Id − S − , t ) , q 1 ( s ) = D + ( s / 4 ) , (13) D_+(t)=\det(\operatorname{Id}-S_{+,t}),\qquad D_-(t)=\det(\operatorname{Id}-S_{-,t}),\qquad q_1(s)=D_+(s/4), \tag*{(13)} D + ( t ) = det ( Id − S + , t ) , D − ( t ) = det ( Id − S − , t ) , q 1 ( s ) = D + ( s /4 ) , (13) for t , s > 0 t,s > 0 t , s > 0 Corollary 7.5 that q 1 ( s ) q_1(s) q 1 ( s ) E 1 , N ( ( y , y + s / ( N σ ( y ) ) ) ) E_{1,N}((y,y+s/(N\sigma(y)))) E 1 , N (( y , y + s / ( N σ ( y )))) N → ∞ N\to\infty N → ∞ y ∈ ( − 2 , 2 ) y\in(-2,2) y ∈ ( − 2 , 2 ) s > 0 s>0 s > 0 ( y , y + s / ( N σ ( y ) ) ) (y,y+s/(N\sigma(y))) ( y , y + s / ( N σ ( y ))) s / ( 2 π ) s/(2\pi) s / ( 2 π ) y y y q 1 ( s ) q_1(s) q 1 ( s ) 1 _1 1 2 π 2\pi 2 π s s s [37 , 48 ] . The asymptotics of D ± D_\pm D ± [16 Theorem 4.8] , confirming predictions of Dyson [13 ] . Ehrhardt proved that
log D ± ( t ) = − t 2 4 + t 2 − log t 8 + ( 1 24 ± 1 4 ) log 2 + 3 2 ζ ′ ( − 1 ) + o ( 1 ) , t → ∞ . (14) \log D_\pm(t)=-\frac{t^2}{4}+\frac{t}{2}-\frac{\log t}{8}+\left(\frac{1}{24}\pm\frac{1}{4}\right)\log2+\frac{3}{2}\zeta'(-1)+o(1),\qquad t\to\infty. \tag*{(14)} log D ± ( t ) = − 4 t 2 + 2 t − 8 log t + ( 24 1 ± 4 1 ) log 2 + 2 3 ζ ′ ( − 1 ) + o ( 1 ) , t → ∞. (14) Setting t = s / 4 t=s/4 t = s /4 (14) (see Lemma 7.1 ), we have
q 1 ( s ) = κ 1 s − 1 / 8 e − F ( s ) ( 1 + o ( 1 ) ) , D + ( s / 4 ) D − ( s / 4 ) = 2 e − s / 4 ( 1 + o ( 1 ) ) , s → ∞ , (15) q_1(s)=\kappa_1s^{-1/8}e^{-F(s)}(1+o(1)),\qquad\frac{D_+(s/4)}{D_-(s/4)}=\sqrt{2}e^{-s/4}(1+o(1)),\qquad s\to\infty, \tag*{(15)} q 1 ( s ) = κ 1 s − 1/8 e − F ( s ) ( 1 + o ( 1 )) , D − ( s /4 ) D + ( s /4 ) = 2 e − s /4 ( 1 + o ( 1 )) , s → ∞ , (15) with κ 1 \kappa_1 κ 1 F F F (3) .
The leading term for the GOE is larger by a factor 2 \sqrt{2} 2 ( 64 / β ) log N \sqrt{(64/\beta)\log N} ( 64/ β ) log N β = 1 \beta=1 β = 1 β \beta β [21 ] .
The following remark records a consequence of Corollary 1.2 for real symmetric generalized Wigner matrices, which was anticipated in [33 ] and [5 ] . We prove it in Section 10.
Remark 1.4. Let W N W_N W N N × N N \times N N × N [33 Definition 1.1] ; in particular, the entries of N W N \sqrt{N}W_N N W N k ≥ 1 k \ge1 k ≥ 1 T ^ k , N \widehat{T}_{k,N} T k , N k k k μ i + 1 − μ i \mu_{i+1}-\mu_i μ i + 1 − μ i μ 1 ≤ ⋯ ≤ μ N \mu_1 \le\cdots\le\mu_N μ 1 ≤ ⋯ ≤ μ N W N W_N W N μ i ∈ I \mu_i \in I μ i ∈ I [33 ] ), and let T ^ k , N = 0 \widehat{T}_{k,N}=0 T k , N = 0 k k k
τ ^ k , N = a N ( N σ I T ^ k , N − b N ) + 9 8 log b N . (17) \widehat{\tau}_{k,N}=a_N\left(N\sigma_I\widehat{T}_{k,N}-b_N\right)+\frac{9}{8}\log b_N. \tag*{(17)} τ k , N = a N ( N σ I T k , N − b N ) + 8 9 log b N . (17) Then (8) holds with τ ^ k , N \widehat{\tau}_{k,N} τ k , N τ k , N ⋄ \tau^\diamond_{k,N} τ k , N ⋄ [33 Corollary 1.6] .
Notation
We use the following notation throughout the paper. For a statement P P P 1 P \mathbf{1}_P 1 P 1 1 1 P P P 0 0 0 A ⊂ R A \subset\mathbb{R} A ⊂ R 1 A \mathbf{1}_A 1 A A ⊂ R A \subset\mathbb{R} A ⊂ R ∣ A ∣ |A| ∣ A ∣ c > 0 c>0 c > 0 c A = { c a : a ∈ A } cA=\{ca:a\in A\} c A = { c a : a ∈ A } ∥ ⋅ ∥ 2 \|\cdot\|_2 ∥ ⋅ ∥ 2 L 2 ( A ) L^2(A) L 2 ( A ) ∥ T ∥ \|T\| ∥ T ∥ T T T L 2 ( A ) L^2(A) L 2 ( A ) L 2 ( A ) L^2(A) L 2 ( A ) L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) A A A t ∈ R t\in\mathbb{R} t ∈ R ( t ) + = max ( t , 0 ) (t)_+=\max(t,0) ( t ) + = max ( t , 0 ) t , y ∈ R t,y\in\mathbb{R} t , y ∈ R ∫ t y \int_t^y ∫ t y ∫ t y = − ∫ y t \int_t^y=-\int_y^t ∫ t y = − ∫ y t y < t y<t y < t § 2.4 , C C C c c c N N N δ \delta δ γ \gamma γ r 0 r_0 r 0 k k k x 1 , … , x k x_1,\ldots,x_k x 1 , … , x k O ( ⋅ ) O(\cdot) O ( ⋅ ) N → ∞ N\to\infty N → ∞
Overview of the proof
We now describe the main steps of the proof. This overview can be read independently of the later sections; all objects in it are defined here, in § 2.2 , or in § 2.3 . The proof has two parts. The first part, which follows the strategy of Feng and Wei for the GUE [21 ] , reduces Theorem 1.1 to three estimates for hole probabilities. The second part proves these estimates in four steps, and Steps 1–3 are new compared with the proof for the GUE. Where the GUE proof uses that the hole probability is a Fredholm determinant, Step 1 uses an exact identity, valid for even and odd N N N N N N
Reduction to hole probabilities. For the Poisson point process with intensity Λ I e − x d x \Lambda_I e^{-x}\,\mathrm{d}x Λ I e − x d x ∏ j = 1 k ( τ j − x j ) + \prod_{j=1}^{k}(\tau_j-x_j)_+ ∏ j = 1 k ( τ j − x j ) + k k k ( τ 1 , … , τ k ) (\tau_1,\ldots,\tau_k) ( τ 1 , … , τ k )
∏ j = 1 k ∫ R ( t − x j ) + Λ I e − t d t = ∏ j = 1 k Λ I e − x j . (18) \prod_{j=1}^{k}\int_{\mathbb{R}}(t-x_j)_+\Lambda_I e^{-t}\,\mathrm{d}t=\prod_{j=1}^{k}\Lambda_I e^{-x_j}. \tag*{(18)} j = 1 ∏ k ∫ R ( t − x j ) + Λ I e − t d t = j = 1 ∏ k Λ I e − x j . (18) By a criterion of the type used in [21 Lemma 1] , which we state in Lemma A.1 , it therefore suffices to show that for every k ≥ 1 k\ge1 k ≥ 1 x 1 , … , x k ∈ R x_1,\ldots,x_k\in\mathbb{R} x 1 , … , x k ∈ R
lim N → ∞ E ∑ i 1 , … , i k ∈ G N ( I ) d i s t i n c t ∏ j = 1 k ( τ i j , N − x j ) + = ∏ j = 1 k Λ I e − x j . (19) \lim_{N\to\infty}\mathbb{E}\sum_{\substack{i_1,\ldots,i_k\in G_N(I)\\ \mathrm{distinct}}}\prod_{j=1}^{k}(\tau_{i_j,N}-x_j)_+=\prod_{j=1}^{k}\Lambda_I e^{-x_j}. \tag*{(19)} N → ∞ lim E i 1 , … , i k ∈ G N ( I ) distinct ∑ j = 1 ∏ k ( τ i j , N − x j ) + = j = 1 ∏ k Λ I e − x j . (19) The positive parts ( τ − x ) + (\tau-x)_+ ( τ − x ) + 1 τ > x \mathbf{1}_{\tau>x} 1 τ > x (19) can be expressed through hole probabilities, as follows. For x ∈ R x\in\mathbb{R} x ∈ R
ℓ N ( x ) = 1 N σ I ( b N + x − 9 8 log b N a N ) , χ N = a N N σ I . (20) \ell_N(x)=\frac{1}{N\sigma_I}\left(b_N+\frac{x-\frac{9}{8}\log b_N}{a_N}\right),\qquad\chi_N=a_NN\sigma_I. \tag*{(20)} ℓ N ( x ) = N σ I 1 ( b N + a N x − 8 9 log b N ) , χ N = a N N σ I . (20) By (7) , we have τ i , N − x = χ N ( λ i + 1 − λ i − ℓ N ( x ) ) \tau_{i,N}-x=\chi_N(\lambda_{i+1}-\lambda_i-\ell_N(x)) τ i , N − x = χ N ( λ i + 1 − λ i − ℓ N ( x )) i ∈ G N ( I ) i\in G_N(I) i ∈ G N ( I ) a N a_N a N b N b_N b N
F ( N σ I ℓ N ( x ) ) = log N + x − 9 8 log b N + o ( 1 ) . (21) F(N\sigma_I\ell_N(x))=\log N+x-\frac{9}{8}\log b_N+o(1). \tag*{(21)} F ( N σ I ℓ N ( x )) = log N + x − 8 9 log b N + o ( 1 ) . (21) We also call the interval ( λ i , λ i + 1 ) (\lambda_i,\lambda_{i+1}) ( λ i , λ i + 1 ) ( λ i , λ i + 1 ) (\lambda_i,\lambda_{i+1}) ( λ i , λ i + 1 ) ℓ > 0 \ell>0 ℓ > 0 y ∈ R y\in\mathbb{R} y ∈ R [ y , y + ℓ ] ⊂ ( λ i , λ i + 1 ) [y,y+\ell]\subset(\lambda_i,\lambda_{i+1}) [ y , y + ℓ ] ⊂ ( λ i , λ i + 1 ) ( λ i + 1 − λ i − ℓ ) + (\lambda_{i+1}-\lambda_i-\ell)_+ ( λ i + 1 − λ i − ℓ ) + y 1 , … , y k ∈ R y_1,\ldots,y_k\in\mathbb{R} y 1 , … , y k ∈ R W N ( y 1 , … , y k ) W_N(y_1,\ldots,y_k) W N ( y 1 , … , y k ) [ y j , y j + ℓ N ( x j ) ] [y_j,y_j+\ell_N(x_j)] [ y j , y j + ℓ N ( x j )] 1 ≤ j ≤ k 1\le j\le k 1 ≤ j ≤ k ( λ i , λ i + 1 ) (\lambda_i,\lambda_{i+1}) ( λ i , λ i + 1 ) i ∈ G N ( I ) i\in G_N(I) i ∈ G N ( I ) y 1 , … , y k y_1,\ldots,y_k y 1 , … , y k N N N ℓ N ( x j ) > 0 \ell_N(x_j)>0 ℓ N ( x j ) > 0 j j j (19) equals
χ N k ∫ y − y + − ℓ N ( x 1 ) ⋯ ∫ y − y + − ℓ N ( x k ) P ( W N ( y 1 , … , y k ) ) d y k ⋯ d y 1 . (22) \chi_N^k\int_{y_-}^{y_+-\ell_N(x_1)}\cdots\int_{y_-}^{y_+-\ell_N(x_k)}\mathbb{P}\bigl(W_N(y_1,\ldots,y_k)\bigr)\,\mathrm{d}y_k\cdots\mathrm{d}y_1. \tag*{(22)} χ N k ∫ y − y + − ℓ N ( x 1 ) ⋯ ∫ y − y + − ℓ N ( x k ) P ( W N ( y 1 , … , y k ) ) d y k ⋯ d y 1 . (22) If two of the intervals [ y j , y j + ℓ N ( x j ) ] [y_j,y_j+\ell_N(x_j)] [ y j , y j + ℓ N ( x j )] P ( W N ( y 1 , … , y k ) ) = 0 \mathbb{P}(\mathcal{W}_N(y_1,\ldots,y_k))=0 P ( W N ( y 1 , … , y k )) = 0 P ( W N ( y 1 , … , y k ) ) \mathbb{P}(\mathcal{W}_N(y_1,\ldots,y_k)) P ( W N ( y 1 , … , y k )) ( y j , y j + ℓ N ( x j ) ) (y_j,y_j+\ell_N(x_j)) ( y j , y j + ℓ N ( x j )) I I I 3 N − 1 / 3 3N^{-1/3} 3 N − 1/3 P ( W N ( y 1 , … , y k ) ) \mathbb{P}(\mathcal{W}_N(y_1,\ldots,y_k)) P ( W N ( y 1 , … , y k )) N − 1 / 3 N^{-1/3} N − 1/3 I I I O ( N − D ) O(N^{-D}) O ( N − D ) D > 0 D>0 D > 0 N − 1 / 3 N^{-1/3} N − 1/3 I I I O ( N 1 / 3 ) O(N^{1/3}) O ( N 1/3 ) ( y , y + s / ( N σ ( y ) ) ) (y,y+s/(N\sigma(y))) ( y , y + s / ( N σ ( y ))) s = γ 0 log N s=\gamma_0\sqrt{\log N} s = γ 0 log N γ 0 > 0 \gamma_0>0 γ 0 > 0 γ 0 2 / 64 > D + 1 / 3 \gamma_0^2/64>D+1/3 γ 0 2 /64 > D + 1/3 γ = γ 0 \gamma=\gamma_0 γ = γ 0 2 q 1 ( γ 0 log N ) ≤ N − γ 0 2 / 64 2q_1(\gamma_0\sqrt{\log N})\leq N^{-\gamma_0^2/64} 2 q 1 ( γ 0 log N ) ≤ N − γ 0 2 /64
It remains to estimate the hole probabilities of unions of at most k k k I I I C log N / N C\sqrt{\log N}/N C log N / N δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) γ > 0 \gamma>0 γ > 0 C > 0 C>0 C > 0
E 1 , N ( ( y , y + s N σ ( y ) ) ) = q 1 ( s ) ( 1 + O ( e C s N − 1 / 2 ) ) (23) E_{1,N}\left(\left(y,y+\frac{s}{N\sigma(y)}\right)\right)=q_1(s)\left(1+O\left(e^{Cs}N^{-1/2}\right)\right) \tag*{(23)} E 1 , N ( ( y , y + N σ ( y ) s ) ) = q 1 ( s ) ( 1 + O ( e C s N − 1/2 ) ) (23) uniformly in y ∈ [ − 2 + δ , 2 − δ ] y\in[-2+\delta,2-\delta] y ∈ [ − 2 + δ , 2 − δ ] 0 < s ≤ γ log N 0<s\leq\gamma\sqrt{\log N} 0 < s ≤ γ log N A 1 , … , A r ⊂ [ − 2 + δ , 2 − δ ] A_1,\ldots,A_r\subset[-2+\delta,2-\delta] A 1 , … , A r ⊂ [ − 2 + δ , 2 − δ ] ∑ j ∣ A j ∣ ≤ γ log N / N \sum_j|A_j|\leq\gamma\sqrt{\log N}/N ∑ j ∣ A j ∣ ≤ γ log N / N r r r r 0 r_0 r 0
E 1 , N ( ⋃ j = 1 r A j ) ≤ e C log N ∏ j = 1 r E 1 , N ( A j ) , (24) E_{1,N}\left(\bigcup_{j=1}^{r}A_j\right)\leq e^{C\sqrt{\log N}}\prod_{j=1}^{r}E_{1,N}(A_j), \tag*{(24)} E 1 , N ( j = 1 ⋃ r A j ) ≤ e C l o g N j = 1 ∏ r E 1 , N ( A j ) , (24) and that if, in addition, dist ( A i , A j ) ≥ N − 1 / 2 \operatorname{dist}(A_i,A_j)\geq N^{-1/2} dist ( A i , A j ) ≥ N − 1/2 i ≠ j i\ne j i = j
E 1 , N ( ⋃ j = 1 r A j ) = ( 1 + O ( e C log N N − 1 / 4 ) ) ∏ j = 1 r E 1 , N ( A j ) . (25) E_{1,N}\left(\bigcup_{j=1}^{r}A_j\right)=\left(1+O\left(e^{C\sqrt{\log N}}N^{-1/4}\right)\right)\prod_{j=1}^{r}E_{1,N}(A_j). \tag*{(25)} E 1 , N ( j = 1 ⋃ r A j ) = ( 1 + O ( e C l o g N N − 1/4 ) ) j = 1 ∏ r E 1 , N ( A j ) . (25) We now explain how these estimates give (1.19). We fix δ > 0 \delta>0 δ > 0 I ⊂ [ − 2 + δ , 2 − δ ] I\subset[-2+\delta,2-\delta] I ⊂ [ − 2 + δ , 2 − δ ] r 0 = k r_0=k r 0 = k N ℓ N ( x ) = ( 8 + o ( 1 ) ) log N / σ I N\ell_N(x)=(8+o(1))\sqrt{\log N}/\sigma_I N ℓ N ( x ) = ( 8 + o ( 1 )) log N / σ I x x x γ > 0 \gamma>0 γ > 0 N ∑ j ℓ N ( x j ) ≤ γ log N N\sum_j\ell_N(x_j)\leq\gamma\sqrt{\log N} N ∑ j ℓ N ( x j ) ≤ γ log N j j j ( y 1 , … , y k ) (y_1,\ldots,y_k) ( y 1 , … , y k ) [ y j , y j + ℓ N ( x j ) ] [y_j,y_j+\ell_N(x_j)] [ y j , y j + ℓ N ( x j )] I I I 3 N − 1 / 3 3N^{-1/3} 3 N − 1/3 P ( W N ( y 1 , … , y k ) ) \mathbb{P}(\mathcal{W}_N(y_1,\ldots,y_k)) P ( W N ( y 1 , … , y k )) ⋃ j ( y j , y j + ℓ N ( x j ) ) \bigcup_j(y_j,y_j+\ell_N(x_j)) ⋃ j ( y j , y j + ℓ N ( x j )) O ( N − D ) O(N^{-D}) O ( N − D ) D > k + 1 D>k+1 D > k + 1 λ N k ≤ ( N log N ) k \lambda_N^k\leq(N\log N)^k λ N k ≤ ( N log N ) k A j = ( y j , y j + ℓ N ( x j ) ) A_j=(y_j,y_j+\ell_N(x_j)) A j = ( y j , y j + ℓ N ( x j )) r = k r=k r = k E 1 , N ( ( y j , y j + ℓ N ( x j ) ) ) E_{1,N}((y_j,y_j+\ell_N(x_j))) E 1 , N (( y j , y j + ℓ N ( x j ))) [ y j , y j + ℓ N ( x j ) ] [y_j,y_j+\ell_N(x_j)] [ y j , y j + ℓ N ( x j )] (24) to bound P ( W N ( y 1 , … , y k ) ) \mathbb{P}(W_N(y_1,\ldots,y_k)) P ( W N ( y 1 , … , y k )) e C log N e^{C\sqrt{\log N}} e C l o g N (23) , (15) , and (21) , and since q 1 q_1 q 1 σ ( y ) ≥ σ I \sigma(y)\geq\sigma_I σ ( y ) ≥ σ I I I I χ N E 1 , N ( ( y j , y j + ℓ N ( x j ) ) ) \chi_N E_{1,N}((y_j,y_j+\ell_N(x_j))) χ N E 1 , N (( y j , y j + ℓ N ( x j ))) O ( log N ) O(\log N) O ( log N ) k k k O ( N − 1 / 3 ) O(N^{-1/3}) O ( N − 1/3 ) O ( e C log N N − 1 / 3 ( log N ) k ) = o ( 1 ) O(e^{C\sqrt{\log N}}N^{-1/3}(\log N)^k)=o(1) O ( e C l o g N N − 1/3 ( log N ) k ) = o ( 1 ) (19) reduces to the statement
lim N → ∞ χ N ∫ y − y + − ℓ N ( x ) E 1 , N ( y , y + ℓ N ( x ) ) d y = Λ I e − x , x ∈ R , (26) \lim_{N\to\infty}\chi_N\int_{y_-}^{y_+-\ell_N(x)}E_{1,N}(y,y+\ell_N(x))\,\mathrm{d}y=\Lambda_I e^{-x},\qquad x\in\mathbb{R}, \tag*{(26)} N → ∞ lim χ N ∫ y − y + − ℓ N ( x ) E 1 , N ( y , y + ℓ N ( x )) d y = Λ I e − x , x ∈ R , (26) which we prove in Section 9. To compute this integral, we apply (23) with s = N σ ( y ) ℓ N ( x ) s=N\sigma(y)\ell_N(x) s = N σ ( y ) ℓ N ( x )
N σ ( y ) ℓ N ( x ) − N σ I ℓ N ( x ) = N σ I ℓ N ( x ) ( σ ( y ) σ I − 1 ) , F ′ ( b N ) b N = a N b N = 2 log N ( 1 + o ( 1 ) ) , (27) \begin{aligned}
N\sigma(y)\ell_N(x)-N\sigma_I\ell_N(x)&=N\sigma_I\ell_N(x)\left(\frac{\sigma(y)}{\sigma_I}-1\right),\\
F'(b_N)b_N&=a_Nb_N=2\log N(1+o(1)), \tag*{(27)}
\end{aligned} N σ ( y ) ℓ N ( x ) − N σ I ℓ N ( x ) F ′ ( b N ) b N = N σ I ℓ N ( x ) ( σ I σ ( y ) − 1 ) , = a N b N = 2 log N ( 1 + o ( 1 )) , (27) by (4) . Set s N = N σ I ℓ N ( x ) s_N=N\sigma_I\ell_N(x) s N = N σ I ℓ N ( x ) Ψ ( y ) = σ ( y ) / σ I \Psi(y)=\sigma(y)/\sigma_I Ψ ( y ) = σ ( y ) / σ I (23) and (15) , we find that the integrand is
χ N q 1 ( s N ) Ψ ( y ) − 1 / 8 exp ( − s N 2 64 ( Ψ ( y ) 2 − 1 ) − s N 8 ( Ψ ( y ) − 1 ) ) ( 1 + o ( 1 ) ) , (28) \chi_Nq_1(s_N)\Psi(y)^{-1/8}\exp\left(-\frac{s_N^2}{64}\left(\Psi(y)^2-1\right)-\frac{s_N}{8}\left(\Psi(y)-1\right)\right)(1+o(1)), \tag*{(28)} χ N q 1 ( s N ) Ψ ( y ) − 1/8 exp ( − 64 s N 2 ( Ψ ( y ) 2 − 1 ) − 8 s N ( Ψ ( y ) − 1 ) ) ( 1 + o ( 1 )) , (28) uniformly in y ∈ [ y − , y + − ℓ N ( x ) ] y\in[y_-,y_+-\ell_N(x)] y ∈ [ y − , y + − ℓ N ( x )] s N 2 / 64 ∼ log N s_N^2/64\sim\log N s N 2 /64 ∼ log N ( log N ) − 1 (\log N)^{-1} ( log N ) − 1 ∂ ∗ I \partial_* I ∂ ∗ I σ ( y ) = σ I \sigma(y)=\sigma_I σ ( y ) = σ I ξ ∈ ∂ ∗ I \xi\in\partial_* I ξ ∈ ∂ ∗ I
σ ( y ) σ I − 1 = ∣ ξ ∣ σ I 2 ∣ y − ξ ∣ + O ( ∣ y − ξ ∣ 2 ) , (29) \frac{\sigma(y)}{\sigma_I}-1=\frac{|\xi|}{\sigma_I^2}|y-\xi|+O(|y-\xi|^2), \tag*{(29)} σ I σ ( y ) − 1 = σ I 2 ∣ ξ ∣ ∣ y − ξ ∣ + O ( ∣ y − ξ ∣ 2 ) , (29) since σ ′ ( ξ ) = − ξ / σ I \sigma'(\xi)=-\xi/\sigma_I σ ′ ( ξ ) = − ξ / σ I ξ \xi ξ (26) is
χ N q 1 ( N σ I ℓ N ( x ) ) σ I 2 2 ∣ ξ ∣ log N ( 1 + o ( 1 ) ) = a N b N 2 log N κ 1 σ I 3 ∣ ξ ∣ e − x ( 1 + o ( 1 ) ) , (30) \chi_Nq_1(N\sigma_I\ell_N(x))\frac{\sigma_I^2}{2|\xi|\log N}(1+o(1))=\frac{a_Nb_N}{2\log N}\kappa_1\frac{\sigma_I^3}{|\xi|}e^{-x}(1+o(1)), \tag*{(30)} χ N q 1 ( N σ I ℓ N ( x )) 2∣ ξ ∣ log N σ I 2 ( 1 + o ( 1 )) = 2 log N a N b N κ 1 ∣ ξ ∣ σ I 3 e − x ( 1 + o ( 1 )) , (30) where we used the asymptotics
q 1 ( N σ I ℓ N ( x ) ) = κ 1 e − x b N N − 1 ( 1 + o ( 1 ) ) , (31) q_1(N\sigma_I\ell_N(x))=\kappa_1e^{-x}b_NN^{-1}(1+o(1)), \tag*{(31)} q 1 ( N σ I ℓ N ( x )) = κ 1 e − x b N N − 1 ( 1 + o ( 1 )) , (31) which follow from (15) and (21) . In this computation, the factor b N 9 / 8 b_N^{9/8} b N 9/8 e − F ( s ) e^{-F(s)} e − F ( s ) s − 1 / 8 = b N − 1 / 8 ( 1 + o ( 1 ) ) s^{-1/8}=b_N^{-1/8}(1+o(1)) s − 1/8 = b N − 1/8 ( 1 + o ( 1 )) (15) and by the width ( log N ) − 1 (\log N)^{-1} ( log N ) − 1 ξ \xi ξ χ N / ( N log N ) = 2 σ I b N − 1 ( 1 + o ( 1 ) ) \chi_N/(N\log N)=2\sigma_Ib_N^{-1}(1+o(1)) χ N / ( N log N ) = 2 σ I b N − 1 ( 1 + o ( 1 )) a N b N / ( 2 log N ) → 1 a_Nb_N/(2\log N)\to1 a N b N / ( 2 log N ) → 1 ξ ∈ ∂ ∗ I \xi\in\partial_* I ξ ∈ ∂ ∗ I Λ I \Lambda_I Λ I
The estimates (23) , (24) , and (25) are the main work of this paper. Two features make (23) delicate. First, at s = N σ I ℓ N ( x ) s=N\sigma_I\ell_N(x) s = N σ I ℓ N ( x ) q 1 ( s ) q_1(s) q 1 ( s ) N − 1 log N N^{-1}\sqrt{\log N} N − 1 log N (31) . So we need a relative error bound for a polynomially small probability. Second, the interval contains of order log N \sqrt{\log N} log N [21 ] using the determinantal structure. The asymptotics of the Fredholm determinant can then be derived from those of Toeplitz determinants [8 ] , and the analogue of (24) holds without the factor e C log N e^{C\sqrt{\log N}} e C l o g N [4 Lemma 3.8] . Neither tool is available for the GOE. The GOE hole probability is the square root of the Fredholm determinant of an operator with a 2 × 2 2\times2 2 × 2 [28 , 44 , 49 ] , and one entry of this kernel contains the sign function sgn ( x − y ) / 2 \operatorname{sgn}(x-y)/2 sgn ( x − y ) /2 ∣ x − y ∣ → ∞ |x-y|\to\infty ∣ x − y ∣ → ∞
Hermite coordinates. For the analysis, it is convenient to rescale the eigenvalues by the factor N / 2 \sqrt{N/2} N /2 (1) , the vector ( N / 2 λ 1 , … , N / 2 λ N ) (\sqrt{N/2}\lambda_1,\ldots,\sqrt{N/2}\lambda_N) ( N /2 λ 1 , … , N /2 λ N ) ∏ i < j ∣ x j − x i ∣ ∏ i e − x i 2 / 2 \prod_{i<j}|x_j-x_i|\prod_i e^{-x_i^2/2} ∏ i < j ∣ x j − x i ∣ ∏ i e − x i 2 /2 { x 1 < ⋯ < x N } \{x_1<\cdots<x_N\} { x 1 < ⋯ < x N } [ − 2 N , 2 N ] [-\sqrt{2N},\sqrt{2N}] [ − 2 N , 2 N ] x = N / 2 λ x=\sqrt{N/2}\,\lambda x = N /2 λ Hermite coordinate . In the rest of the overview, the variables x x x y y y z z z t t t J J J R \mathbb{R} R P N ( J ) P_N(J) P N ( J ) N / 2 λ i ∉ J \sqrt{N/2}\lambda_i\notin J N /2 λ i ∈ / J i i i P N ( N / 2 A ) = E 1 , N ( A ) P_N(\sqrt{N/2}\,A)=E_{1,N}(A) P N ( N /2 A ) = E 1 , N ( A ) A ⊂ R A\subset\mathbb{R} A ⊂ R δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ]
B N ( δ ) = [ − ( 1 − δ ) 2 N , ( 1 − δ ) 2 N ] , (32) B_N(\delta)=\left[-(1-\delta)\sqrt{2N},(1-\delta)\sqrt{2N}\right], \tag*{(32)} B N ( δ ) = [ − ( 1 − δ ) 2 N , ( 1 − δ ) 2 N ] , (32) and we note that N / 2 [ − 2 + δ , 2 − δ ] = B N ( δ / 2 ) \sqrt{N/2}\,[-2+\delta,2-\delta]=B_N(\delta/2) N /2 [ − 2 + δ , 2 − δ ] = B N ( δ /2 ) δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ] B N ( δ ) B_N(\delta) B N ( δ ) δ \delta δ x x x π / 2 N − x 2 \pi/\sqrt{2N-x^2} π / 2 N − x 2
The Hermite functions h 0 , h 1 , … h_0,h_1,\ldots h 0 , h 1 , … L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) x ↦ x n e − x 2 / 2 x\mapsto x^n e^{-x^2/2} x ↦ x n e − x 2 /2 n ≥ 0 n\ge0 n ≥ 0 L 2 ( R ) L^2(\mathbb{R}) L 2 ( R )
K N ( x , y ) = ∑ n = 0 N − 1 h n ( x ) h n ( y ) , ( K N f ) ( y ) = ∫ R K N ( y , x ) f ( x ) d x , (33) K_N(x,y)=\sum_{n=0}^{N-1}h_n(x)h_n(y),\qquad(K_Nf)(y)=\int_{\mathbb{R}}K_N(y,x)f(x)\,dx, \tag*{(33)} K N ( x , y ) = n = 0 ∑ N − 1 h n ( x ) h n ( y ) , ( K N f ) ( y ) = ∫ R K N ( y , x ) f ( x ) d x , (33) for x , y ∈ R x,y\in\mathbb{R} x , y ∈ R f ∈ L 2 ( R ) f\in L^2(\mathbb{R}) f ∈ L 2 ( R ) f f f K N K_N K N L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) h 0 , … , h N − 1 h_0,\ldots,h_{N-1} h 0 , … , h N − 1 J ⊂ R J\subset\mathbb{R} J ⊂ R K J K_J K J L 2 ( J ) L^2(J) L 2 ( J ) K N ( x , y ) K_N(x,y) K N ( x , y ) x , y ∈ J x,y\in J x , y ∈ J D N ( J ) = det ( Id − K J ) D_N(J)=\det(\operatorname{Id}-K_J) D N ( J ) = det ( Id − K J )
Remark 1.3(c), with its eigenvalues multiplied by N / 2 \sqrt{N}/2 N /2 J J J [2 ] . The eigenvalues of K J K_J K J [ 0 , 1 ) [0,1) [ 0 , 1 ) Id − K J \operatorname{Id}-K_J Id − K J
Step 1: An exact formula and the endpoint matrix. Let J = ⋃ j = 1 r ( a j , b j ) J=\bigcup_{j=1}^{r}(a_j,b_j) J = ⋃ j = 1 r ( a j , b j ) a 1 < b 1 < a 2 < ⋯ < a r < b r a_1<b_1<a_2<\cdots<a_r<b_r a 1 < b 1 < a 2 < ⋯ < a r < b r t ∈ R t\in\mathbb{R} t ∈ R ε t ( x ) = sgn ( x − t ) / 2 \varepsilon_t(x)=\operatorname{sgn}(x-t)/2 ε t ( x ) = sgn ( x − t ) /2
g t ( y ) = ( K N ε t ) ( y ) − ε t ( y ) , y ∈ R . (34) g_t(y)=(K_N\varepsilon_t)(y)-\varepsilon_t(y),\qquad y\in\mathbb{R}. \tag*{(34)} g t ( y ) = ( K N ε t ) ( y ) − ε t ( y ) , y ∈ R . (34) The function g t g_t g t K N K_N K N ε t \varepsilon_t ε t 1 ≤ i ≤ r 1\leq i\leq r 1 ≤ i ≤ r G i = ( Id − K J ) − 1 ( 1 J g a i ) ∈ L 2 ( J ) G_i=(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_Jg_{a_i})\in L^2(J) G i = ( Id − K J ) − 1 ( 1 J g a i ) ∈ L 2 ( J ) G i = 1 J g a i + K N ( 1 J G i ) G_i=\mathbf{1}_Jg_{a_i}+K_N(\mathbf{1}_JG_i) G i = 1 J g a i + K N ( 1 J G i ) J J J G i G_i G i J J J t t t J J J G i ( t + ) G_i(t^+) G i ( t + ) G i ( t − ) G_i(t^-) G i ( t − ) G i G_i G i t t t r × r r\times r r × r A N ( J ) \mathcal{A}_N(J) A N ( J )
A N ( J ) i j = 1 i = j − ( G i ( b j − ) − G i ( a j + ) ) , 1 ≤ i , j ≤ r . (35) \mathcal{A}_N(J)_{ij}=\mathbf{1}_{i=j}-\left(G_i(b_j^-)-G_i(a_j^+)\right),\qquad1\leq i,j\leq r. \tag*{(35)} A N ( J ) ij = 1 i = j − ( G i ( b j − ) − G i ( a j + ) ) , 1 ≤ i , j ≤ r . (35) We measure the size of J J J microscopic length L ( J ) = 2 N ∣ J ∣ L(J)=\sqrt{2N}|J| L ( J ) = 2 N ∣ J ∣ J = N / 2 A J=\sqrt{N/2}\,A J = N /2 A L ( J ) = N ∣ A ∣ L(J)=N|A| L ( J ) = N ∣ A ∣ L ( J ) L(J) L ( J ) J J J J J J N N N δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ] r 0 ≥ 1 r_0\geq1 r 0 ≥ 1 C > 0 C>0 C > 0 r ≤ r 0 r\leq r_0 r ≤ r 0 J ⊂ B N ( δ ) J\subset\mathcal{B}_N(\delta) J ⊂ B N ( δ ) L ( J ) ≤ N 1 / 8 L(J)\leq N^{1/8} L ( J ) ≤ N 1/8
P N ( J ) 2 = D N ( J ) ( det A N ( J ) + O ( e C L ( J ) N − 1 / 2 ) ) , ∣ det A N ( J ) ∣ ≤ C e C L ( J ) . (36) \mathcal{P}_N(J)^2=\mathcal{D}_N(J)\left(\det\mathcal{A}_N(J)+O\left(e^{CL(J)}N^{-1/2}\right)\right),\qquad\left|\det\mathcal{A}_N(J)\right|\leq Ce^{CL(J)}. \tag*{(36)} P N ( J ) 2 = D N ( J ) ( det A N ( J ) + O ( e C L ( J ) N − 1/2 ) ) , ∣ det A N ( J ) ∣ ≤ C e C L ( J ) . (36) The estimate (36) is the combination of an exact identity, Corollary 3.3 , with Proposition 6.2 . The exact identity expresses P N ( J ) 2 \mathcal{P}_N(J)^2 P N ( J ) 2 N ≥ 2 N\geq2 N ≥ 2 D N ( J ) \mathcal{D}_N(J) D N ( J ) ( r + 2 ) × ( r + 2 ) (r+2)\times(r+2) ( r + 2 ) × ( r + 2 ) A N ( J ) \mathcal{A}_N(J) A N ( J ) P N ( J ) \mathcal{P}_N(J) P N ( J ) [7 ] , and the square of a Pfaffian is a determinant. The Pfaffian involves the sign function on the complement of J J J r r r J J J ( r + 2 ) × ( r + 2 ) (r+2)\times(r+2) ( r + 2 ) × ( r + 2 ) h N − 1 h_{N-1} h N − 1 h N h_N h N (46) below, we show that all entries of the matrix are O ( e C L ( J ) ) O(e^{CL(J)}) O ( e C L ( J ) ) O ( e C L ( J ) N − 1 / 2 ) O(e^{CL(J)}N^{-1/2}) O ( e C L ( J ) N − 1/2 ) (36) .
For even N N N [48 (33)] , who manipulated Fredholm determinants of operators with matrix kernels and justified these manipulations in detail for the sine kernel. We derive the exact identity differently, starting from de Bruijn’s formula and using only finite-dimensional linear algebra. Our derivation applies to even and odd N N N Theorem 1.1 concerns all N N N N N N
( ε f ) ( x ) = 1 2 ∫ R sgn ( x − z ) f ( z ) d z , p N ( y ) = ∫ R K N ( x , y ) d x , x , y ∈ R , (37) (\varepsilon f)(x)=\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-z)f(z)\,\mathrm{d}z,\qquad p_N(y)=\int_{\mathbb{R}}K_N(x,y)\,\mathrm{d}x,\qquad x,y\in\mathbb{R}, \tag*{(37)} ( ε f ) ( x ) = 2 1 ∫ R sgn ( x − z ) f ( z ) d z , p N ( y ) = ∫ R K N ( x , y ) d x , x , y ∈ R , (37) for all N ≥ 1 N\ge1 N ≥ 1 f f f ∫ R h N − 1 ( x ) d x ≠ 0 \int_{\mathbb{R}}h_{N-1}(x)\,\mathrm{d}x\ne0 ∫ R h N − 1 ( x ) d x = 0 N N N N N N
c N = ( N 2 ) − 1 / 4 ( ∫ R h N − 1 ( x ) d x ) − 1 , ω N = ( N 2 ) 1 / 4 ε h N + c N . (38) c_N=\left(\frac{N}{2}\right)^{-1/4}\left(\int_{\mathbb{R}}h_{N-1}(x)\,\mathrm{d}x\right)^{-1},\qquad\omega_N=\left(\frac{N}{2}\right)^{1/4}\varepsilon h_N+c_N. \tag*{(38)} c N = ( 2 N ) − 1/4 ( ∫ R h N − 1 ( x ) d x ) − 1 , ω N = ( 2 N ) 1/4 ε h N + c N . (38) In the bulk, the two terms of ω N \omega_N ω N Lemma 2.6 (b) that their sum is exactly a multiple of 1 − p N 1-p_N 1 − p N
ω N = c N ( 1 − p N ) , N odd . (39) \omega_N=c_N(1-p_N),\qquad N\text{ odd}. \tag*{(39)} ω N = c N ( 1 − p N ) , N odd . (39) We also show that p N ( y ) = 1 + O ( N − 1 / 2 ) p_N(y)=1+O(N^{-1/2}) p N ( y ) = 1 + O ( N − 1/2 ) (39) shows that ω N = O ( N − 1 / 2 ) \omega_N=O(N^{-1/2}) ω N = O ( N − 1/2 ) [32 Lemma 8.14 and Appendix D] .
Step 2: The sign function becomes local. Observe that the entries of A N ( J ) \mathcal{A}_N(J) A N ( J ) g a i g_{a_i} g a i J J J g t g_t g t t t t K N K_N K N φ N = ( N / 2 ) 1 / 4 h N \varphi_N=(N/2)^{1/4}h_N φ N = ( N /2 ) 1/4 h N ψ N = ( N / 2 ) 1 / 4 h N − 1 \psi_N=(N/2)^{1/4}h_{N-1} ψ N = ( N /2 ) 1/4 h N − 1 Lemma 2.7 the exact identity
( K N ε t ) ( y ) = ( ε φ N ) ( y ) ( ε ψ N ) ( y ) + ∫ t y K N ( x , y ) d x , t , y ∈ R . (40) (K_N\varepsilon_t)(y)=(\varepsilon\varphi_N)(y)(\varepsilon\psi_N)(y)+\int_t^y K_N(x,y)\,\mathrm{d}x,\qquad t,y\in\mathbb{R}. \tag*{(40)} ( K N ε t ) ( y ) = ( ε φ N ) ( y ) ( ε ψ N ) ( y ) + ∫ t y K N ( x , y ) d x , t , y ∈ R . (40) In the bulk, the product ( ε φ N ) ( ε ψ N ) (\varepsilon\varphi_N)(\varepsilon\psi_N) ( ε φ N ) ( ε ψ N ) O ( N − 1 / 2 ) O(N^{-1/2}) O ( N − 1/2 ) p N p_N p N ε φ N = − c N p N \varepsilon\varphi_N=-c_Np_N ε φ N = − c N p N N N N (39) , and ε ψ N = − p N / ∫ R φ N \varepsilon\psi_N=-p_N/\int_{\mathbb{R}}\varphi_N ε ψ N = − p N / ∫ R φ N N N N Lemma 2.6 (a). This factor is bounded in the bulk, since p N = 1 + O ( N − 1 / 2 ) p_N=1+O(N^{-1/2}) p N = 1 + O ( N − 1/2 ) c N c_N c N ( ∫ R φ N ) − 1 \left(\int_{\mathbb{R}}\varphi_N\right)^{-1} ( ∫ R φ N ) − 1 ∫ 0 y φ N \int_0^y\varphi_N ∫ 0 y φ N ∫ 0 y ψ N \int_0^y\psi_N ∫ 0 y ψ N O ( N − 1 / 2 ) O(N^{-1/2}) O ( N − 1/2 ) t t t y y y sgn ( y − t ) / 2 \operatorname{sgn}(y-t)/2 sgn ( y − t ) /2 N ∣ y − t ∣ \sqrt{N}|y-t| N ∣ y − t ∣ Corollary 4.7 , we deduce from (40) that
∣ g t ( y ) ∣ ≤ C 1 + N ∣ y − t ∣ + C N − 1 / 4 , t , y ∈ B N ( δ ) . (41) |g_t(y)|\le\frac{C}{1+\sqrt{N}|y-t|}+CN^{-1/4},\qquad t,y\in B_N(\delta). \tag*{(41)} ∣ g t ( y ) ∣ ≤ 1 + N ∣ y − t ∣ C + C N − 1/4 , t , y ∈ B N ( δ ) . (41) The term C N − 1 / 4 CN^{-1/4} C N − 1/4
g s i n e ( u ) = − sgn ( u ) ∫ ∣ u ∣ ∞ S ( v ) d v , u ∈ R , (42) g_{\mathrm{sine}}(u)=-\operatorname{sgn}(u)\int_{|u|}^{\infty}S(v)\,\mathrm{d}v,\qquad u\in\mathbb{R}, \tag*{(42)} g sine ( u ) = − sgn ( u ) ∫ ∣ u ∣ ∞ S ( v ) d v , u ∈ R , (42) and we show that
∣ g t ( t + u ϱ N ( t ) ) − g s i n e ( u ) ∣ ≤ C ( 1 + ∣ u ∣ ) N − 1 / 2 , t ∈ B N ( δ ) , 0 < ∣ u ∣ ≤ N 1 / 4 , (43) \left|g_t\left(t+\frac{u}{\varrho_N(t)}\right)-g_{\mathrm{sine}}(u)\right|\leq C(1+|u|)N^{-1/2},\qquad t\in B_N(\delta),\ 0<|u|\leq N^{1/4}, \tag*{(43)} g t ( t + ϱ N ( t ) u ) − g sine ( u ) ≤ C ( 1 + ∣ u ∣ ) N − 1/2 , t ∈ B N ( δ ) , 0 < ∣ u ∣ ≤ N 1/4 , (43) where we write ϱ N ( t ) = 2 N − t 2 \varrho_N(t)=\sqrt{2N-t^2} ϱ N ( t ) = 2 N − t 2 Figure 1.1 . The functions p N p_N p N ε φ N \varepsilon\varphi_N ε φ N ε ψ N \varepsilon\psi_N ε ψ N R \mathbb{R} R [ 0 , y ] [0,y] [ 0 , y ] ∫ R h n \int_{\mathbb{R}}h_n ∫ R h n ± 2 N \pm\sqrt{2N} ± 2 N
Figure 1.1. The function g t = K N ε t − ε t g_t=K_N\varepsilon_t-\varepsilon_t g t = K N ε t − ε t N = 60 N=60 N = 60 t = 2 t=2 t = 2 K N ε t K_N\varepsilon_t K N ε t ε t \varepsilon_t ε t ± 2 N \pm\sqrt{2N} ± 2 N t t t g t ( t + u / ϱ N ( t ) ) g_t(t+u/\varrho_N(t)) g t ( t + u / ϱ N ( t )) ϱ N ( t ) = 2 N − t 2 \varrho_N(t)=\sqrt{2N-t^2} ϱ N ( t ) = 2 N − t 2 g s i n e ( u ) g_{\mathrm{sine}}(u) g sine ( u )
The bound (41) is what makes the GOE hole probability factorize. We now assume, in addition to the assumptions of (36) , that the distance between any two intervals of J J J 1 / 2 1/\sqrt{2} 1/ 2 N − 1 / 2 N^{-1/2} N − 1/2 λ \lambda λ L ( J ) ≤ γ log N L(J)\leq\gamma\sqrt{\log N} L ( J ) ≤ γ log N γ > 0 \gamma>0 γ > 0 g a i ( y ) = O ( N − 1 / 4 ) g_{a_i}(y)=O(N^{-1/4}) g a i ( y ) = O ( N − 1/4 ) y y y ( a j , b j ) (a_j,b_j) ( a j , b j ) j ≠ i j\ne i j = i (41) . Further, ∣ K N ( x , y ) ∣ ≤ C |K_N(x,y)|\leq C ∣ K N ( x , y ) ∣ ≤ C x x x y y y K N ( x , x ) K_N(x,x) K N ( x , x ) N \sqrt{N} N (46) below, we show in Lemma 6.8 that the off-diagonal entries of A N ( J ) A_N(J) A N ( J ) O ( e C L ( J ) N − 1 / 4 ) O(e^{CL(J)}N^{-1/4}) O ( e C L ( J ) N − 1/4 ) A N ( J ) j j A_N(J)_{jj} A N ( J ) j j O ( e C L ( J ) N − 1 / 2 ) O(e^{CL(J)}N^{-1/2}) O ( e C L ( J ) N − 1/2 ) J j = ( a j , b j ) J_j=(a_j,b_j) J j = ( a j , b j ) 1 × 1 1\times1 1 × 1 J j J_j J j x 0 = a j x_0=a_j x 0 = a j s = 2 ϱ N ( a j ) ( b j − a j ) ≤ C L ( J ) s=2\varrho_N(a_j)(b_j-a_j)\leq CL(J) s = 2 ϱ N ( a j ) ( b j − a j ) ≤ C L ( J ) det A N ( J j ) ≥ c e − C L ( J ) \det A_N(J_j)\geq ce^{-CL(J)} det A N ( J j ) ≥ c e − C L ( J )
det A N ( J ) = ( 1 + O ( e C L ( J ) N − 1 / 4 ) ) ∏ j = 1 r A N ( ( a j , b j ) ) . (44) \det A_N(J)=\left(1+O(e^{CL(J)}N^{-1/4})\right)\prod_{j=1}^{r}A_N((a_j,b_j)). \tag*{(44)} det A N ( J ) = ( 1 + O ( e C L ( J ) N − 1/4 ) ) j = 1 ∏ r A N (( a j , b j )) . (44) Next, the operator K J K_J K J K J j K_{J_j} K J j K N ( x , y ) K_N(x,y) K N ( x , y ) x x x y y y J J J C ∣ J ∣ ≤ C L ( J ) N − 1 / 2 C|J| \le CL(J)N^{-1/2} C ∣ J ∣ ≤ C L ( J ) N − 1/2
D N ( J ) = ( 1 + O ( e C L ( J ) N − 1 ) ) ∏ j = 1 r D N ( J j ) . (45) \mathcal{D}_N(J) = \left(1+O\left(e^{CL(J)}N^{-1}\right)\right)\prod_{j=1}^{r}\mathcal{D}_N(J_j). \tag*{(45)} D N ( J ) = ( 1 + O ( e C L ( J ) N − 1 ) ) j = 1 ∏ r D N ( J j ) . (45) To prove (1.25), we set J = N / 2 ⋃ j A j J=\sqrt{N/2}\bigcup_j A_j J = N /2 ⋃ j A j J ‾ ⊂ B N ( δ / 2 ) \overline{J}\subset B_N(\delta/2) J ⊂ B N ( δ /2 ) L ( J ) = N ∑ j ∣ A j ∣ ≤ γ log N ≤ N 1 / 8 L(J)=N\sum_j|A_j|\le\gamma\sqrt{\log N}\le N^{1/8} L ( J ) = N ∑ j ∣ A j ∣ ≤ γ log N ≤ N 1/8 J J J J j J_j J j det A N ( J j ) \det A_N(J_j) det A N ( J j )
Step 3: An inverse bound from the uncertainty principle. The estimates in Steps 1 and 2 involve the norm of ( Id − K J ) − 1 (\operatorname{Id}-K_J)^{-1} ( Id − K J ) − 1 L 2 ( J ) L^2(J) L 2 ( J ) K J K_J K J [ 0 , 1 ) [0,1) [ 0 , 1 ) ∥ ( Id − K J ) − 1 ∥ ≤ D N ( J ) − 1 \|(\operatorname{Id}-K_J)^{-1}\|\le\mathcal{D}_N(J)^{-1} ∥ ( Id − K J ) − 1 ∥ ≤ D N ( J ) − 1 J J J e c L ( J ) 2 e^{cL(J)^2} e c L ( J ) 2 N N N L ( J ) L(J) L ( J ) log N \sqrt{\log N} log N
∥ ( Id − K J ) − 1 ∥ ≤ C e C L ( J ) , ∣ J ∣ ≤ 1 , (46) \|(\operatorname{Id}-K_J)^{-1}\|\le Ce^{CL(J)},\qquad|J|\le1, \tag*{(46)} ∥ ( Id − K J ) − 1 ∥ ≤ C e C L ( J ) , ∣ J ∣ ≤ 1 , (46) which we prove in Proposition 5.1 using a quantitative uncertainty principle of Nazarov [29 , 39 ] . To state it, we define the Fourier transform of f ∈ L 1 ( R ) ∩ L 2 ( R ) f\in L^1(\mathbb{R})\cap L^2(\mathbb{R}) f ∈ L 1 ( R ) ∩ L 2 ( R )
( F f ) ( ξ ) = 1 2 π ∫ R f ( x ) e − i x ξ d x , ξ ∈ R , (47) (\mathcal{F}f)(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)e^{-ix\xi}\,\mathrm{d}x,\qquad\xi\in\mathbb{R}, \tag*{(47)} ( F f ) ( ξ ) = 2 π 1 ∫ R f ( x ) e − i x ξ d x , ξ ∈ R , (47) and we extend F \mathcal{F} F L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) A , Σ ⊂ R A,\Sigma\subset\mathbb{R} A , Σ ⊂ R f ∈ L 2 ( R ) f\in L^2(\mathbb{R}) f ∈ L 2 ( R )
∥ f ∥ 2 2 ≤ C e C ∣ A ∣ ∣ Σ ∣ ( ∫ R ∖ A ∣ f ( x ) ∣ 2 d x + ∫ R ∖ Σ ∣ ( F f ) ( ξ ) ∣ 2 d ξ ) . (48) \|f\|_2^2\le Ce^{C|A||\Sigma|}\left(\int_{\mathbb{R}\setminus A}|f(x)|^2\,\mathrm{d}x+\int_{\mathbb{R}\setminus\Sigma}|(\mathcal{F}f)(\xi)|^2\,\mathrm{d}\xi\right). \tag*{(48)} ∥ f ∥ 2 2 ≤ C e C ∣ A ∣∣Σ∣ ( ∫ R ∖ A ∣ f ( x ) ∣ 2 d x + ∫ R ∖ Σ ∣ ( F f ) ( ξ ) ∣ 2 d ξ ) . (48) We let V N V_N V N h 0 , … , h N − 1 h_0,\ldots,h_{N-1} h 0 , … , h N − 1 h n h_n h n F \mathcal{F} F F \mathcal{F} F V N V_N V N h 0 , … , h N − 1 h_0,\ldots,h_{N-1} h 0 , … , h N − 1 f f f ∣ f ∣ 2 |f|^2 ∣ f ∣ 2 [ − 8 N , 8 N ] [-8\sqrt{N},8\sqrt{N}] [ − 8 N , 8 N ] e − N ∥ f ∥ 2 2 e^{-N}\|f\|_2^2 e − N ∥ f ∥ 2 2 A = J A=J A = J Σ = [ − 8 N , 8 N ] \Sigma=[-8\sqrt{N},8\sqrt{N}] Σ = [ − 8 N , 8 N ] f ∈ V N f\in V_N f ∈ V N ∣ J ∣ ∣ Σ ∣ = 8 2 L ( J ) |J||\Sigma|=8\sqrt{2}L(J) ∣ J ∣∣Σ∣ = 8 2 L ( J )
∥ f ∥ 2 2 ≤ C e C L ( J ) ( ∥ f ∥ 2 2 − ∥ 1 J f ∥ 2 2 + e − N ∥ f ∥ 2 2 ) , f ∈ V N . (49) \|f\|_2^2\le Ce^{CL(J)}\left(\|f\|_2^2-\|1_Jf\|_2^2+e^{-N}\|f\|_2^2\right),\qquad f\in V_N. \tag*{(49)} ∥ f ∥ 2 2 ≤ C e C L ( J ) ( ∥ f ∥ 2 2 − ∥ 1 J f ∥ 2 2 + e − N ∥ f ∥ 2 2 ) , f ∈ V N . (49) Since ∣ J ∣ ≤ 1 |J|\le1 ∣ J ∣ ≤ 1 L ( J ) ≤ 2 N L(J)\le\sqrt{2N} L ( J ) ≤ 2 N e − N e^{-N} e − N e − C L ( J ) e^{-CL(J)} e − C L ( J ) ∥ 1 J f ∥ 2 2 ≤ ( 1 − c e − C L ( J ) ) ∥ f ∥ 2 2 \|1_Jf\|_2^2\le(1-ce^{-CL(J)})\|f\|_2^2 ∥ 1 J f ∥ 2 2 ≤ ( 1 − c e − C L ( J ) ) ∥ f ∥ 2 2 f ∈ V N f\in V_N f ∈ V N ∥ K J ∥ \|K_J\| ∥ K J ∥ ∥ 1 J f ∥ 2 2 / ∥ f ∥ 2 2 \lVert1_Jf\rVert_2^2/\lVert f\rVert_2^2 ∥ 1 J f ∥ 2 2 / ∥ f ∥ 2 2 f ∈ V N f \in V_N f ∈ V N (46) . For a bounded interval J ′ ⊂ R J' \subset\mathbb{R} J ′ ⊂ R S J ′ S_{J'} S J ′ L 2 ( J ′ ) L^2(J') L 2 ( J ′ ) S ( u − v ) S(u-v) S ( u − v ) S ( u − v ) S(u-v) S ( u − v ) L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] V N V_N V N A = J ′ A=J' A = J ′ Σ = [ − 1 , 1 ] \Sigma=[-1,1] Σ = [ − 1 , 1 ] ∣ A ∣ ∣ Σ ∣ = 2 ∣ J ′ ∣ |A||\Sigma|=2|J'| ∣ A ∣∣Σ∣ = 2∣ J ′ ∣ ∥ ( Id − S J ′ ) − 1 ∥ ≤ C e C ∣ J ′ ∣ \lVert(\operatorname{Id}-S_{J'})^{-1}\rVert\le Ce^{C|J'|} ∥( Id − S J ′ ) − 1 ∥ ≤ C e C ∣ J ′ ∣
Step 4: Comparison with the sine kernel. We now explain the proof of the one-interval estimate (23) . We fix a point y 0 ∈ [ − 2 + δ , 2 − δ ] y_0 \in[-2+\delta,2-\delta] y 0 ∈ [ − 2 + δ , 2 − δ ] 0 < s ≤ γ log N 0<s\le\gamma\sqrt{\log N} 0 < s ≤ γ log N
x 0 = N / 2 y 0 , ϱ = ϱ N ( x 0 ) , J = ( x 0 , x 0 + s 2 ϱ ) . (50) x_0=\sqrt{N/2}\,y_0,\qquad\varrho=\varrho_N(x_0),\qquad J=\left(x_0,x_0+\frac{s}{2\varrho}\right). \tag*{(50)} x 0 = N /2 y 0 , ϱ = ϱ N ( x 0 ) , J = ( x 0 , x 0 + 2 ϱ s ) . (50) Then ϱ = N / 2 σ ( y 0 ) \varrho=\sqrt{N/2}\,\sigma(y_0) ϱ = N /2 σ ( y 0 ) J = N / 2 ( y 0 , y 0 + s / ( N σ ( y 0 ) ) ) J=\sqrt{N/2}\left(y_0,y_0+s/(N\sigma(y_0))\right) J = N /2 ( y 0 , y 0 + s / ( N σ ( y 0 )) ) L ( J ) = s / σ ( y 0 ) ≤ C s L(J)=s/\sigma(y_0)\le Cs L ( J ) = s / σ ( y 0 ) ≤ C s (36) to J J J r = 1 r=1 r = 1 δ / 4 \delta/4 δ /4 δ \delta δ O ( e C s N − 1 / 2 ) O(e^{Cs}N^{-1/2}) O ( e C s N − 1/2 ) [41 , 47 ] , that
∣ 1 ϱ K N ( x 0 + u ϱ , x 0 + v ϱ ) − S ( u − v ) ∣ ≤ C N − 1 / 2 , ∣ u ∣ , ∣ v ∣ ≤ 2 N 1 / 4 . (51) \left|\frac{1}{\varrho}K_N\left(x_0+\frac{u}{\varrho},x_0+\frac{v}{\varrho}\right)-S(u-v)\right|\le CN^{-1/2},\qquad|u|,|v|\le2N^{1/4}. \tag*{(51)} ϱ 1 K N ( x 0 + ϱ u , x 0 + ϱ v ) − S ( u − v ) ≤ C N − 1/2 , ∣ u ∣ , ∣ v ∣ ≤ 2 N 1/4 . (51) This shows that after the change of variables x = x 0 + u / ϱ x=x_0+u/\varrho x = x 0 + u / ϱ K J K_J K J S ( 0 , s / 2 ) S_{(0,s/2)} S ( 0 , s /2 ) ℓ > 0 \ell>0 ℓ > 0 A s i n e ( ℓ ) A_{\mathrm{sine}}(\ell) A sine ( ℓ ) r = 1 r=1 r = 1 K N K_N K N J J J g a 1 ( x 0 + ⋅ / ϱ ) g_{a_1}(x_0+\cdot/\varrho) g a 1 ( x 0 + ⋅ / ϱ ) S ( u − v ) S(u-v) S ( u − v ) L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) ( 0 , ℓ ) (0,\ell) ( 0 , ℓ ) g s i n e g_{\mathrm{sine}} g sine A s i n e ( s / 2 ) A_{\mathrm{sine}}(s/2) A sine ( s /2 ) det A N ( J ) \det A_N(J) det A N ( J ) (51) , the convergence of g x 0 ( x 0 + u / ϱ ) g_{x_0}(x_0+u/\varrho) g x 0 ( x 0 + u / ϱ ) g s i n e ( u ) g_{\mathrm{sine}}(u) g sine ( u ) ( Id − K J ) − 1 (\operatorname{Id}-K_J)^{-1} ( Id − K J ) − 1 ( Id − S ( 0 , s / 2 ) ) − 1 (\operatorname{Id}-S_{(0,s/2)})^{-1} ( Id − S ( 0 , s /2 ) ) − 1
D N ( J ) = det ( Id − S ( 0 , s / 2 ) ) ( 1 + O ( e C s N − 1 / 2 ) ) , D_N(J)=\det(\operatorname{Id}-S_{(0,s/2)})(1+O(e^{Cs}N^{-1/2})), D N ( J ) = det ( Id − S ( 0 , s /2 ) ) ( 1 + O ( e C s N − 1/2 )) , det A N ( J ) = A s i n e ( s / 2 ) + O ( e C s N − 1 / 2 ) . (52) \det A_N(J)=A_{\mathrm{sine}}(s/2)+O(e^{Cs}N^{-1/2}). \tag*{(52)} det A N ( J ) = A sine ( s /2 ) + O ( e C s N − 1/2 ) . (52) We translate ( 0 , s / 2 ) (0,s/2) ( 0 , s /2 ) ( − s / 4 , s / 4 ) (-s/4,s/4) ( − s /4 , s /4 ) S ( u − v ) S(u-v) S ( u − v ) S + , s / 4 S_{+,s/4} S + , s /4 S − , s / 4 S_{-,s/4} S − , s /4 det ( Id − S ( 0 , s / 2 ) ) = D + ( s / 4 ) D − ( s / 4 ) \det(\operatorname{Id}-S_{(0,s/2)})=D_+(s/4)D_-(s/4) det ( Id − S ( 0 , s /2 ) ) = D + ( s /4 ) D − ( s /4 ) [48 (19), (25), (30)] , we have A s i n e ( s / 2 ) = D + ( s / 4 ) / D − ( s / 4 ) A_{\mathrm{sine}}(s/2)=D_+(s/4)/D_-(s/4) A sine ( s /2 ) = D + ( s /4 ) / D − ( s /4 ) (15) and the positivity and continuity of D ± D_\pm D ± c e − s / 4 ce^{-s/4} c e − s /4 s > 0 s>0 s > 0 det A N ( J ) ≥ c e − C L ( J ) \det A_N(J)\ge ce^{-CL(J)} det A N ( J ) ≥ c e − C L ( J ) (36) is small compared to det A N ( J ) \det A_N(J) det A N ( J ) s ≤ γ log N s\le\gamma\sqrt{\log N} s ≤ γ log N
P N ( J ) 2 = D + ( s / 4 ) D − ( s / 4 ) ⋅ D + ( s / 4 ) D − ( s / 4 ) ( 1 + O ( e C s N − 1 / 2 ) ) = q 1 ( s ) 2 ( 1 + O ( e C s N − 1 / 2 ) ) . (53) \begin{aligned}
\mathcal{P}_N(J)^2=D_+(s/4)D_-(s/4)\cdot\frac{D_+(s/4)}{D_-(s/4)}(1+O(e^{Cs}N^{-1/2})) \\
&=q_1(s)^2(1+O(e^{Cs}N^{-1/2})). \tag*{(53)}
\end{aligned} P N ( J ) 2 = D + ( s /4 ) D − ( s /4 ) ⋅ D − ( s /4 ) D + ( s /4 ) ( 1 + O ( e C s N − 1/2 )) = q 1 ( s ) 2 ( 1 + O ( e C s N − 1/2 )) . (53) By the description of J J J P N ( J ) \mathcal{P}_N(J) P N ( J ) (23) with y = y 0 y=y_0 y = y 0 (23) .
Finally, we prove the upper bound (24) , which requires no separation assumption. Let J j = N 2 A j J_j=\frac{\sqrt{N}}{2}A_j J j = 2 N A j 1 ≤ j ≤ r 1\le j\le r 1 ≤ j ≤ r J = ⋃ j J j J=\bigcup_j J_j J = ⋃ j J j J ‾ ⊂ B N ( δ / 2 ) \overline{J}\subset B_N(\delta/2) J ⊂ B N ( δ /2 ) L ( J ) ≤ γ log N ≤ N 1 / 8 L(J)\le\gamma\sqrt{\log N}\le N^{1/8} L ( J ) ≤ γ log N ≤ N 1/8 (36) with δ / 2 \delta/2 δ /2 δ \delta δ P N ( J ) 2 ≤ C e C L ( J ) D N ( J ) \mathcal{P}_N(J)^2\le Ce^{CL(J)}\mathcal{D}_N(J) P N ( J ) 2 ≤ C e C L ( J ) D N ( J )
Next, the operator Id − K J \operatorname{Id}-K_J Id − K J 0 ≤ K J < Id 0\le K_J<\operatorname{Id} 0 ≤ K J < Id D N ( J ) ≤ ∏ j D N ( J j ) \mathcal{D}_N(J)\le\prod_j\mathcal{D}_N(J_j) D N ( J ) ≤ ∏ j D N ( J j )
Finally, we write J j J_j J j y 0 y_0 y 0 A j A_j A j s = N σ ( y 0 ) ∣ A j ∣ ≤ 2 γ log N s=N\sigma(y_0)|A_j|\le2\gamma\sqrt{\log N} s = N σ ( y 0 ) ∣ A j ∣ ≤ 2 γ log N (36) for the single interval J j J_j J j det A N ( J j ) ≥ c e − C L ( J j ) \det\mathcal{A}_N(J_j)\ge ce^{-CL(J_j)} det A N ( J j ) ≥ c e − C L ( J j ) D N ( J j ) ≤ C e C L ( J j ) P N ( J j ) 2 \mathcal{D}_N(J_j)\le Ce^{CL(J_j)}\mathcal{P}_N(J_j)^2 D N ( J j ) ≤ C e C L ( J j ) P N ( J j ) 2 L ( J ) ≤ γ log N L(J)\le\gamma\sqrt{\log N} L ( J ) ≤ γ log N (24) .
Organization of the paper
Section 2 collects classical facts about Hermite functions and proves the exact identities, including (39) and (40) . Section 3 proves the exact identity for P N ( J ) 2 \mathcal{P}_N(J)^2 P N ( J ) 2 (46) . Section 6 reduces this identity to the endpoint matrix and shows that the endpoint matrix of separated intervals is nearly diagonal. Section 7 proves the one-interval estimate (23) , and Section 8 proves the estimates (24) and (25) . Section 9 computes the first moment (26) , and Section 10 completes the proofs of Theorem 1.1, Corollary 1.2, and Remark 1.4. Appendix A contains the criterion for Poisson convergence. The main logical dependencies between the sections are shown in Figure 1.2 .
Figure 1.2. The main logical dependencies between the sections. An arrow from one section to another means that the second section uses results of the first. The classical facts and exact identities of Section 2 are used throughout.
Hermite functions and exact identities
In this section, we fix our notation and collect the facts about Hermite functions that will be used in the rest of the paper. The classical facts are recalled in § 3.2 . The identities in § 3.4 are elementary, but our argument depends on them. They express every global quantity that we need, such as the integral of the Hermite kernel over a half-line, through quantities in the bulk of the spectrum. This allows us to avoid any asymptotic analysis of Hermite functions near the spectral edges.
Conventions
For real-valued measurable functions f f f g g g R \mathbb{R} R
⟨ f , g ⟩ = ∫ R f ( x ) g ( x ) d x (54) \langle f,g\rangle= \int_{\mathbb{R}} f(x)g(x)\,\mathrm{d}x \tag*{(54)} ⟨ f , g ⟩ = ∫ R f ( x ) g ( x ) d x (54) whenever the integral converges absolutely. All functions in this paper are real-valued, except for the Fourier transforms. For a measurable set A ⊂ R A \subset\mathbb{R} A ⊂ R 1 A \mathbf{1}_A 1 A ∣ A ∣ |A| ∣ A ∣ A c = R ∖ A A^{c} = \mathbb{R} \setminus A A c = R ∖ A Y Y Y v v v ⟨ Y , ⋅ ⟩ \langle Y,\cdot\rangle ⟨ Y , ⋅ ⟩ v ⊗ Y v \otimes Y v ⊗ Y
( v ⊗ Y ) f = ⟨ Y , f ⟩ v . (55) (v \otimes Y)f = \langle Y,f\rangle v. \tag*{(55)} ( v ⊗ Y ) f = ⟨ Y , f ⟩ v . (55) We set sgn ( x ) = x / ∣ x ∣ \operatorname{sgn}(x) = x/|x| sgn ( x ) = x /∣ x ∣ x ≠ 0 x \ne0 x = 0 sgn ( 0 ) = 0 \operatorname{sgn}(0) = 0 sgn ( 0 ) = 0 t ∈ R t \in\mathbb{R} t ∈ R
ε t ( x ) = 1 2 sgn ( x − t ) , (56) \varepsilon_t(x) = \frac{1}{2}\operatorname{sgn}(x-t), \tag*{(56)} ε t ( x ) = 2 1 sgn ( x − t ) , (56) and for f ∈ L 1 ( R ) f \in L^1(\mathbb{R}) f ∈ L 1 ( R ) ε f \varepsilon f ε f
( ε f ) ( x ) = 1 2 ∫ R sgn ( x − z ) f ( z ) d z . (57) (\varepsilon f)(x) = \frac{1}{2}\int_{\mathbb{R}} \operatorname{sgn}(x-z)f(z)\,\mathrm{d}z. \tag*{(57)} ( ε f ) ( x ) = 2 1 ∫ R sgn ( x − z ) f ( z ) d z . (57) We call ε \varepsilon ε sign operator . The following properties are immediate from its definition. First, if f ∈ L 1 ( R ) f \in L^1(\mathbb{R}) f ∈ L 1 ( R ) ε f \varepsilon f ε f ( ε f ) ′ = f (\varepsilon f)' = f ( ε f ) ′ = f
lim x → ± ∞ ( ε f ) ( x ) = ± 1 2 ∫ R f ( z ) d z . (58) \lim_{x\to\pm\infty}(\varepsilon f)(x) = \pm\frac{1}{2}\int_{\mathbb{R}} f(z)\,\mathrm{d}z. \tag*{(58)} x → ± ∞ lim ( ε f ) ( x ) = ± 2 1 ∫ R f ( z ) d z . (58) Further, if f ∈ L 1 ( R ) f \in L^1(\mathbb{R}) f ∈ L 1 ( R ) f ′ ∈ L 1 ( R ) f' \in L^1(\mathbb{R}) f ′ ∈ L 1 ( R ) f ( x ) → 0 f(x) \to0 f ( x ) → 0 ∣ x ∣ → ∞ |x| \to\infty ∣ x ∣ → ∞ ε ( f ′ ) = f \varepsilon(f')=f ε ( f ′ ) = f f , g ∈ L 1 ( R ) f,g \in L^1(\mathbb{R}) f , g ∈ L 1 ( R ) ⟨ f , ε g ⟩ = − ⟨ ε f , g ⟩ \langle f,\varepsilon g\rangle=-\langle\varepsilon f,g\rangle ⟨ f , ε g ⟩ = − ⟨ ε f , g ⟩
Throughout the paper, we let C C C c c c N N N
Classical facts
For every integer n ≥ 0 n \ge0 n ≥ 0 Hermite function
h n ( x ) = ( 2 n n ! π ) − 1 / 2 ( − 1 ) n e x 2 / 2 d n d x n e − x 2 , (59) h_n(x) = (2^n n!\sqrt{\pi})^{-1/2}(-1)^n e^{x^2/2}\frac{\mathrm{d}^n}{\mathrm{d}x^n}e^{-x^2}, \tag*{(59)} h n ( x ) = ( 2 n n ! π ) − 1/2 ( − 1 ) n e x 2 /2 d x n d n e − x 2 , (59) and we set h − 1 = 0 h_{-1}=0 h − 1 = 0 [47 ] . The functions ( h n ) n ≥ 0 (h_n)_{n\ge0} ( h n ) n ≥ 0 L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) h n h_n h n n n n e − x 2 / 2 e^{-x^2/2} e − x 2 /2 h n ( − x ) = ( − 1 ) n h n ( x ) h_n(-x)=(-1)^n h_n(x) h n ( − x ) = ( − 1 ) n h n ( x ) n ≥ 0 n \ge0 n ≥ 0
x h n ( x ) = n + 1 2 h n + 1 ( x ) + n 2 h n − 1 ( x ) , h n ′ ( x ) = n 2 h n − 1 ( x ) − n + 1 2 h n + 1 ( x ) . (60) \begin{aligned}
xh_n(x) &= \sqrt{\frac{n+1}{2}}h_{n+1}(x)+\sqrt{\frac{n}{2}}h_{n-1}(x),\\
h_n'(x) &= \sqrt{\frac{n}{2}}h_{n-1}(x)-\sqrt{\frac{n+1}{2}}h_{n+1}(x). \tag*{(60)}
\end{aligned} x h n ( x ) h n ′ ( x ) = 2 n + 1 h n + 1 ( x ) + 2 n h n − 1 ( x ) , = 2 n h n − 1 ( x ) − 2 n + 1 h n + 1 ( x ) . (60) For f ∈ L 1 ( R ) ∩ L 2 ( R ) f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R}) f ∈ L 1 ( R ) ∩ L 2 ( R )
( F f ) ( ξ ) = 1 2 π ∫ R f ( x ) e − i x ξ d x (61) (\mathcal{F}f)(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} f(x)e^{-ix\xi}\,\mathrm{d}x \tag*{(61)} ( F f ) ( ξ ) = 2 π 1 ∫ R f ( x ) e − i x ξ d x (61) denote its Fourier transform, extended as usual to a unitary operator on L 2 ( R ) L^2(\mathbb{R}) L 2 ( R )
F h n = ( − i ) n h n . (62) \mathcal{F}h_n=(-i)^n h_n. \tag*{(62)} F h n = ( − i ) n h n . (62) We also recall that h 2 m + 1 ( 0 ) = 0 h_{2m+1}(0)=0 h 2 m + 1 ( 0 ) = 0
h 2 m ( 0 ) = ( − 1 ) m π − 1 / 4 ( 2 m ) ! 2 m m ! . (63) h_{2m}(0)=(-1)^m\pi^{-1/4}\frac{\sqrt{(2m)!}}{2^m m!}. \tag*{(63)} h 2 m ( 0 ) = ( − 1 ) m π − 1/4 2 m m ! ( 2 m )! . (63) for all m ≥ 0 m \ge0 m ≥ 0 ξ = 0 \xi=0 ξ = 0 ∫ R h 2 m + 1 = 0 \int_{\mathbb{R}}h_{2m+1}=0 ∫ R h 2 m + 1 = 0
∫ R h 2 m ( x ) d x = 2 π ( − 1 ) m h 2 m ( 0 ) = 2 π 1 / 4 ( 2 m ) ! 2 m m ! . (64) \int_{\mathbb{R}}h_{2m}(x)\,\mathrm{d}x=\sqrt{2\pi}(-1)^m h_{2m}(0)=\sqrt{2}\pi^{1/4}\frac{\sqrt{(2m)!}}{2^m m!}. \tag*{(64)} ∫ R h 2 m ( x ) d x = 2 π ( − 1 ) m h 2 m ( 0 ) = 2 π 1/4 2 m m ! ( 2 m )! . (64) We set V 0 = { 0 } V_0=\{0\} V 0 = { 0 } N ≥ 1 N \ge1 N ≥ 1 V N V_N V N h 0 , … , h N − 1 h_0,\ldots,h_{N-1} h 0 , … , h N − 1
K N ( x , y ) = ∑ n = 0 N − 1 h n ( x ) h n ( y ) . (65) K_N(x,y)=\sum_{n=0}^{N-1}h_n(x)h_n(y). \tag*{(65)} K N ( x , y ) = n = 0 ∑ N − 1 h n ( x ) h n ( y ) . (65) We also let K N K_N K N K N ( x , y ) K_N(x,y) K N ( x , y ) L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) V N V_N V N f f f K N f K_Nf K N f K N f ∈ V N K_Nf \in V_N K N f ∈ V N
K N ( x , y ) = N 2 h N ( x ) h N − 1 ( y ) − h N − 1 ( x ) h N ( y ) x − y , x ≠ y , (66) K_N(x,y)=\sqrt{\frac{N}{2}}\frac{h_N(x)h_{N-1}(y)-h_{N-1}(x)h_N(y)}{x-y},\qquad x\ne y, \tag*{(66)} K N ( x , y ) = 2 N x − y h N ( x ) h N − 1 ( y ) − h N − 1 ( x ) h N ( y ) , x = y , (66) which follows from the first identity in (2.7). Further, for x ∈ R x\in\mathbb{R} x ∈ R K N ( x , x ) = ∑ n < N h n ( x ) 2 K_N(x,x)=\sum_{n<N}h_n(x)^2 K N ( x , x ) = ∑ n < N h n ( x ) 2
∣ K N ( x , y ) ∣ ≤ K N ( x , x ) 1 / 2 K N ( y , y ) 1 / 2 , ∣ f ( x ) ∣ 2 ≤ K N ( x , x ) ∥ f ∥ 2 2 for f ∈ V N . (67) \lvert K_N(x,y)\rvert\le K_N(x,x)^{1/2}K_N(y,y)^{1/2},\qquad\lvert f(x)\rvert^2\le K_N(x,x)\lVert f\rVert_2^2\quad\text{for }f\in V_N. \tag*{(67)} ∣ K N ( x , y )∣ ≤ K N ( x , x ) 1/2 K N ( y , y ) 1/2 , ∣ f ( x ) ∣ 2 ≤ K N ( x , x ) ∥ f ∥ 2 2 for f ∈ V N . (67) We will need a crude bound on Hermite functions far outside the spectrum.
Lemma 2.1. For all integers N ≥ 1 N\ge1 N ≥ 1 0 ≤ n ≤ N 0\le n\le N 0 ≤ n ≤ N x ∈ R x\in\mathbb{R} x ∈ R ∣ x ∣ ≥ 8 N \lvert x\rvert\ge8\sqrt{N} ∣ x ∣ ≥ 8 N h n ( x ) 2 ≤ e − x 2 / 4 h_n(x)^2\le e^{-x^2/4} h n ( x ) 2 ≤ e − x 2 /4
Proof. Let P n P_n P n n n n
h n ( x ) = P n ( x ) e − x 2 / 2 ( 2 n n ! π ) 1 / 2 . (68) h_n(x)=\frac{P_n(x)e^{-x^2/2}}{(2^n n!\sqrt{\pi})^{1/2}}. \tag*{(68)} h n ( x ) = ( 2 n n ! π ) 1/2 P n ( x ) e − x 2 /2 . (68) These polynomials have the generating function (see [47 ] )
∑ n ≥ 0 P n ( x ) z n n ! = e 2 x z − z 2 . (69) \sum_{n\ge0}P_n(x)\frac{z^n}{n!}=e^{2xz-z^2}. \tag*{(69)} n ≥ 0 ∑ P n ( x ) n ! z n = e 2 x z − z 2 . (69) By Cauchy’s estimate on the circle ∣ z ∣ = N \lvert z\rvert=\sqrt{N} ∣ z ∣ = N
∣ P n ( x ) ∣ ≤ n ! N − n / 2 e 2 ∣ x ∣ N + N . (70) \lvert P_n(x)\rvert\le n!N^{-n/2}e^{2\lvert x\rvert\sqrt{N}+N}. \tag*{(70)} ∣ P n ( x )∣ ≤ n ! N − n /2 e 2 ∣ x ∣ N + N . (70) It follows that
h n ( x ) 2 = P n ( x ) 2 e − x 2 2 n n ! π ≤ n ! N n e 4 ∣ x ∣ N + 2 N − x 2 ≤ e 4 ∣ x ∣ N + 2 N − x 2 . (71) h_n(x)^2=\frac{P_n(x)^2e^{-x^2}}{2^n n!\sqrt{\pi}}\le\frac{n!}{N^n}e^{4\lvert x\rvert\sqrt{N}+2N-x^2}\le e^{4\lvert x\rvert\sqrt{N}+2N-x^2}. \tag*{(71)} h n ( x ) 2 = 2 n n ! π P n ( x ) 2 e − x 2 ≤ N n n ! e 4 ∣ x ∣ N + 2 N − x 2 ≤ e 4 ∣ x ∣ N + 2 N − x 2 . (71) In the last inequality, we used
n ! ≤ n n ≤ N n (72) n!\le n^n\le N^n \tag*{(72)} n ! ≤ n n ≤ N n (72) for 1 ≤ n ≤ N 1\le n\le N 1 ≤ n ≤ N n = 0 n=0 n = 0 ∣ x ∣ ≥ 8 N \lvert x\rvert\ge8\sqrt{N} ∣ x ∣ ≥ 8 N 4 ∣ x ∣ N ≤ x 2 / 2 4\lvert x\rvert\sqrt{N}\le x^2/2 4 ∣ x ∣ N ≤ x 2 /2 2 N ≤ x 2 / 32 2N\le x^2/32 2 N ≤ x 2 /32 − x 2 / 4 -x^2/4 − x 2 /4 □ \square □
The GOE in Hermite coordinates
Recall that H N H_N H N N × N N \times N N × N (1) , with eigenvalues λ 1 < ⋯ < λ N \lambda_1 < \cdots< \lambda_N λ 1 < ⋯ < λ N N / 2 λ i \sqrt{N/2}\lambda_i N /2 λ i (1) , the vector ( N / 2 λ 1 , … , N / 2 λ N ) (\sqrt{N/2}\lambda_1,\ldots,\sqrt{N/2}\lambda_N) ( N /2 λ 1 , … , N /2 λ N )
1 Z ~ N ∏ 1 ≤ i < j ≤ N ∣ x j − x i ∣ ∏ i = 1 N e − x i 2 / 2 (73) \frac{1}{\widetilde{Z}_N}\prod_{1\le i<j\le N}|x_j-x_i|\prod_{i=1}^{N}e^{-x_i^2/2} \tag*{(73)} Z N 1 1 ≤ i < j ≤ N ∏ ∣ x j − x i ∣ i = 1 ∏ N e − x i 2 /2 (73) on { x 1 < ⋯ < x N } \{x_1<\cdots<x_N\} { x 1 < ⋯ < x N } Z ~ N \widetilde{Z}_N Z N
x = N 2 λ (74) x=\sqrt{\frac{N}{2}}\lambda \tag*{(74)} x = 2 N λ (74) the Hermite coordinate , and the variable λ \lambda λ eigenvalue coordinate . In Hermite coordinates, the semicircle law is supported on [ − 2 N , 2 N ] [-\sqrt{2N},\sqrt{2N}] [ − 2 N , 2 N ] x 0 x_0 x 0 π − 1 2 N − x 0 2 \pi^{-1}\sqrt{2N-x_0^2} π − 1 2 N − x 0 2
Definition 2.2. For a measurable set J ⊂ R J\subset\mathbb{R} J ⊂ R K J K_J K J L 2 ( J ) L^2(J) L 2 ( J ) K N ( x , y ) K_N(x,y) K N ( x , y ) x , y ∈ J x,y\in J x , y ∈ J
P N ( J ) = P ( N 2 λ i ∉ J for all i ∈ { 1 , … , N } ) , D N ( J ) = det ( Id − K J ) . (75) P_N(J)=\mathbb{P}\left(\sqrt{\frac{N}{2}}\lambda_i\notin J\text{ for all }i\in\{1,\ldots,N\}\right),\qquad D_N(J)=\det(\operatorname{Id}-K_J). \tag*{(75)} P N ( J ) = P ( 2 N λ i ∈ / J for all i ∈ { 1 , … , N } ) , D N ( J ) = det ( Id − K J ) . (75) We call P N ( J ) P_N(J) P N ( J ) GOE hole probability of J J J
The operator K J K_J K J ∫ J K N ( x , x ) d x < ∞ \int_J K_N(x,x)\,\mathrm{d}x<\infty ∫ J K N ( x , x ) d x < ∞ x ↦ K N ( x , x ) x\mapsto K_N(x,x) x ↦ K N ( x , x ) V N V_N V N ∥ K J ∥ < 1 \lVert K_J\rVert<1 ∥ K J ∥ < 1 D N ( J ) D_N(J) D N ( J ) ( 0 , 1 ] (0,1] ( 0 , 1 ] ∏ i < j ∣ x j − x i ∣ 2 ∏ i e − x i 2 \prod_{i<j}|x_j-x_i|^2\prod_i e^{-x_i^2} ∏ i < j ∣ x j − x i ∣ 2 ∏ i e − x i 2 J J J [2 ] ), but we will not use this interpretation. Expanding K N K_N K N ( h n ) n < N (h_n)_{n<N} ( h n ) n < N
D N ( J ) = det ( 1 m = n − ⟨ h m , 1 J h n ⟩ ) m , n = 0 N − 1 . (76) D_N(J)=\det\left(\mathbf{1}_{m=n}-\langle h_m,\mathbf{1}_J h_n\rangle\right)_{m,n=0}^{N-1}. \tag*{(76)} D N ( J ) = det ( 1 m = n − ⟨ h m , 1 J h n ⟩ ) m , n = 0 N − 1 . (76) Finally, for every measurable set A ⊂ R A\subset\mathbb{R} A ⊂ R E 1 , N ( A ) = P N ( N / 2 A ) E_{1,N}(A)=P_N(\sqrt{N/2}A) E 1 , N ( A ) = P N ( N /2 A ) E 1 , N ( A ) E_{1,N}(A) E 1 , N ( A ) (11) .
Top modes and exact identities
We now define the functions that measure how the GOE differs from the GUE. For every integer N ≥ 2 N\ge2 N ≥ 2 top modes
φ N = ( N 2 ) 1 / 4 h N , ψ N = ( N 2 ) 1 / 4 h N − 1 . (77) \varphi_N=\left(\frac{N}{2}\right)^{1/4}h_N,\qquad\psi_N=\left(\frac{N}{2}\right)^{1/4}h_{N-1}. \tag*{(77)} φ N = ( 2 N ) 1/4 h N , ψ N = ( 2 N ) 1/4 h N − 1 . (77) With this normalization, φ N ( x ) ψ N ( y ) = N / 2 h N ( x ) h N − 1 ( y ) \varphi_N(x)\psi_N(y)=\sqrt{N/2}\,h_N(x)h_{N-1}(y) φ N ( x ) ψ N ( y ) = N /2 h N ( x ) h N − 1 ( y ) c N c_N c N ω N \omega_N ω N
c N = { ( ∫ R ψ N ) − 1 if N is odd , 0 if N is even , ω N = ε φ N + c N , (78) c_N=\begin{cases}
\left(\int_{\mathbb{R}}\psi_N\right)^{-1} & \text{if }N\text{ is odd},\\
0 & \text{if }N\text{ is even},
\end{cases}
\qquad\omega_N=\varepsilon\varphi_N+c_N, \tag*{(78)} c N = { ( ∫ R ψ N ) − 1 0 if N is odd , if N is even , ω N = ε φ N + c N , (78) and the function
p N ( y ) = ∫ R K N ( x , y ) d x = ( K N 1 R ) ( y ) . (79) p_N(y)=\int_{\mathbb{R}}K_N(x,y)\,\mathrm{d}x=(K_N\mathbf{1}_{\mathbb{R}})(y). \tag*{(79)} p N ( y ) = ∫ R K N ( x , y ) d x = ( K N 1 R ) ( y ) . (79) For odd N N N N − 1 N-1 N − 1 ∫ ψ N ≠ 0 \int\psi_N\ne0 ∫ ψ N = 0 c N c_N c N p N ∈ V N p_N\in V_N p N ∈ V N
Lemma 2.3. For all N ≥ 2 N\ge2 N ≥ 2 x , y ∈ R x,y\in\mathbb{R} x , y ∈ R
∂ x K N ( x , y ) + ∂ y K N ( x , y ) = − φ N ( x ) ψ N ( y ) − ψ N ( x ) φ N ( y ) . (80) \partial_xK_N(x,y)+\partial_yK_N(x,y)=-\varphi_N(x)\psi_N(y)-\psi_N(x)\varphi_N(y). \tag*{(80)} ∂ x K N ( x , y ) + ∂ y K N ( x , y ) = − φ N ( x ) ψ N ( y ) − ψ N ( x ) φ N ( y ) . (80) Proof. By the second identity in (2.7), we have
∑ n = 0 N − 1 h n ′ ( x ) h n ( y ) = ∑ n = 1 N − 1 n 2 h n − 1 ( x ) h n ( y ) − ∑ n = 1 N n 2 h n ( x ) h n − 1 ( y ) , (81) \sum_{n=0}^{N-1}h_n'(x)h_n(y)=\sum_{n=1}^{N-1}\sqrt{\frac{n}{2}}\,h_{n-1}(x)h_n(y)-\sum_{n=1}^{N}\sqrt{\frac{n}{2}}\,h_n(x)h_{n-1}(y), \tag*{(81)} n = 0 ∑ N − 1 h n ′ ( x ) h n ( y ) = n = 1 ∑ N − 1 2 n h n − 1 ( x ) h n ( y ) − n = 1 ∑ N 2 n h n ( x ) h n − 1 ( y ) , (81) where we shifted the index in the second sum. We add the same identity with x x x y y y n ≤ N − 1 n\le N-1 n ≤ N − 1 n = N n=N n = N
− N / 2 ( h N ( x ) h N − 1 ( y ) + h N − 1 ( x ) h N ( y ) ) , (82) -\sqrt{N/2}\left(h_N(x)h_{N-1}(y)+h_{N-1}(x)h_N(y)\right), \tag*{(82)} − N /2 ( h N ( x ) h N − 1 ( y ) + h N − 1 ( x ) h N ( y ) ) , (82) which is the right side of (2.27).
Lemma 2.4. For all N ≥ 2 N\ge2 N ≥ 2
p N ′ = − ( ∫ R φ N ) ψ N − ( ∫ R ψ N ) φ N (83) p_N'=-\left(\int_{\mathbb{R}}\varphi_N\right)\psi_N-\left(\int_{\mathbb{R}}\psi_N\right)\varphi_N \tag*{(83)} p N ′ = − ( ∫ R φ N ) ψ N − ( ∫ R ψ N ) φ N (83) and
p N ( 0 ) = 2 ∑ 0 ≤ m ≤ ( N − 1 ) / 2 ( − 1 ) m ( 2 m m ) 4 − m , ∣ p N ( 0 ) − 1 ∣ ≤ 2 N − 1 / 2 . (84) p_N(0)=\sqrt{2}\sum_{0\le m\le(N-1)/2}(-1)^m\binom{2m}{m}4^{-m},\qquad|p_N(0)-1|\le2N^{-1/2}. \tag*{(84)} p N ( 0 ) = 2 0 ≤ m ≤ ( N − 1 ) /2 ∑ ( − 1 ) m ( m 2 m ) 4 − m , ∣ p N ( 0 ) − 1∣ ≤ 2 N − 1/2 . (84) Moreover, if N N N ∫ ψ N = 0 \int\psi_N=0 ∫ ψ N = 0 1 ≤ ∫ φ N ≤ 2 1\le\int\varphi_N\le2 1 ≤ ∫ φ N ≤ 2 N N N ∫ φ N = 0 \int\varphi_N=0 ∫ φ N = 0 1 ≤ ∫ ψ N ≤ 2 1\le\int\psi_N\le2 1 ≤ ∫ ψ N ≤ 2 c N ∈ [ 1 / 2 , 1 ] c_N\in[1/2,1] c N ∈ [ 1/2 , 1 ] N N N
Proof. Integrating (2.27) in x x x R \mathbb{R} R
∫ R ∂ x K N ( x , y ) d x = 0 , (85) \int_{\mathbb{R}}\partial_xK_N(x,y)\,\mathrm{d}x=0, \tag*{(85)} ∫ R ∂ x K N ( x , y ) d x = 0 , (85) we obtain (83) .
Next, since h n ( 0 ) = 0 h_n(0)=0 h n ( 0 ) = 0 n n n
p N ( 0 ) = ∑ 2 m < N h 2 m ( 0 ) ∫ h 2 m . (86) p_N(0)=\sum_{2m<N}h_{2m}(0)\int h_{2m}. \tag*{(86)} p N ( 0 ) = 2 m < N ∑ h 2 m ( 0 ) ∫ h 2 m . (86) Inserting (63) and (64) into this identity, we obtain the formula in (84) . We now turn to the bound. Set
ω m = ( 2 m m ) 4 − m . (87) \omega_m=\binom{2m}{m}4^{-m}. \tag*{(87)} ω m = ( m 2 m ) 4 − m . (87) An elementary induction using ω m + 1 = ω m ( 2 m + 1 ) / ( 2 m + 2 ) \omega_{m+1}=\omega_m(2m+1)/(2m+2) ω m + 1 = ω m ( 2 m + 1 ) / ( 2 m + 2 )
1 2 m ≤ ω m ≤ 1 2 m + 1 , m ≥ 1. (88) \frac{1}{2\sqrt{m}}\leq\omega_m\leq\frac{1}{\sqrt{2m+1}},\qquad m\geq1. \tag*{(88)} 2 m 1 ≤ ω m ≤ 2 m + 1 1 , m ≥ 1. (88) In particular, ( ω m ) m ≥ 0 (\omega_m)_{m\geq0} ( ω m ) m ≥ 0 0 0 0 ∑ m ≥ 0 ( − 1 ) m ω m \sum_{m\geq0}(-1)^m\omega_m ∑ m ≥ 0 ( − 1 ) m ω m
( 1 − 4 z ) − 1 / 2 = ∑ m ≥ 0 ω m ( 4 z ) m (89) (1-4z)^{-1/2}=\sum_{m\geq0}\omega_m(4z)^m \tag*{(89)} ( 1 − 4 z ) − 1/2 = m ≥ 0 ∑ ω m ( 4 z ) m (89) at z = − 1 / 4 z=-1/4 z = − 1/4 2 − 1 / 2 2^{-1/2} 2 − 1/2 ∣ p N ( 0 ) − 1 ∣ ≤ 2 ω M + 1 |p_N(0)-1|\leq\sqrt{2}\,\omega_{M+1} ∣ p N ( 0 ) − 1∣ ≤ 2 ω M + 1 M = ⌊ ( N − 1 ) / 2 ⌋ M=\lfloor(N-1)/2\rfloor M = ⌊( N − 1 ) /2 ⌋ (88) implies that
2 ω M + 1 ≤ 2 ( 2 M + 3 ) − 1 / 2 ≤ 2 N − 1 / 2 . (90) \sqrt{2}\,\omega_{M+1}\leq\sqrt{2}(2M+3)^{-1/2}\leq2N^{-1/2}. \tag*{(90)} 2 ω M + 1 ≤ 2 ( 2 M + 3 ) − 1/2 ≤ 2 N − 1/2 . (90) Finally, we have
∫ h n = 0 (91) \int h_n=0 \tag*{(91)} ∫ h n = 0 (91) for odd n n n N = 2 m N=2m N = 2 m (64) gives
∫ φ N = m 1 / 4 2 π 1 / 4 ω m 1 / 2 , (92) \int\varphi_N=m^{1/4}\sqrt{2}\,\pi^{1/4}\omega_m^{1/2}, \tag*{(92)} ∫ φ N = m 1/4 2 π 1/4 ω m 1/2 , (92) and (88) shows that this number belongs to [ π 1 / 4 , 2 1 / 4 π 1 / 4 ] ⊂ [ 1 , 2 ] [\pi^{1/4},2^{1/4}\pi^{1/4}]\subset[1,2] [ π 1/4 , 2 1/4 π 1/4 ] ⊂ [ 1 , 2 ] ( N / 2 ) 1 / 4 = ( m + 1 / 2 ) 1 / 4 (N/2)^{1/4}=(m+1/2)^{1/4} ( N /2 ) 1/4 = ( m + 1/2 ) 1/4 ∫ ψ N ∈ [ 1 , 2 ] \int\psi_N\in[1,2] ∫ ψ N ∈ [ 1 , 2 ] N = 2 m + 1 N=2m+1 N = 2 m + 1
The next lemma records which antiderivatives of Hermite functions lie in V N V_N V N
Lemma 2.5. Let n ≥ 0 n\geq0 n ≥ 0 V n V_n V n { f ∈ V n + 1 : ∫ R f = 0 } \{f\in V_{n+1}:\int_{\mathbb{R}}f=0\} { f ∈ V n + 1 : ∫ R f = 0 }
(a) If n n n ε h n ∈ V n \varepsilon h_n\in V_n ε h n ∈ V n
(b) If N N N K N ( ε h N ) = 0 K_N(\varepsilon h_N)=0 K N ( ε h N ) = 0
(c) If N N N ε h N ∈ V N \varepsilon h_N \in V_N ε h N ∈ V N K N ( ε h N − 1 ) = 0 K_N(\varepsilon h_{N-1}) = 0 K N ( ε h N − 1 ) = 0
Proof. By (60) , the derivative maps V n V_n V n V n + 1 V_{n+1} V n + 1 V n V_n V n n n n { f ∈ V n + 1 : ∫ f = 0 } \{f \in V_{n+1} : \int f = 0\} { f ∈ V n + 1 : ∫ f = 0 } n n n ∫ h 0 ≠ 0 \int h_0 \ne0 ∫ h 0 = 0
To prove (a), suppose that n n n h n ∈ V n + 1 h_n \in V_{n+1} h n ∈ V n + 1 ∫ h n = 0 \int h_n = 0 ∫ h n = 0 h n = g ′ h_n = g' h n = g ′ g ∈ V n g \in V_n g ∈ V n g g g ε h n \varepsilon h_n ε h n h n h_n h n
We now prove (b) and (c), using the identity ⟨ h m , ε f ⟩ = − ⟨ ε h m , f ⟩ \langle h_m,\varepsilon f\rangle= -\langle\varepsilon h_m,f\rangle ⟨ h m , ε f ⟩ = − ⟨ ε h m , f ⟩ m < N m < N m < N m m m ε h m ∈ V m \varepsilon h_m \in V_m ε h m ∈ V m ε h m \varepsilon h_m ε h m h N h_N h N h N − 1 h_{N-1} h N − 1 m ≤ N − 2 m \le N - 2 m ≤ N − 2 m m m ε h m \varepsilon h_m ε h m
Suppose first that N N N m < N m < N m < N m ≤ N − 1 m \le N - 1 m ≤ N − 1 ⟨ ε h m , h N ⟩ = 0 \langle\varepsilon h_m,h_N\rangle= 0 ⟨ ε h m , h N ⟩ = 0 m m m ε h m h N \varepsilon h_m h_N ε h m h N ⟨ ε h m , h N ⟩ = 0 \langle\varepsilon h_m,h_N\rangle= 0 ⟨ ε h m , h N ⟩ = 0 K N ( ε h N ) = 0 K_N(\varepsilon h_N) = 0 K N ( ε h N ) = 0
Finally, suppose that N N N ε h N ∈ V N \varepsilon h_N \in V_N ε h N ∈ V N h N − 1 h_{N-1} h N − 1 ⟨ ε h m , h N − 1 ⟩ = 0 \langle\varepsilon h_m,h_{N-1}\rangle= 0 ⟨ ε h m , h N − 1 ⟩ = 0 m < N m < N m < N
The following lemma contains the cancellation between the two terms in ω N \omega_N ω N N N N (78) .
Lemma 2.6. Let N ≥ 2 N \ge2 N ≥ 2
(a) If N N N ε ψ N = − p N / ∫ φ N \varepsilon\psi_N = -p_N/\int\varphi_N ε ψ N = − p N / ∫ φ N ω N = ε φ N \omega_N = \varepsilon\varphi_N ω N = ε φ N
(b) If N N N ε φ N = − c N p N \varepsilon\varphi_N = -c_Np_N ε φ N = − c N p N ω N = c N ( 1 − p N ) \omega_N = c_N(1-p_N) ω N = c N ( 1 − p N )
(c) In both cases, K N ω N = 0 K_N\omega_N = 0 K N ω N = 0 ⟨ ψ N , ω N ⟩ = 0 \langle\psi_N,\omega_N\rangle= 0 ⟨ ψ N , ω N ⟩ = 0 ⟨ ψ N , ε φ N ⟩ = − c N ∫ ψ N \langle\psi_N,\varepsilon\varphi_N\rangle= -c_N\int\psi_N ⟨ ψ N , ε φ N ⟩ = − c N ∫ ψ N
Proof. Suppose first that N N N
p N ′ = − ( ∫ φ N ) ψ N (93) p'_N = -\left(\int\varphi_N\right)\psi_N \tag*{(93)} p N ′ = − ( ∫ φ N ) ψ N (93) so the derivative of ε ψ N + p N / ∫ φ N \varepsilon\psi_N + p_N/\int\varphi_N ε ψ N + p N / ∫ φ N y → ∞ y \to\infty y → ∞ p N ( y ) → 0 p_N(y) \to0 p N ( y ) → 0 p N ∈ V N p_N \in V_N p N ∈ V N (58) gives
( ε ψ N ) ( y ) → ( 1 / 2 ) ∫ ψ N = 0. (94) (\varepsilon\psi_N)(y) \to(1/2)\int\psi_N = 0. \tag*{(94)} ( ε ψ N ) ( y ) → ( 1/2 ) ∫ ψ N = 0. (94) Therefore,
ε ψ N + p N / ∫ φ N = 0 , (95) \varepsilon\psi_N + p_N/\int\varphi_N = 0, \tag*{(95)} ε ψ N + p N / ∫ φ N = 0 , (95) and (a) follows, since c N = 0 c_N = 0 c N = 0
Next, suppose that N N N p N ′ = − c N − 1 φ N p'_N = -c_N^{-1}\varphi_N p N ′ = − c N − 1 φ N ε φ N + c N p N \varepsilon\varphi_N + c_Np_N ε φ N + c N p N y → ∞ y \to\infty y → ∞ p N ( y ) → 0 p_N(y) \to0 p N ( y ) → 0
( ε φ N ) ( y ) → ( 1 / 2 ) ∫ φ N = 0 , (96) (\varepsilon\varphi_N)(y) \to(1/2)\int\varphi_N = 0, \tag*{(96)} ( ε φ N ) ( y ) → ( 1/2 ) ∫ φ N = 0 , (96) which gives (b). We now prove (c). When N N N
K N ω N = K N ε φ N = 0. (97) K_N\omega_N=K_N\varepsilon\varphi_N=0. \tag*{(97)} K N ω N = K N ε φ N = 0. (97) For odd N N N
p N = K N 1 R ∈ V N , (98) p_N=K_N\mathbf{1}_{\mathbb{R}}\in V_N, \tag*{(98)} p N = K N 1 R ∈ V N , (98) so K N p N = p N K_Np_N=p_N K N p N = p N
K N ω N = c N ( p N − K N p N ) = 0. (99) K_N\omega_N=c_N(p_N-K_Np_N)=0. \tag*{(99)} K N ω N = c N ( p N − K N p N ) = 0. (99) Since ψ N ∈ V N \psi_N\in V_N ψ N ∈ V N ω N \omega_N ω N
⟨ ψ N , ω N ⟩ = ⟨ K N ψ N , ω N ⟩ = ⟨ ψ N , K N ω N ⟩ = 0. (100) \langle\psi_N,\omega_N\rangle=\langle K_N\psi_N,\omega_N\rangle=\langle\psi_N,K_N\omega_N\rangle=0. \tag*{(100)} ⟨ ψ N , ω N ⟩ = ⟨ K N ψ N , ω N ⟩ = ⟨ ψ N , K N ω N ⟩ = 0. (100) Finally, we have
⟨ ψ N , ε φ N ⟩ = ⟨ ψ N , ω N ⟩ − c N ∫ ψ N = − c N ∫ ψ N . (101) \langle\psi_N,\varepsilon\varphi_N\rangle=\langle\psi_N,\omega_N\rangle-c_N\int\psi_N=-c_N\int\psi_N. \tag*{(101)} ⟨ ψ N , ε φ N ⟩ = ⟨ ψ N , ω N ⟩ − c N ∫ ψ N = − c N ∫ ψ N . (101) Observe that for odd N N N ε φ N \varepsilon\varphi_N ε φ N − c N -c_N − c N p N p_N p N ω N \omega_N ω N
The last identity of this section concerns the sign function. It shows that, up to the product ( ε φ N ) ( ε ψ N ) (\varepsilon\varphi_N)(\varepsilon\psi_N) ( ε φ N ) ( ε ψ N ) K N K_N K N
Lemma 2.7. For all N ≥ 2 N\ge2 N ≥ 2 y ∈ R y\in\mathbb{R} y ∈ R
1 2 ∫ R sgn ( x − y ) K N ( x , y ) d x = ( ε φ N ) ( y ) ( ε ψ N ) ( y ) . (102) \frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-y)K_N(x,y)\,\mathrm{d}x=(\varepsilon\varphi_N)(y)(\varepsilon\psi_N)(y). \tag*{(102)} 2 1 ∫ R sgn ( x − y ) K N ( x , y ) d x = ( ε φ N ) ( y ) ( ε ψ N ) ( y ) . (102) Consequently, for all t , y ∈ R t,y\in\mathbb{R} t , y ∈ R
( K N ε t ) ( y ) = ( ε φ N ) ( y ) ( ε ψ N ) ( y ) + ∫ t y K N ( x , y ) d x , (103) (K_N\varepsilon_t)(y)=(\varepsilon\varphi_N)(y)(\varepsilon\psi_N)(y)+\int_t^y K_N(x,y)\,\mathrm{d}x, \tag*{(103)} ( K N ε t ) ( y ) = ( ε φ N ) ( y ) ( ε ψ N ) ( y ) + ∫ t y K N ( x , y ) d x , (103) where the integral is oriented, so that ∫ t y = − ∫ y t \int_t^y=-\int_y^t ∫ t y = − ∫ y t y < t y<t y < t
Proof. Let G ( y ) G(y) G ( y ) (102) . Then
2 G ( y ) = ∫ y ∞ K N ( x , y ) d x − ∫ − ∞ y K N ( x , y ) d x . (104) 2G(y)=\int_y^\infty K_N(x,y)\,\mathrm{d}x-\int_{-\infty}^y K_N(x,y)\,\mathrm{d}x. \tag*{(104)} 2 G ( y ) = ∫ y ∞ K N ( x , y ) d x − ∫ − ∞ y K N ( x , y ) d x . (104) Differentiating this identity, we have
G ′ ( y ) = − K N ( y , y ) + 1 2 ∫ R sgn ( x − y ) ∂ y K N ( x , y ) d x . (105) G'(y)=-K_N(y,y)+\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-y)\,\partial_y K_N(x,y)\,\mathrm{d}x. \tag*{(105)} G ′ ( y ) = − K N ( y , y ) + 2 1 ∫ R sgn ( x − y ) ∂ y K N ( x , y ) d x . (105) We now insert (80) into the integral. Integrating over ( y , ∞ ) (y,\infty) ( y , ∞ ) ( − ∞ , y ) (-\infty,y) ( − ∞ , y ) − ∂ x K N ( x , y ) -\partial_x K_N(x,y) − ∂ x K N ( x , y ) K N ( y , y ) K_N(y,y) K N ( y , y )
− 1 2 ∫ R sgn ( x − y ) ( φ N ( x ) ψ N ( y ) + ψ N ( x ) φ N ( y ) ) d x = ψ N ( y ) ( ε φ N ) ( y ) + φ N ( y ) ( ε ψ N ) ( y ) , (106) -\frac{1}{2}\int_{\mathbb{R}}\operatorname{sgn}(x-y)\left(\varphi_N(x)\psi_N(y)+\psi_N(x)\varphi_N(y)\right)\,\mathrm{d}x=\psi_N(y)(\varepsilon\varphi_N)(y)+\varphi_N(y)(\varepsilon\psi_N)(y), \tag*{(106)} − 2 1 ∫ R sgn ( x − y ) ( φ N ( x ) ψ N ( y ) + ψ N ( x ) φ N ( y ) ) d x = ψ N ( y ) ( ε φ N ) ( y ) + φ N ( y ) ( ε ψ N ) ( y ) , (106) where we used
∫ sgn ( x − y ) f ( x ) d x = − 2 ( ε f ) ( y ) . (107) \int\operatorname{sgn}(x-y)f(x)\,\mathrm{d}x=-2(\varepsilon f)(y). \tag*{(107)} ∫ sgn ( x − y ) f ( x ) d x = − 2 ( ε f ) ( y ) . (107) Therefore, G ′ ( y ) = ( ( ε φ N ) ( ε ψ N ) ) ′ ( y ) G'(y)=\left((\varepsilon\varphi_N)(\varepsilon\psi_N)\right)'(y) G ′ ( y ) = ( ( ε φ N ) ( ε ψ N ) ) ′ ( y ) y → ∞ y\to\infty y → ∞ G ( y ) → 0 G(y)\to0 G ( y ) → 0
( ε φ N ) ( y ) ( ε ψ N ) ( y ) → ( 1 / 4 ) ∫ φ N ∫ ψ N = 0 , (108) (\varepsilon\varphi_N)(y)(\varepsilon\psi_N)(y)\to(1/4)\int\varphi_N\int\psi_N=0, \tag*{(108)} ( ε φ N ) ( y ) ( ε ψ N ) ( y ) → ( 1/4 ) ∫ φ N ∫ ψ N = 0 , (108) since one of the two integrals vanishes by Lemma 2.4 . This completes the proof of (102) .
We now prove (103) , whose two sides are continuously differentiable in t t t − K N ( t , y ) -K_N(t,y) − K N ( t , y ) K N K_N K N t = y t=y t = y G ( y ) G(y) G ( y ) ( ε φ N ) ( y ) ( ε ψ N ) ( y ) (\varepsilon\varphi_N)(y)(\varepsilon\psi_N)(y) ( ε φ N ) ( y ) ( ε ψ N ) ( y ) (103) follows from (102) .
An exact formula for the hole probability
In this section, we express P N ( J ) 2 \mathcal{P}_{N}(J)^2 P N ( J ) 2 D N ( J ) \mathcal{D}_{N}(J) D N ( J ) J J J N ≥ 2 N \ge2 N ≥ 2 J J J N N N [48 (33)] , who manipulated Fredholm determinants of operators with matrix kernels. They gave a detailed justification of these manipulations for the sine kernel in [48 Section VIII] . Our derivation is different. It starts from de Bruijn’s integration formula and uses only finite-dimensional linear algebra, so no regularization of the sign kernel is required. It treats N N N
Statement
Throughout this section, we fix N ≥ 2 N \ge2 N ≥ 2 r ≥ 1 r \ge1 r ≥ 1 a 1 < b 1 < a 2 < b 2 < ⋯ < a r < b r a_1 < b_1 < a_2 < b_2 < \cdots< a_r < b_r a 1 < b 1 < a 2 < b 2 < ⋯ < a r < b r
J j = ( a j , b j ) , J = ⋃ j = 1 r J j . (109) J_j = (a_j,b_j), \qquad J = \bigcup_{j=1}^{r} J_j. \tag*{(109)} J j = ( a j , b j ) , J = j = 1 ⋃ r J j . (109) For 1 ≤ j ≤ r 1 \le j \le r 1 ≤ j ≤ r
k j = K N ( ⋅ , b j ) − K N ( ⋅ , a j ) , w j = 1 J c ε a j . (110) k_j = K_N(\cdot,b_j) - K_N(\cdot,a_j), \qquad w_j = \mathbf{1}_{J^c}\varepsilon_{a_j}. \tag*{(110)} k j = K N ( ⋅ , b j ) − K N ( ⋅ , a j ) , w j = 1 J c ε a j . (110) Observe that k j ∈ V N k_j \in V_N k j ∈ V N ⟨ k j , f ⟩ = f ( b j ) − f ( a j ) \langle k_j,f\rangle= f(b_j) - f(a_j) ⟨ k j , f ⟩ = f ( b j ) − f ( a j ) f ∈ V N f \in V_N f ∈ V N sgn ( x − a j ) = sgn ( x − b j ) \operatorname{sgn}(x-a_j) = \operatorname{sgn}(x-b_j) sgn ( x − a j ) = sgn ( x − b j ) x ∉ [ a j , b j ] x \notin[a_j,b_j] x ∈ / [ a j , b j ] w j = 1 J c ε b j w_j = \mathbf{1}_{J^c}\varepsilon_{b_j} w j = 1 J c ε b j w j w_j w j J J J − 1 / 2 -1/2 − 1/2 J c ∩ ( − ∞ , a j ) J^c \cap(-\infty,a_j) J c ∩ ( − ∞ , a j ) 1 / 2 1/2 1/2 J c ∩ ( b j , ∞ ) J^c \cap(b_j,\infty) J c ∩ ( b j , ∞ )
Y N ( J ) = 1 J c ( ω N − ε ( 1 J φ N ) ) , (111) Y_N(J) = \mathbf{1}_{J^c}\bigl(\omega_N - \varepsilon(\mathbf{1}_J\varphi_N)\bigr), \tag*{(111)} Y N ( J ) = 1 J c ( ω N − ε ( 1 J φ N ) ) , (111) where we recall that ω N \omega_N ω N T N ( J ) T_N(J) T N ( J ) V N V_N V N
T N ( J ) = Id − K N 1 J − ∑ j = 1 r k j ⊗ w j + ψ N ⊗ Y N ( J ) , (112) T_N(J) = \operatorname{Id} - K_N\mathbf{1}_J - \sum_{j=1}^{r} k_j \otimes w_j + \psi_N \otimes Y_N(J), \tag*{(112)} T N ( J ) = Id − K N 1 J − j = 1 ∑ r k j ⊗ w j + ψ N ⊗ Y N ( J ) , (112) where K N 1 J K_N1_J K N 1 J f ↦ K N ( 1 J f ) f \mapsto K_N(1_Jf) f ↦ K N ( 1 J f ) (55) . Finally, we define
X N ♯ ( J ) = − 1 2 ∑ j = 1 r k j − 1 2 ( ∫ J φ N ) ψ N , d N ( J ) = 1 − c N ∫ J ψ N , X_N^\sharp(J)=-\frac{1}{2}\sum_{j=1}^{r}k_j-\frac{1}{2}\left(\int_J\varphi_N\right)\psi_N,\qquad d_N(J)=1-c_N\int_J\psi_N, X N ♯ ( J ) = − 2 1 j = 1 ∑ r k j − 2 1 ( ∫ J φ N ) ψ N , d N ( J ) = 1 − c N ∫ J ψ N , Y N ♯ ( J ) = − 2 c N ( 1 J ε ψ N + 1 J c ε ( 1 J ψ N ) ) . (113) Y_N^\sharp(J)=-2c_N\left(1_J\varepsilon\psi_N+1_{J^c}\varepsilon(1_J\psi_N)\right). \tag*{(113)} Y N ♯ ( J ) = − 2 c N ( 1 J ε ψ N + 1 J c ε ( 1 J ψ N ) ) . (113) Note that X N ♯ ( J ) ∈ V N X_N^\sharp(J)\in V_N X N ♯ ( J ) ∈ V N Y N ♯ ( J ) Y_N^\sharp(J) Y N ♯ ( J ) N N N c N = 0 c_N=0 c N = 0 Y N ♯ ( J ) = 0 Y_N^\sharp(J)=0 Y N ♯ ( J ) = 0 d N ( J ) = 1 d_N(J)=1 d N ( J ) = 1
Proposition 3.1. Let T N ( J ) T_N(J) T N ( J ) V N ⊕ R V_N\oplus\mathbb{R} V N ⊕ R
T N ( J ) ( f , a ) = ( T N ( J ) f + a X N ♯ ( J ) , ⟨ Y N ♯ ( J ) , f ⟩ + a d N ( J ) ) . (114) T_N(J)(f,a)=\left(T_N(J)f+aX_N^\sharp(J),\langle Y_N^\sharp(J),f\rangle+ad_N(J)\right). \tag*{(114)} T N ( J ) ( f , a ) = ( T N ( J ) f + a X N ♯ ( J ) , ⟨ Y N ♯ ( J ) , f ⟩ + a d N ( J ) ) . (114) Then
P N ( J ) 2 = det T N ( J ) . (115) \mathcal{P}_N(J)^2=\det T_N(J). \tag*{(115)} P N ( J ) 2 = det T N ( J ) . (115) In particular, if N N N P N ( J ) 2 = det T N ( J ) \mathcal{P}_N(J)^2=\det T_N(J) P N ( J ) 2 = det T N ( J )
To state the consequence of Proposition 3.1 that we will use, we write
Q N ( J ) = Id − K N 1 J , (116) Q_N(J)=\operatorname{Id}-K_N1_J, \tag*{(116)} Q N ( J ) = Id − K N 1 J , (116) viewed as an operator on V N V_N V N ( h n ) n < N (h_n)_{n<N} ( h n ) n < N (76) , so det Q N ( J ) = D N ( J ) > 0 \det Q_N(J)=D_N(J)>0 det Q N ( J ) = D N ( J ) > 0 Q N ( J ) Q_N(J) Q N ( J ) X 0 , … , X r ∈ V N X_0,\ldots,X_r\in V_N X 0 , … , X r ∈ V N Y 0 , … , Y r Y_0,\ldots,Y_r Y 0 , … , Y r
X 0 = − ψ N , X j = k j , Y 0 = Y N ( J ) , Y j = w j , 1 ≤ j ≤ r , (117) X_0=-\psi_N,\qquad X_j=k_j,\qquad Y_0=Y_N(J),\qquad Y_j=w_j,\qquad1\leq j\leq r, \tag*{(117)} X 0 = − ψ N , X j = k j , Y 0 = Y N ( J ) , Y j = w j , 1 ≤ j ≤ r , (117) so that T N ( J ) = Q N ( J ) − ∑ α = 0 r X α ⊗ Y α T_N(J)=Q_N(J)-\sum_{\alpha=0}^{r}X_\alpha\otimes Y_\alpha T N ( J ) = Q N ( J ) − ∑ α = 0 r X α ⊗ Y α { 0 , 1 , … , r } \{0,1,\ldots,r\} { 0 , 1 , … , r } { 0 , 1 , … , r + 1 } \{0,1,\ldots,r+1\} { 0 , 1 , … , r + 1 }
Definition 3.2. We define the ( r + 1 ) × ( r + 1 ) (r+1)\times(r+1) ( r + 1 ) × ( r + 1 ) M N ( J ) M_N(J) M N ( J ) ( r + 2 ) × ( r + 2 ) (r+2)\times(r+2) ( r + 2 ) × ( r + 2 ) M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J )
M N ( J ) α β = 1 α = β − ⟨ Y α , Q N ( J ) − 1 X β ⟩ , 0 ≤ α , β ≤ r , (118) M_N(J)_{\alpha\beta}=1_{\alpha=\beta}-\left\langle Y_\alpha,Q_N(J)^{-1}X_\beta\right\rangle,\qquad0\leq\alpha,\beta\leq r, \tag*{(118)} M N ( J ) α β = 1 α = β − ⟨ Y α , Q N ( J ) − 1 X β ⟩ , 0 ≤ α , β ≤ r , (118) and
M N ♯ ( J ) = ( M N ( J ) v N ( J ) z N ( J ) T d N ( J ) ) , (119) M_N^\sharp(J)=
\begin{pmatrix}
M_N(J) & v_N(J)\\
z_N(J)^{\mathsf T} & d_N(J)
\end{pmatrix}, \tag*{(119)} M N ♯ ( J ) = ( M N ( J ) z N ( J ) T v N ( J ) d N ( J ) ) , (119) where v N ( J ) , z N ( J ) ∈ R r + 1 v_N(J),z_N(J)\in\mathbb{R}^{r+1} v N ( J ) , z N ( J ) ∈ R r + 1
v N ( J ) 0 = 1 2 ∫ J φ N , v N ( J ) j = − 1 2 , z N ( J ) β = ⟨ Y N ♯ ( J ) , Q N ( J ) − 1 X β ⟩ , (120) v_N(J)_0=\frac{1}{2}\int_J\varphi_N,\qquad v_N(J)_j=-\frac{1}{2},\qquad z_N(J)_\beta=\left\langle Y_N^\sharp(J),Q_N(J)^{-1}X_\beta\right\rangle, \tag*{(120)} v N ( J ) 0 = 2 1 ∫ J φ N , v N ( J ) j = − 2 1 , z N ( J ) β = ⟨ Y N ♯ ( J ) , Q N ( J ) − 1 X β ⟩ , (120) for 1 ≤ j ≤ r 1\leq j\leq r 1 ≤ j ≤ r
Corollary 3.3. For all N ≥ 2 N \ge2 N ≥ 2 J J J
P N ( J ) 2 = D N ( J ) det M N ♯ ( J ) . (121) \mathcal{P}_N(J)^2=\mathcal{D}_N(J)\det M_N^{\sharp}(J). \tag*{(121)} P N ( J ) 2 = D N ( J ) det M N ♯ ( J ) . (121) If N N N z N ( J ) = 0 z_N(J)=0 z N ( J ) = 0 d N ( J ) = 1 d_N(J)=1 d N ( J ) = 1 det M N ♯ ( J ) = det M N ( J ) \det M_N^{\sharp}(J)=\det M_N(J) det M N ♯ ( J ) = det M N ( J )
Proof. We write Q = Q N ( J ) Q=Q_N(J) Q = Q N ( J ) X ~ α = Q − 1 X α \widetilde{X}_\alpha=Q^{-1}X_\alpha X α = Q − 1 X α
X N ♯ ( J ) = ∑ α v N ( J ) α X α . (122) X_N^{\sharp}(J)=\sum_\alpha v_N(J)_\alpha X_\alpha. \tag*{(122)} X N ♯ ( J ) = α ∑ v N ( J ) α X α . (122) Therefore, T N ( J ) = ( Q ⊕ 1 ) T ′ T_N(J)=(Q\oplus1)T' T N ( J ) = ( Q ⊕ 1 ) T ′
T ′ ( f , a ) = ( f − ∑ α = 0 r X ~ α ( ⟨ Y α , f ⟩ − a v N ( J ) α ) , a + ⟨ Y N ♯ ( J ) , f ⟩ + a ( d N ( J ) − 1 ) ) . (123) T'(f,a)=\left(f-\sum_{\alpha=0}^{r}\widetilde{X}_\alpha\left(\langle Y_\alpha,f\rangle-a\,v_N(J)_\alpha\right),a+\langle Y_N^{\sharp}(J),f\rangle+a(d_N(J)-1)\right). \tag*{(123)} T ′ ( f , a ) = ( f − α = 0 ∑ r X α ( ⟨ Y α , f ⟩ − a v N ( J ) α ) , a + ⟨ Y N ♯ ( J ) , f ⟩ + a ( d N ( J ) − 1 ) ) . (123) We now write T ′ T' T ′ r + 2 r+2 r + 2 U β = ( X ~ β , 0 ) U_\beta=(\widetilde{X}_\beta,0) U β = ( X β , 0 ) β ≤ r \beta\le r β ≤ r U r + 1 = ( 0 , 1 ) U_{r+1}=(0,1) U r + 1 = ( 0 , 1 )
Φ α ( f , a ) = ⟨ Y α , f ⟩ − a v N ( J ) α (124) \Phi_\alpha(f,a)=\langle Y_\alpha,f\rangle-a\,v_N(J)_\alpha \tag*{(124)} Φ α ( f , a ) = ⟨ Y α , f ⟩ − a v N ( J ) α (124) for α ≤ r \alpha\le r α ≤ r
Φ r + 1 ( f , a ) = − ⟨ Y N ♯ ( J ) , f ⟩ − a ( d N ( J ) − 1 ) , (125) \Phi_{r+1}(f,a)=-\langle Y_N^{\sharp}(J),f\rangle-a(d_N(J)-1), \tag*{(125)} Φ r + 1 ( f , a ) = − ⟨ Y N ♯ ( J ) , f ⟩ − a ( d N ( J ) − 1 ) , (125) and we let U ⊗ Φ U\otimes\Phi U ⊗ Φ u ↦ Φ ( u ) U u\mapsto\Phi(u)U u ↦ Φ ( u ) U
T ′ = Id − ∑ α = 0 r ( X ~ α , 0 ) ⊗ Φ α − ( 0 , 1 ) ⊗ Φ r + 1 , (126) T'=\operatorname{Id}-\sum_{\alpha=0}^{r}(\widetilde{X}_\alpha,0)\otimes\Phi_\alpha-(0,1)\otimes\Phi_{r+1}, \tag*{(126)} T ′ = Id − α = 0 ∑ r ( X α , 0 ) ⊗ Φ α − ( 0 , 1 ) ⊗ Φ r + 1 , (126) so Sylvester’s determinant identity implies that det T ′ \det T' det T ′ ( r + 2 ) × ( r + 2 ) (r+2)\times(r+2) ( r + 2 ) × ( r + 2 ) 1 α = β − Φ α ( U β ) 1_{\alpha=\beta}-\Phi_\alpha(U_\beta) 1 α = β − Φ α ( U β ) M N ♯ ( J ) M_N^{\sharp}(J) M N ♯ ( J ) det ( Q ⊕ 1 ) = D N ( J ) \det(Q\oplus1)=\mathcal{D}_N(J) det ( Q ⊕ 1 ) = D N ( J ) c N = 0 c_N=0 c N = 0 N N N
De Bruijn’s formula
Let A ⊂ R A\subset\mathbb{R} A ⊂ R
Ω A ( f , g ) = ⟨ 1 A f , ε ( 1 A g ) ⟩ , f , g ∈ V N + 1 , (127) \Omega_A(f,g)=\langle1_Af,\varepsilon(1_Ag)\rangle,\qquad f,g\in V_{N+1}, \tag*{(127)} Ω A ( f , g ) = ⟨ 1 A f , ε ( 1 A g )⟩ , f , g ∈ V N + 1 , (127) and, for odd N N N V N ⊕ R V_N\oplus\mathbb{R} V N ⊕ R
Ω ^ A ( ( f , a ) , ( g , b ) ) = Ω A ( f , g ) − b 2 ⟨ 1 A , f ⟩ + a 2 ⟨ 1 A , g ⟩ . (128) \widehat{\Omega}_A((f,a),(g,b))=\Omega_A(f,g)-\frac{b}{2}\langle1_A,f\rangle+\frac{a}{2}\langle1_A,g\rangle. \tag*{(128)} Ω A (( f , a ) , ( g , b )) = Ω A ( f , g ) − 2 b ⟨ 1 A , f ⟩ + 2 a ⟨ 1 A , g ⟩ . (128) With a slight abuse of notation, we also let Ω A \Omega_A Ω A Ω A \Omega_A Ω A ( h n ) n < N (h_n)_{n<N} ( h n ) n < N V N V_N V N Ω ^ A \widehat{\Omega}_A Ω A Ω ^ A \widehat{\Omega}_A Ω A V N ⊕ R V_N\oplus\mathbb{R} V N ⊕ R ( h n , 0 ) (h_n,0) ( h n , 0 ) n < N n<N n < N ( 0 , 1 ) (0,1) ( 0 , 1 )
Lemma 3.4. If N N N det Ω ^ R ≠ 0 \det\widehat{\Omega}_{\mathbb{R}} \ne0 det Ω R = 0 P N ( J ) 2 = det Ω J c / det Ω R \mathcal{P}_N(J)^2 = \det\Omega_{J^c}/\det\Omega_{\mathbb{R}} P N ( J ) 2 = det Ω J c / det Ω R N N N det Ω ^ R ≠ 0 \det\widehat{\Omega}_{\mathbb{R}} \ne0 det Ω R = 0 P N ( J ) 2 = det Ω ^ J c / det Ω ^ R \mathcal{P}_N(J)^2 = \det\widehat{\Omega}_{J^c}/\det\widehat{\Omega}_{\mathbb{R}} P N ( J ) 2 = det Ω J c / det Ω R
Proof. Since h n h_n h n n n n e − x 2 / 2 e^{-x^2/2} e − x 2 /2 ς N ≠ 0 \varsigma_N \ne0 ς N = 0
det ( h m ( x i ) ) m = 0 , … , N − 1 i = 1 , … , N = ς N ∏ i < j ( x j − x i ) ∏ i e − x i 2 / 2 . (129) \det(h_m(x_i))_{\substack{m=0,\ldots,N-1\\i=1,\ldots,N}}=\varsigma_N\prod_{i<j}(x_j-x_i)\prod_i e^{-x_i^2/2}. \tag*{(129)} det ( h m ( x i ) ) m = 0 , … , N − 1 i = 1 , … , N = ς N i < j ∏ ( x j − x i ) i ∏ e − x i 2 /2 . (129) The product ∏ i < j ( x j − x i ) \prod_{i<j}(x_j-x_i) ∏ i < j ( x j − x i ) { x 1 < ⋯ < x N } \{x_1<\cdots<x_N\} { x 1 < ⋯ < x N } (73) , we find that for every measurable A A A
P ( N 2 λ i ∈ A for all i ) = 1 ς N Z ~ N ∫ x 1 < ⋯ < x N x i ∈ A det ( h m ( x i ) ) d x 1 ⋯ d x N . (130) \mathbb{P}\left(\sqrt{\frac{N}{2}}\lambda_i\in A\text{ for all }i\right)=\frac{1}{\varsigma_N\widetilde{Z}_N}\int_{\substack{x_1<\cdots<x_N\\x_i\in A}}\det(h_m(x_i))\,\mathrm{d}x_1\cdots\mathrm{d}x_N. \tag*{(130)} P ( 2 N λ i ∈ A for all i ) = ς N Z N 1 ∫ x 1 < ⋯ < x N x i ∈ A det ( h m ( x i )) d x 1 ⋯ d x N . (130) For even N N N [7 ] states that the integral on the right is the Pfaffian of the matrix with entries ∬ A × A sgn ( y − x ) h m ( x ) h n ( y ) d x d y \iint_{A\times A}\operatorname{sgn}(y-x)h_m(x)h_n(y)\,\mathrm{d}x\,\mathrm{d}y ∬ A × A sgn ( y − x ) h m ( x ) h n ( y ) d x d y − 2 Ω A ( h m , h n ) -2\Omega_A(h_m,h_n) − 2 Ω A ( h m , h n ) N N N [3 Theorem 6.1] or [44 (2.2)] ) states that the integral is the Pfaffian of the same matrix bordered by the column ( ∫ A h m ) m < N (\int_A h_m)_{m<N} ( ∫ A h m ) m < N − 2 Ω ^ A -2\widehat{\Omega}_A − 2 Ω A
Taking A = R A=\mathbb{R} A = R A = J c A=J^c A = J c A = R A=\mathbb{R} A = R − 2 -2 − 2
The derivative trick
To compute the determinants in Lemma 3.4, we replace the first argument of Ω A \Omega_A Ω A ∂ f = f ′ \partial f=f' ∂ f = f ′ ε \varepsilon ε Ω R ( ∂ f , g ) = − ⟨ f , g ⟩ \Omega_{\mathbb{R}}(\partial f,g)=-\langle f,g\rangle Ω R ( ∂ f , g ) = − ⟨ f , g ⟩ ∂ f \partial f ∂ f V N V_N V N K N ∂ f K_N\partial f K N ∂ f ∂ f \partial f ∂ f h N h_N h N A = R A=\mathbb{R} A = R A = J c A=J^c A = J c (111) that γ N ( J ) = 1 J c ( ω N − ε ( 1 J φ N ) ) \gamma_N(J)=\mathbf{1}_{J^c}(\omega_N-\varepsilon(\mathbf{1}_J\varphi_N)) γ N ( J ) = 1 J c ( ω N − ε ( 1 J φ N ))
Lemma 3.5. For all f , g ∈ V N f,g\in V_N f , g ∈ V N we have
Ω R ( K N ∂ f , g ) = − ⟨ f , g ⟩ − ⟨ ψ N , f ⟩ ⟨ ε φ N , g ⟩ , (131) \Omega_{\mathbb{R}}(K_N\partial f,g)=-\langle f,g\rangle-\langle\psi_N,f\rangle\langle\varepsilon\varphi_N,g\rangle, \tag*{(131)} Ω R ( K N ∂ f , g ) = − ⟨ f , g ⟩ − ⟨ ψ N , f ⟩ ⟨ ε φ N , g ⟩ , (131) Ω J c ( K N ∂ f , g ) = − ⟨ f , T N ( J ) g ⟩ + c N ⟨ ψ N , f ⟩ ⟨ 1 J c , g ⟩ . (132) \Omega_{J^c}(K_N\partial f,g)=-\langle f,T_N(J)g\rangle+c_N\langle\psi_N,f\rangle\langle\mathbf{1}_{J^c},g\rangle. \tag*{(132)} Ω J c ( K N ∂ f , g ) = − ⟨ f , T N ( J ) g ⟩ + c N ⟨ ψ N , f ⟩ ⟨ 1 J c , g ⟩ . (132) Proof. By (60) , we have ∂ f ∈ V N + 1 \partial f\in V_{N+1} ∂ f ∈ V N + 1
⟨ h N , ∂ f ⟩ = − ⟨ ∂ h N , f ⟩ = − N 2 ⟨ h N − 1 , f ⟩ . (133) \langle h_N,\partial f\rangle=-\langle\partial h_N,f\rangle=-\sqrt{\frac{N}{2}}\langle h_{N-1},f\rangle. \tag*{(133)} ⟨ h N , ∂ f ⟩ = − ⟨ ∂ h N , f ⟩ = − 2 N ⟨ h N − 1 , f ⟩ . (133) Therefore,
K N ∂ f = ∂ f + N 2 ⟨ h N − 1 , f ⟩ h N = ∂ f + ⟨ ψ N , f ⟩ φ N . (134) K_N\partial f=\partial f+\sqrt{\frac{N}{2}}\langle h_{N-1},f\rangle h_N=\partial f+\langle\psi_N,f\rangle\varphi_N. \tag*{(134)} K N ∂ f = ∂ f + 2 N ⟨ h N − 1 , f ⟩ h N = ∂ f + ⟨ ψ N , f ⟩ φ N . (134) We first consider A = R A=\mathbb{R} A = R ∂ ε g = g \partial\varepsilon g=g ∂ ε g = g
Ω R ( ∂ f , g ) = ⟨ ∂ f , ε g ⟩ = − ⟨ f , g ⟩ . (135) \Omega_{\mathbb{R}}(\partial f,g)=\langle\partial f,\varepsilon g\rangle=-\langle f,g\rangle. \tag*{(135)} Ω R ( ∂ f , g ) = ⟨ ∂ f , ε g ⟩ = − ⟨ f , g ⟩ . (135) Also, Ω R ( φ N , g ) = − ⟨ ε φ N , g ⟩ \Omega_{\mathbb{R}}(\varphi_N,g)=-\langle\varepsilon\varphi_N,g\rangle Ω R ( φ N , g ) = − ⟨ ε φ N , g ⟩ Ω R ( ⋅ , g ) \Omega_{\mathbb{R}}(\cdot,g) Ω R ( ⋅ , g ) (134) and inserting these identities, we obtain (131) .
We now consider A = J c A=J^c A = J c
Ψ g = ε g − 1 J c ε ( 1 J c g ) = 1 J ε g + 1 J c ε ( 1 J g ) , (136) \Psi_g=\varepsilon g-\mathbf{1}_{J^c}\varepsilon(\mathbf{1}_{J^c}g)=\mathbf{1}_J\varepsilon g+\mathbf{1}_{J^c}\varepsilon(\mathbf{1}_Jg), \tag*{(136)} Ψ g = ε g − 1 J c ε ( 1 J c g ) = 1 J ε g + 1 J c ε ( 1 J g ) , (136) so that
Ω J c ( ∂ f , g ) = ⟨ ∂ f , ε g ⟩ − ⟨ ∂ f , Ψ g ⟩ = − ⟨ f , g ⟩ − ⟨ ∂ f , Ψ g ⟩ . (137) \Omega_{J^c}(\partial f,g)=\langle\partial f,\varepsilon g\rangle-\langle\partial f,\Psi_g\rangle=-\langle f,g\rangle-\langle\partial f,\Psi_g\rangle. \tag*{(137)} Ω J c ( ∂ f , g ) = ⟨ ∂ f , ε g ⟩ − ⟨ ∂ f , Ψ g ⟩ = − ⟨ f , g ⟩ − ⟨ ∂ f , Ψ g ⟩ . (137) The restriction of Ψ g \Psi_g Ψ g J j J_j J j J j ‾ \overline{J_j} J j g g g J j J_j J j J c J^c J c ε ( 1 J g ) \varepsilon(\mathbf{1}_Jg) ε ( 1 J g ) 1 J g \mathbf{1}_Jg 1 J g a j a_j a j b j b_j b j
Ψ g ( a j − ) − Ψ g ( a j + ) = ( ε ( 1 J c g ) ) ( a j ) − ( ε g ) ( a j ) = − ( ε ( 1 J c g ) ) ( a j ) = ⟨ w j , g ⟩ , (138) \Psi_g(a_j^-)-\Psi_g(a_j^+)=(\varepsilon(\mathbf{1}_J^cg))(a_j)-(\varepsilon g)(a_j)=-(\varepsilon(\mathbf{1}_{J^c}g))(a_j)=\langle w_j,g\rangle, \tag*{(138)} Ψ g ( a j − ) − Ψ g ( a j + ) = ( ε ( 1 J c g )) ( a j ) − ( ε g ) ( a j ) = − ( ε ( 1 J c g )) ( a j ) = ⟨ w j , g ⟩ , (138) where the last equality follows from ( ε h ) ( t ) = − ⟨ ε t , h ⟩ (\varepsilon h)(t)=-\langle\varepsilon_t,h\rangle ( ε h ) ( t ) = − ⟨ ε t , h ⟩ 1 J c ε a j = w j \mathbf{1}_{J^c}\varepsilon_{a_j}=w_j 1 J c ε a j = w j
Ψ g ( b j − ) − Ψ g ( b j + ) = ( ε ( 1 J c g ) ) ( b j ) = − ⟨ w j , g ⟩ , (139) \Psi_g(b_j^-)-\Psi_g(b_j^+)=(\varepsilon(\mathbf{1}_{J^c}g))(b_j)=-\langle w_j,g\rangle, \tag*{(139)} Ψ g ( b j − ) − Ψ g ( b j + ) = ( ε ( 1 J c g )) ( b j ) = − ⟨ w j , g ⟩ , (139) where we used 1 J c ε b j = w j \mathbf{1}_{J^c}\varepsilon_{b_j}=w_j 1 J c ε b j = w j
Integrating by parts on each interval where Ψ g \Psi_g Ψ g
⟨ ∂ f , Ψ g ⟩ = ∑ j = 1 r ( f ( a j ) − f ( b j ) ) ⟨ w j , g ⟩ − ⟨ 1 J f , g ⟩ = − ∑ j = 1 r ⟨ f , k j ⟩ ⟨ w j , g ⟩ − ⟨ f , K N ( 1 J g ) ⟩ , (140) \begin{aligned}
\langle\partial f,\Psi_g\rangle
&=\sum_{j=1}^{r}(f(a_j)-f(b_j))\langle w_j,g\rangle-\langle\mathbf{1}_Jf,g\rangle\\
&=-\sum_{j=1}^{r}\langle f,k_j\rangle\langle w_j,g\rangle-\langle f,K_N(\mathbf{1}_Jg)\rangle, \tag*{(140)}
\end{aligned} ⟨ ∂ f , Ψ g ⟩ = j = 1 ∑ r ( f ( a j ) − f ( b j )) ⟨ w j , g ⟩ − ⟨ 1 J f , g ⟩ = − j = 1 ∑ r ⟨ f , k j ⟩ ⟨ w j , g ⟩ − ⟨ f , K N ( 1 J g )⟩ , (140) where in the last step we used f ∈ V N f\in V_N f ∈ V N ε \varepsilon ε
Ω J c ( φ N , g ) = − ⟨ 1 J c ε ( 1 J c φ N ) , g ⟩ = − ⟨ γ N ( J ) , g ⟩ + c N ⟨ 1 J c , g ⟩ , (141) \Omega_{J^c}(\varphi_N,g)=-\langle\mathbf{1}_{J^c}\varepsilon(\mathbf{1}_{J^c}\varphi_N),g\rangle=-\langle\gamma_N(J),g\rangle+c_N\langle\mathbf{1}_{J^c},g\rangle, \tag*{(141)} Ω J c ( φ N , g ) = − ⟨ 1 J c ε ( 1 J c φ N ) , g ⟩ = − ⟨ γ N ( J ) , g ⟩ + c N ⟨ 1 J c , g ⟩ , (141) where we used
ε ( 1 J c φ N ) = ε φ N − ε ( 1 J φ N ) = ω N − c N − ε ( 1 J φ N ) . (142) \varepsilon(\mathbf{1}_{J^c}\varphi_N)=\varepsilon\varphi_N-\varepsilon(\mathbf{1}_J\varphi_N)=\omega_N-c_N-\varepsilon(\mathbf{1}_J\varphi_N). \tag*{(142)} ε ( 1 J c φ N ) = ε φ N − ε ( 1 J φ N ) = ω N − c N − ε ( 1 J φ N ) . (142) Applying Ω J c ( ⋅ , g ) \Omega_{J^c}(\cdot,g) Ω J c ( ⋅ , g ) (134) and combining the identities above, we have
Ω J c ( K N ∂ f , g ) = − ⟨ f , g ⟩ + ⟨ f , K N ( 1 J g ) ⟩ + ∑ j = 1 r ⟨ f , k j ⟩ ⟨ w j , g ⟩ − ⟨ ψ N , f ⟩ ( ⟨ γ N ( J ) , g ⟩ − c N ⟨ 1 J c , g ⟩ ) , (143) \begin{aligned}
\Omega_{J^c}(K_N\partial f,g)
&=-\langle f,g\rangle+\langle f,K_N(\mathbf{1}_Jg)\rangle+\sum_{j=1}^{r}\langle f,k_j\rangle\langle w_j,g\rangle\\
&\quad-\langle\psi_N,f\rangle\bigl(\langle\gamma_N(J),g\rangle-c_N\langle\mathbf{1}_{J^c},g\rangle\bigr), \tag*{(143)}
\end{aligned} Ω J c ( K N ∂ f , g ) = − ⟨ f , g ⟩ + ⟨ f , K N ( 1 J g )⟩ + j = 1 ∑ r ⟨ f , k j ⟩ ⟨ w j , g ⟩ − ⟨ ψ N , f ⟩ ( ⟨ γ N ( J ) , g ⟩ − c N ⟨ 1 J c , g ⟩ ) , (143) which is (132) by the definition (112) .
Proof of Proposition 3.1
Let B B B N × N N \times N N × N B ℓ m = ⟨ h ℓ , ∂ h m ⟩ B_{\ell m} = \langle h_\ell, \partial h_m\rangle B ℓ m = ⟨ h ℓ , ∂ h m ⟩
K N ∂ h m = ∑ ℓ B ℓ m h ℓ , (144) K_N\partial h_m = \sum_{\ell} B_{\ell m}h_\ell, \tag*{(144)} K N ∂ h m = ℓ ∑ B ℓ m h ℓ , (144) so for every measurable A A A Ω A ( K N ∂ h m , h n ) \Omega_A(K_N\partial h_m,h_n) Ω A ( K N ∂ h m , h n ) B T Ω A B^{\mathsf{T}}\Omega_A B T Ω A
Suppose first that N N N c N = 0 c_N = 0 c N = 0 − B T Ω R -B^{\mathsf{T}}\Omega_R − B T Ω R − B T Ω J c -B^{\mathsf{T}}\Omega_{J^c} − B T Ω J c Id + ψ N ⊗ ε φ N \operatorname{Id} + \psi_N \otimes\varepsilon\varphi_N Id + ψ N ⊗ ε φ N T N ( J ) T_N(J) T N ( J ) ( h n ) n < N (h_n)_{n<N} ( h n ) n < N 1 + ⟨ ε φ N , ψ N ⟩ = 1 1 + \langle\varepsilon\varphi_N,\psi_N\rangle= 1 1 + ⟨ ε φ N , ψ N ⟩ = 1 c N = 0 c_N = 0 c N = 0 det B ≠ 0 \det B \ne0 det B = 0
P N ( J ) 2 = det Ω J c det Ω R = det ( − B T Ω J c ) det ( − B T Ω R ) = det T N ( J ) . (145) \mathcal{P}_N(J)^2 = \frac{\det\Omega_{J^c}}{\det\Omega_R} = \frac{\det(-B^{\mathsf{T}}\Omega_{J^c})}{\det(-B^{\mathsf{T}}\Omega_R)} = \det T_N(J). \tag*{(145)} P N ( J ) 2 = det Ω R det Ω J c = det ( − B T Ω R ) det ( − B T Ω J c ) = det T N ( J ) . (145) Suppose now that N N N
ν = 2 c N ( N / 2 ) 1 / 4 . (146) \nu= 2c_N(N/2)^{1/4}. \tag*{(146)} ν = 2 c N ( N /2 ) 1/4 . (146) We define the vectors
u m = ( K N ∂ h m , − ν 1 m = N − 1 ) , 0 ≤ m < N , u N = ( ν h N − 1 , 0 ) , (147) u_m = (K_N\partial h_m,-\nu1_{m=N-1}),\qquad0 \le m < N,\qquad u_N = (\nu h_{N-1},0), \tag*{(147)} u m = ( K N ∂ h m , − ν 1 m = N − 1 ) , 0 ≤ m < N , u N = ( ν h N − 1 , 0 ) , (147) in V N ⊕ R V_N \oplus\mathbb{R} V N ⊕ R e n = ( h n , 0 ) e_n = (h_n,0) e n = ( h n , 0 ) n < N n < N n < N e N = ( 0 , 1 ) e_N = (0,1) e N = ( 0 , 1 ) Γ \Gamma Γ
u α = ∑ β Γ α β e β . (148) u_\alpha= \sum_\beta\Gamma_{\alpha\beta}e_\beta. \tag*{(148)} u α = β ∑ Γ α β e β . (148) Then the matrix with entries − Ω ^ A ( u α , e β ) -\widehat{\Omega}_A(u_\alpha,e_\beta) − Ω A ( u α , e β ) − Γ Ω ^ A -\Gamma\widehat{\Omega}_A − Γ Ω A A ∈ { R , J c } A \in\{R,J^c\} A ∈ { R , J c } ν 1 m = N − 1 / 2 = c N ⟨ ψ N , h m ⟩ \nu1_{m=N-1}/2 = c_N\langle\psi_N,h_m\rangle ν 1 m = N − 1 /2 = c N ⟨ ψ N , h m ⟩
We begin with the case m , n < N m,n<N m , n < N (132) gives
− Ω ^ J c ( u m , e n ) = − Ω J c ( K N ∂ h m , h n ) + c N ⟨ ψ N , h m ⟩ ⟨ 1 J c , h n ⟩ = ⟨ h m , T N ( J ) h n ⟩ , (149) -\widehat{\Omega}_{J^c}(u_m,e_n) = -\Omega_{J^c}(K_N\partial h_m,h_n) + c_N\langle\psi_N,h_m\rangle\langle1_{J^c},h_n\rangle= \langle h_m,T_N(J)h_n\rangle, \tag*{(149)} − Ω J c ( u m , e n ) = − Ω J c ( K N ∂ h m , h n ) + c N ⟨ ψ N , h m ⟩ ⟨ 1 J c , h n ⟩ = ⟨ h m , T N ( J ) h n ⟩ , (149) and in the same way, (131) gives − Ω ^ R ( u m , e n ) = ⟨ h m , ( Id + ψ N ⊗ ω N ) h n ⟩ -\widehat{\Omega}_R(u_m,e_n) = \langle h_m,(\operatorname{Id}+\psi_N\otimes\omega_N)h_n\rangle − Ω R ( u m , e n ) = ⟨ h m , ( Id + ψ N ⊗ ω N ) h n ⟩ ω N = ε φ N + c N \omega_N=\varepsilon\varphi_N+c_N ω N = ε φ N + c N
Next, let m < N m<N m < N n = N n=N n = N (128) gives − Ω ^ A ( u m , e N ) = ⟨ 1 A , K N ∂ h m ⟩ / 2 -\widehat{\Omega}_A(u_m,e_N)=\langle1_A,K_N\partial h_m\rangle/2 − Ω A ( u m , e N ) = ⟨ 1 A , K N ∂ h m ⟩ /2 A = R A=R A = R
⟨ p N , ∂ h m ⟩ / 2 = − ⟨ p N ′ , h m ⟩ / 2 = 0 , (150) \langle p_N,\partial h_m\rangle/2 = -\langle p_N',h_m\rangle/2 = 0, \tag*{(150)} ⟨ p N , ∂ h m ⟩ /2 = − ⟨ p N ′ , h m ⟩ /2 = 0 , (150) where we used (83) , ∫ J φ N = 0 \int_J\varphi_N=0 ∫ J φ N = 0 φ N \varphi_N φ N V N V_N V N A = J c A=J^c A = J c A = R A=R A = R (134) to obtain
1 2 ⟨ 1 J c , K N ∂ h m ⟩ = − 1 2 ⟨ 1 J , ∂ h m + ⟨ ψ N , h m ⟩ φ N ⟩ = − 1 2 ∑ j = 1 r ⟨ k j , h m ⟩ − 1 2 ⟨ ψ N , h m ⟩ ∫ J φ N = ⟨ h m , X N ♯ ( J ) ⟩ . (151) \frac{1}{2}\langle1_{J^c},K_N\partial h_m\rangle
= -\frac{1}{2}\langle1_J,\partial h_m+\langle\psi_N,h_m\rangle\varphi_N\rangle
= -\frac{1}{2}\sum_{j=1}^{r}\langle k_j,h_m\rangle-\frac{1}{2}\langle\psi_N,h_m\rangle\int_J\varphi_N
= \langle h_m,X_N^{\sharp}(J)\rangle. \tag*{(151)} 2 1 ⟨ 1 J c , K N ∂ h m ⟩ = − 2 1 ⟨ 1 J , ∂ h m + ⟨ ψ N , h m ⟩ φ N ⟩ = − 2 1 j = 1 ∑ r ⟨ k j , h m ⟩ − 2 1 ⟨ ψ N , h m ⟩ ∫ J φ N = ⟨ h m , X N ♯ ( J )⟩ . (151) It remains to compute the entries with α = N \alpha= N α = N n < N n < N n < N − Ω ^ A ( u N , e n ) = − ν Ω A ( h N − 1 , h n ) -\widehat{\Omega}_A(u_N,e_n) = -\nu\Omega_A(h_{N-1},h_n) − Ω A ( u N , e n ) = − ν Ω A ( h N − 1 , h n ) A = R A = \mathbb{R} A = R Lemma 2.5 (c) implies that
Ω R ( h N − 1 , h n ) = − ⟨ ε h N − 1 , h n ⟩ = 0 , (152) \Omega_{\mathbb{R}}(h_{N-1},h_n) = -\langle\varepsilon h_{N-1},h_n\rangle= 0, \tag*{(152)} Ω R ( h N − 1 , h n ) = − ⟨ ε h N − 1 , h n ⟩ = 0 , (152) so this entry vanishes. To treat A = J c A = J^c A = J c 1 J c = 1 − 1 J \mathbf{1}_{J^c} = 1 - \mathbf{1}_J 1 J c = 1 − 1 J ε \varepsilon ε f , g ∈ V N f,g \in V_N f , g ∈ V N
Ω J c ( f , g ) − Ω R ( f , g ) = ⟨ 1 J ε f + 1 J c ε ( 1 J f ) , g ⟩ . (153) \Omega_{J^c}(f,g) - \Omega_{\mathbb{R}}(f,g) = \langle\mathbf{1}_J\varepsilon f + \mathbf{1}_{J^c}\varepsilon(\mathbf{1}_Jf),g\rangle. \tag*{(153)} Ω J c ( f , g ) − Ω R ( f , g ) = ⟨ 1 J ε f + 1 J c ε ( 1 J f ) , g ⟩ . (153) Taking f = h N − 1 f = h_{N-1} f = h N − 1 ν h N − 1 = 2 c N ψ N \nu h_{N-1} = 2c_N\psi_N ν h N − 1 = 2 c N ψ N
− Ω ^ J c ( u N , e n ) = ⟨ Y N ♯ ( J ) , h n ⟩ . (154) -\widehat{\Omega}_{J^c}(u_N,e_n) = \langle Y_N^\sharp(J),h_n\rangle. \tag*{(154)} − Ω J c ( u N , e n ) = ⟨ Y N ♯ ( J ) , h n ⟩ . (154) Finally, we have
− Ω ^ A ( u N , e N ) = ( ν / 2 ) ⟨ 1 A , h N − 1 ⟩ = c N ⟨ 1 A , ψ N ⟩ , (155) -\widehat{\Omega}_A(u_N,e_N) = (\nu/2)\langle\mathbf{1}_A,h_{N-1}\rangle= c_N\langle\mathbf{1}_A,\psi_N\rangle, \tag*{(155)} − Ω A ( u N , e N ) = ( ν /2 ) ⟨ 1 A , h N − 1 ⟩ = c N ⟨ 1 A , ψ N ⟩ , (155) which takes the value 1 1 1 A = R A = \mathbb{R} A = R d N ( J ) d_N(J) d N ( J ) A = J c A = J^c A = J c − Γ Ω ^ J c -\Gamma\widehat{\Omega}_{J^c} − Γ Ω J c T N ( J ) T_N(J) T N ( J ) ( e β ) (e_\beta) ( e β ) − Γ Ω ^ R -\Gamma\widehat{\Omega}_{\mathbb{R}} − Γ Ω R ( Id + ψ N ⊗ ω N ) ⊕ 1 (\operatorname{Id} + \psi_N \otimes\omega_N) \oplus1 ( Id + ψ N ⊗ ω N ) ⊕ 1 1 + ⟨ ω N , ψ N ⟩ = 1 1 + \langle\omega_N,\psi_N\rangle= 1 1 + ⟨ ω N , ψ N ⟩ = 1 Lemma 2.6 (c). In particular, det Γ ≠ 0 \det\Gamma\ne0 det Γ = 0
P N ( J ) 2 = det Ω ^ J c det Ω ^ R = det ( − Γ Ω ^ J c ) det ( − Γ Ω ^ R ) = det T N ( J ) . (156) \mathcal{P}_N(J)^2 = \frac{\det\widehat{\Omega}_{J^c}}{\det\widehat{\Omega}_{\mathbb{R}}} = \frac{\det(-\Gamma\widehat{\Omega}_{J^c})}{\det(-\Gamma\widehat{\Omega}_{\mathbb{R}})} = \det T_N(J). \tag*{(156)} P N ( J ) 2 = det Ω R det Ω J c = det ( − Γ Ω R ) det ( − Γ Ω J c ) = det T N ( J ) . (156) This completes the proof.
Estimates in the bulk
In this section, we prove the asymptotic estimates for Hermite functions that we need. They all concern points in the bulk of the spectrum, and they follow from the Plancherel–Rotach asymptotics together with the identities of Section 2.4. The main results are Lemma 4.3, which compares the Hermite kernel with the sine kernel on growing windows, and Lemma 4.6 and Corollary 4.7, which control integrals of the kernel against step functions.
For δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ]
B N ( δ ) = [ − ( 1 − δ ) 2 N , ( 1 − δ ) 2 N ] . (157) \mathcal{B}_N(\delta) = \left[-(1-\delta)\sqrt{2N},(1-\delta)\sqrt{2N}\right]. \tag*{(157)} B N ( δ ) = [ − ( 1 − δ ) 2 N , ( 1 − δ ) 2 N ] . (157) and for x 0 ∈ B N ( δ ) x_0 \in\mathcal{B}_N(\delta) x 0 ∈ B N ( δ ) local scale
ϱ N ( x 0 ) = 2 N − x 0 2 . (158) \varrho_N(x_0)=\sqrt{2N-x_0^2}. \tag*{(158)} ϱ N ( x 0 ) = 2 N − x 0 2 . (158) Note that 2 δ N ≤ ϱ N ( x 0 ) ≤ 2 N \sqrt{2\delta N}\leq\varrho_N(x_0)\leq\sqrt{2N} 2 δ N ≤ ϱ N ( x 0 ) ≤ 2 N x 0 ∈ B N ( δ ) x_0\in\mathcal{B}_N(\delta) x 0 ∈ B N ( δ ) sine kernel
S ( t ) = sin t π t , t ≠ 0 , S ( 0 ) = 1 π . (159) S(t)=\frac{\sin t}{\pi t},\qquad t\ne0,\qquad S(0)=\frac{1}{\pi}. \tag*{(159)} S ( t ) = π t sin t , t = 0 , S ( 0 ) = π 1 . (159) Throughout this section, we fix δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ] C C C c c c N 0 N_0 N 0 δ \delta δ N ≥ N 0 N\geq N_0 N ≥ N 0
Plancherel–Rotach asymptotics
For n ≥ 1 n\geq1 n ≥ 1 ∣ x ∣ < 2 n + 1 |x|<\sqrt{2n+1} ∣ x ∣ < 2 n + 1 θ n ( x ) ∈ ( 0 , π ) \theta_n(x)\in(0,\pi) θ n ( x ) ∈ ( 0 , π ) x = 2 n + 1 cos θ n ( x ) x=\sqrt{2n+1}\cos\theta_n(x) x = 2 n + 1 cos θ n ( x )
α n ( x ) = 2 1 / 4 π n 1 / 4 ( sin θ n ( x ) ) − 1 / 2 , \alpha_n(x)=\frac{2^{1/4}}{\sqrt{\pi}\,n^{1/4}}\left(\sin\theta_n(x)\right)^{-1/2}, α n ( x ) = π n 1/4 2 1/4 ( sin θ n ( x ) ) − 1/2 , Θ n ( x ) = ( n 2 + 1 4 ) ( sin 2 θ n ( x ) − 2 θ n ( x ) ) + 3 π 4 . (160) \Theta_n(x)=\left(\frac{n}{2}+\frac{1}{4}\right)\left(\sin2\theta_n(x)-2\theta_n(x)\right)+\frac{3\pi}{4}. \tag*{(160)} Θ n ( x ) = ( 2 n + 4 1 ) ( sin 2 θ n ( x ) − 2 θ n ( x ) ) + 4 3 π . (160) Lemma 4.1. There exists C > 0 C>0 C > 0 n n n ∣ n − N ∣ ≤ 2 |n-N|\leq2 ∣ n − N ∣ ≤ 2 x ∈ B N ( δ ) x\in\mathcal{B}_N(\delta) x ∈ B N ( δ )
h n ( x ) = α n ( x ) ( sin Θ n ( x ) + ρ n ( x ) ) , ∣ ρ n ( x ) ∣ ≤ C N . (161) h_n(x)=\alpha_n(x)\left(\sin\Theta_n(x)+\rho_n(x)\right),\qquad|\rho_n(x)|\leq\frac{C}{N}. \tag*{(161)} h n ( x ) = α n ( x ) ( sin Θ n ( x ) + ρ n ( x ) ) , ∣ ρ n ( x ) ∣ ≤ N C . (161) Moreover, the following estimates hold for such n n n
(i) We have Θ n ′ ( x ) = 2 n + 1 − x 2 \Theta_n'(x)=\sqrt{2n+1-x^2} Θ n ′ ( x ) = 2 n + 1 − x 2 ∣ Θ n ′ ′ ( x ) ∣ ≤ C |\Theta_n''(x)|\le C ∣ Θ n ′′ ( x ) ∣ ≤ C ∣ θ n ′ ( x ) ∣ ≤ C N − 1 / 2 |\theta_n'(x)|\le CN^{-1/2} ∣ θ n ′ ( x ) ∣ ≤ C N − 1/2
(ii) We have C − 1 N − 1 / 4 ≤ α n ( x ) ≤ C N − 1 / 4 C^{-1}N^{-1/4}\le\alpha_n(x)\le CN^{-1/4} C − 1 N − 1/4 ≤ α n ( x ) ≤ C N − 1/4 ∣ α n ′ ( x ) ∣ ≤ C N − 3 / 4 |\alpha_n'(x)|\le CN^{-3/4} ∣ α n ′ ( x ) ∣ ≤ C N − 3/4
(iii) The functions defined by a = α N − 1 / α N a=\alpha_{N-1}/\alpha_N a = α N − 1 / α N ϑ = Θ N − 1 − Θ N \vartheta=\Theta_{N-1}-\Theta_N ϑ = Θ N − 1 − Θ N ∣ a ( x ) − 1 ∣ ≤ C / N |a(x)-1|\le C/N ∣ a ( x ) − 1∣ ≤ C / N ∣ a ′ ( x ) ∣ ≤ C N − 3 / 2 |a'(x)|\le CN^{-3/2} ∣ a ′ ( x ) ∣ ≤ C N − 3/2 ∣ ϑ ( x ) − θ N ( x ) ∣ ≤ C / N |\vartheta(x)-\theta_N(x)|\le C/N ∣ ϑ ( x ) − θ N ( x ) ∣ ≤ C / N ∣ ϑ ′ ( x ) ∣ ≤ C N − 1 / 2 |\vartheta'(x)|\le CN^{-1/2} ∣ ϑ ′ ( x ) ∣ ≤ C N − 1/2
Proof. When N ≥ N 0 N\ge N_0 N ≥ N 0 ∣ x ∣ ≤ ( 1 − δ / 2 ) 2 n + 1 |x|\le(1-\delta/2)\sqrt{2n+1} ∣ x ∣ ≤ ( 1 − δ /2 ) 2 n + 1 ∣ n − N ∣ ≤ 2 |n-N|\le2 ∣ n − N ∣ ≤ 2 x ∈ B N ( δ ) x\in B_N(\delta) x ∈ B N ( δ ) θ n ( x ) \theta_n(x) θ n ( x ) ( 0 , π ) (0,\pi) ( 0 , π ) δ \delta δ [41 ] in the form of [47 Theorem 8.22.9] , multiplied by the normalizing factor ( 2 n n ! π ) − 1 / 2 (2^n n!\sqrt{\pi})^{-1/2} ( 2 n n ! π ) − 1/2
For (i), we have
θ n ′ ( x ) = − ( 2 n + 1 − x 2 ) − 1 / 2 (162) \theta_n'(x)=-(2n+1-x^2)^{-1/2} \tag*{(162)} θ n ′ ( x ) = − ( 2 n + 1 − x 2 ) − 1/2 (162) and
∂ θ ( ( ( 2 n + 1 ) / 4 ) ( sin 2 θ − 2 θ ) ) = − ( 2 n + 1 ) sin 2 θ . (163) \partial_\theta\left(((2n+1)/4)(\sin2\theta-2\theta)\right)=-(2n+1)\sin^2\theta. \tag*{(163)} ∂ θ ( (( 2 n + 1 ) /4 ) ( sin 2 θ − 2 θ ) ) = − ( 2 n + 1 ) sin 2 θ . (163) Therefore,
Θ n ′ ( x ) = 2 n + 1 sin θ n ( x ) = 2 n + 1 − x 2 , (164) \Theta_n'(x)=\sqrt{2n+1}\sin\theta_n(x)=\sqrt{2n+1-x^2}, \tag*{(164)} Θ n ′ ( x ) = 2 n + 1 sin θ n ( x ) = 2 n + 1 − x 2 , (164) and
Θ n ′ ′ ( x ) = − x ( 2 n + 1 − x 2 ) − 1 / 2 . (165) \Theta_n''(x)=-x(2n+1-x^2)^{-1/2}. \tag*{(165)} Θ n ′′ ( x ) = − x ( 2 n + 1 − x 2 ) − 1/2 . (165) These are bounded as claimed, since 2 n + 1 − x 2 ≥ c N 2n+1-x^2\ge cN 2 n + 1 − x 2 ≥ c N
The bounds on α n \alpha_n α n sin θ n ( x ) ∈ [ c , 1 ] \sin\theta_n(x)\in[c,1] sin θ n ( x ) ∈ [ c , 1 ]
( log α n ) ′ ( x ) = x / ( 2 ( 2 n + 1 − x 2 ) ) , (166) (\log\alpha_n)'(x)=x/(2(2n+1-x^2)), \tag*{(166)} ( log α n ) ′ ( x ) = x / ( 2 ( 2 n + 1 − x 2 )) , (166) which is bounded by C N − 1 / 2 CN^{-1/2} C N − 1/2
For (iii), the identity for ( log α n ) ′ (\log\alpha_n)' ( log α n ) ′
( log a ) ′ = x / ( 2 ( 2 N − 1 − x 2 ) ) − x / ( 2 ( 2 N + 1 − x 2 ) ) , (167) (\log a)'=x/(2(2N-1-x^2))-x/(2(2N+1-x^2)), \tag*{(167)} ( log a ) ′ = x / ( 2 ( 2 N − 1 − x 2 )) − x / ( 2 ( 2 N + 1 − x 2 )) , (167) which is bounded by C N − 3 / 2 CN^{-3/2} C N − 3/2 a − 1 a-1 a − 1 ( N / ( N − 1 ) ) 1 / 4 = 1 + O ( N − 1 ) (N/(N-1))^{1/4}=1+O(N^{-1}) ( N / ( N − 1 ) ) 1/4 = 1 + O ( N − 1 ) sin θ N − 1 ( x ) / sin θ N ( x ) = 1 + O ( N − 1 ) \sin\theta_{N-1}(x)/\sin\theta_N(x)=1+O(N^{-1}) sin θ N − 1 ( x ) / sin θ N ( x ) = 1 + O ( N − 1 )
ϑ ′ = 2 N − 1 − x 2 − 2 N + 1 − x 2 , (168) \vartheta'=\sqrt{2N-1-x^2}-\sqrt{2N+1-x^2}, \tag*{(168)} ϑ ′ = 2 N − 1 − x 2 − 2 N + 1 − x 2 , (168) which is bounded by C N − 1 / 2 CN^{-1/2} C N − 1/2
Finally, we regard n n n n n n x x x
∂ n θ n ( x ) = cos θ n ( x ) / ( ( 2 n + 1 ) sin θ n ( x ) ) , (169) \partial_n\theta_n(x)=\cos\theta_n(x)/((2n+1)\sin\theta_n(x)), \tag*{(169)} ∂ n θ n ( x ) = cos θ n ( x ) / (( 2 n + 1 ) sin θ n ( x )) , (169) we have
∂ n Θ n ( x ) = 1 2 ( sin 2 θ n − 2 θ n ) − ( 2 n + 1 ) sin 2 θ n ⋅ cos θ n ( 2 n + 1 ) sin θ n = − θ n ( x ) . (170) \partial_n\Theta_n(x)=\frac{1}{2}(\sin2\theta_n-2\theta_n)-(2n+1)\sin^2\theta_n\cdot\frac{\cos\theta_n}{(2n+1)\sin\theta_n}=-\theta_n(x). \tag*{(170)} ∂ n Θ n ( x ) = 2 1 ( sin 2 θ n − 2 θ n ) − ( 2 n + 1 ) sin 2 θ n ⋅ ( 2 n + 1 ) sin θ n cos θ n = − θ n ( x ) . (170) Integrating this identity, and using the bound ∣ ∂ n θ n ( x ) ∣ ≤ C / N |\partial_n\theta_n(x)| \le C/N ∣ ∂ n θ n ( x ) ∣ ≤ C / N n ∈ [ N − 1 , N ] n \in[N-1,N] n ∈ [ N − 1 , N ]
ϑ ( x ) = ∫ N − 1 N θ t ( x ) d t = θ N ( x ) + O ( N − 1 ) . (171) \vartheta(x) = \int_{N-1}^{N} \theta_t(x)\,\mathrm{d}t = \theta_N(x) + O(N^{-1}). \tag*{(171)} ϑ ( x ) = ∫ N − 1 N θ t ( x ) d t = θ N ( x ) + O ( N − 1 ) . (171) Lemma 4.2. For all x , y ∈ B N ( δ ) x,y \in B_N(\delta) x , y ∈ B N ( δ )
(a) We have ∣ h n ( x ) ∣ ≤ C N − 1 / 4 |h_n(x)| \le CN^{-1/4} ∣ h n ( x ) ∣ ≤ C N − 1/4 ∣ n − N ∣ ≤ 2 |n-N| \le2 ∣ n − N ∣ ≤ 2 ∣ φ N ( x ) ∣ + ∣ ψ N ( x ) ∣ ≤ C |\varphi_N(x)| + |\psi_N(x)| \le C ∣ φ N ( x ) ∣ + ∣ ψ N ( x ) ∣ ≤ C
(b) We have K N ( x , x ) ≤ C N K_N(x,x) \le C\sqrt{N} K N ( x , x ) ≤ C N K N + 1 ( x , x ) ≤ C N K_{N+1}(x,x) \le C\sqrt{N} K N + 1 ( x , x ) ≤ C N
(c) We have ∣ K N ( x , y ) ∣ ≤ C N ( 1 + N ∣ x − y ∣ ) − 1 |K_N(x,y)| \le C\sqrt{N}(1+\sqrt{N}|x-y|)^{-1} ∣ K N ( x , y ) ∣ ≤ C N ( 1 + N ∣ x − y ∣ ) − 1
(d) We have ∣ ∂ x K N ( x , y ) ∣ ≤ C N |\partial_xK_N(x,y)| \le CN ∣ ∂ x K N ( x , y ) ∣ ≤ C N
Proof. The bounds in (a) follow from Lemma 4.1. For (b), letting y → x y \to x y → x (66) and using the second identity in (60) for h N ′ h'_N h N ′ h N − 1 ′ h'_{N-1} h N − 1 ′
K N ( x , x ) = N 2 ( h N − 1 ( x ) 2 + h N ( x ) 2 ) − N ( N + 1 ) 2 h N + 1 ( x ) h N − 1 ( x ) − N ( N − 1 ) 2 h N − 2 ( x ) h N ( x ) . (172) K_N(x,x) = \frac{N}{2}\left(h_{N-1}(x)^2+h_N(x)^2\right)-\frac{\sqrt{N(N+1)}}{2}h_{N+1}(x)h_{N-1}(x)-\frac{\sqrt{N(N-1)}}{2}h_{N-2}(x)h_N(x). \tag*{(172)} K N ( x , x ) = 2 N ( h N − 1 ( x ) 2 + h N ( x ) 2 ) − 2 N ( N + 1 ) h N + 1 ( x ) h N − 1 ( x ) − 2 N ( N − 1 ) h N − 2 ( x ) h N ( x ) . (172) By (a), each term on the right side is bounded by C N ⋅ N − 1 / 2 CN \cdot N^{-1/2} C N ⋅ N − 1/2 N + 1 N+1 N + 1 N N N K N + 1 ( x , x ) K_{N+1}(x,x) K N + 1 ( x , x ) h n h_n h n ∣ n − N ∣ ≤ 2 |n-N| \le2 ∣ n − N ∣ ≤ 2
To prove (c), we first suppose that ∣ x − y ∣ ≥ N − 1 / 2 |x-y| \ge N^{-1/2} ∣ x − y ∣ ≥ N − 1/2 (66) and (a) give
∣ K N ( x , y ) ∣ ≤ C N 1 / 2 ⋅ N − 1 / 2 ⋅ ∣ x − y ∣ − 1 , (173) |K_N(x,y)| \le CN^{1/2}\cdot N^{-1/2}\cdot|x-y|^{-1}, \tag*{(173)} ∣ K N ( x , y ) ∣ ≤ C N 1/2 ⋅ N − 1/2 ⋅ ∣ x − y ∣ − 1 , (173) which implies the claim. In the remaining case ∣ x − y ∣ < N − 1 / 2 |x-y| < N^{-1/2} ∣ x − y ∣ < N − 1/2 (67) and (b).
For (d), we write
∂ x K N ( x , y ) = ∑ n < N h n ′ ( x ) h n ( y ) , (174) \partial_xK_N(x,y) = \sum_{n<N} h'_n(x)h_n(y), \tag*{(174)} ∂ x K N ( x , y ) = n < N ∑ h n ′ ( x ) h n ( y ) , (174) and the Cauchy–Schwarz inequality gives
∣ ∂ x K N ( x , y ) ∣ ≤ ( ∑ n = 0 N − 1 h n ′ ( x ) 2 ) 1 / 2 K N ( y , y ) 1 / 2 . (175) |\partial_xK_N(x,y)| \le\left(\sum_{n=0}^{N-1} h'_n(x)^2\right)^{1/2}K_N(y,y)^{1/2}. \tag*{(175)} ∣ ∂ x K N ( x , y ) ∣ ≤ ( n = 0 ∑ N − 1 h n ′ ( x ) 2 ) 1/2 K N ( y , y ) 1/2 . (175) To bound the sum on the right side, we use (60) , which gives
h n ′ ( x ) 2 ≤ n h n − 1 ( x ) 2 + ( n + 1 ) h n + 1 ( x ) 2 , (176) h'_n(x)^2 \le n h_{n-1}(x)^2 + (n+1)h_{n+1}(x)^2, \tag*{(176)} h n ′ ( x ) 2 ≤ n h n − 1 ( x ) 2 + ( n + 1 ) h n + 1 ( x ) 2 , (176) so the sum is bounded by
2 N K N + 1 ( x , x ) ≤ C N 3 / 2 , (177) 2NK_{N+1}(x,x) \le CN^{3/2}, \tag*{(177)} 2 N K N + 1 ( x , x ) ≤ C N 3/2 , (177) where we used (b) in the last step. Using (b) again, we have
∣ ∂ x K N ( x , y ) ∣ ≤ C N 3 / 4 N 1 / 4 . (178) |\partial_x K_N(x,y)| \le C N^{3/4}N^{1/4}. \tag*{(178)} ∣ ∂ x K N ( x , y ) ∣ ≤ C N 3/4 N 1/4 . (178) The local sine kernel
Lemma 4.3. For all x 0 ∈ B N ( δ ) x_0 \in B_N(\delta) x 0 ∈ B N ( δ ) u , v ∈ R u,v \in\mathbb{R} u , v ∈ R ∣ u ∣ , ∣ v ∣ ≤ 2 N 1 / 4 |u|, |v| \le2N^{1/4} ∣ u ∣ , ∣ v ∣ ≤ 2 N 1/4 ϱ = ϱ N ( x 0 ) \varrho= \varrho_N(x_0) ϱ = ϱ N ( x 0 )
∣ 1 ϱ K N ( x 0 + u ϱ , x 0 + v ϱ ) − S ( u − v ) ∣ ≤ C N − 1 / 2 . (179) \left|\frac{1}{\varrho}K_N\left(x_0+\frac{u}{\varrho},x_0+\frac{v}{\varrho}\right)-S(u-v)\right| \le C N^{-1/2}. \tag*{(179)} ϱ 1 K N ( x 0 + ϱ u , x 0 + ϱ v ) − S ( u − v ) ≤ C N − 1/2 . (179) Proof. Set x = x 0 + u / ϱ x=x_0+u/\varrho x = x 0 + u / ϱ y = x 0 + v / ϱ y=x_0+v/\varrho y = x 0 + v / ϱ
K ( u , v ) = ϱ − 1 K N ( x , y ) . (180) \mathcal{K}(u,v)=\varrho^{-1}K_N(x,y). \tag*{(180)} K ( u , v ) = ϱ − 1 K N ( x , y ) . (180) Observe that
∣ x − x 0 ∣ + ∣ y − x 0 ∣ ≤ C N − 1 / 4 . (181) |x-x_0|+|y-x_0|\le C N^{-1/4}. \tag*{(181)} ∣ x − x 0 ∣ + ∣ y − x 0 ∣ ≤ C N − 1/4 . (181) Hence x , y ∈ B N ( δ / 2 ) x,y \in B_N(\delta/2) x , y ∈ B N ( δ /2 ) N ≥ N 0 N \ge N_0 N ≥ N 0 δ / 2 \delta/2 δ /2 δ \delta δ
We first suppose that ∣ u − v ∣ ≥ N − 1 / 2 |u-v|\ge N^{-1/2} ∣ u − v ∣ ≥ N − 1/2 x − y = ( u − v ) / ϱ x-y=(u-v)/\varrho x − y = ( u − v ) / ϱ
K ( u , v ) = N 2 h N ( x ) h N − 1 ( y ) − h N − 1 ( x ) h N ( y ) u − v . (182) \mathcal{K}(u,v)=\sqrt{\frac{N}{2}}\frac{h_N(x)h_{N-1}(y)-h_{N-1}(x)h_N(y)}{u-v}. \tag*{(182)} K ( u , v ) = 2 N u − v h N ( x ) h N − 1 ( y ) − h N − 1 ( x ) h N ( y ) . (182) We insert (4.5). The terms involving ρ N \rho_N ρ N ρ N − 1 \rho_{N-1} ρ N − 1 K ( u , v ) \mathcal{K}(u,v) K ( u , v )
C N 1 / 2 ⋅ N − 1 / 2 N − 1 ∣ u − v ∣ − 1 ≤ C N − 1 / 2 . (183) C N^{1/2}\cdot N^{-1/2}N^{-1}|u-v|^{-1}\le C N^{-1/2}. \tag*{(183)} C N 1/2 ⋅ N − 1/2 N − 1 ∣ u − v ∣ − 1 ≤ C N − 1/2 . (183) We set z 1 = Θ N ( x ) z_1=\Theta_N(x) z 1 = Θ N ( x ) z 2 = Θ N ( y ) z_2=\Theta_N(y) z 2 = Θ N ( y )
α N ( x ) α N ( y ) ( a ( y ) sin z 1 sin ( z 2 + ϑ ( y ) ) − a ( x ) sin ( z 1 + ϑ ( x ) ) sin z 2 ) . (184) \alpha_N(x)\alpha_N(y)\left(a(y)\sin z_1\sin\left(z_2+\vartheta(y)\right)-a(x)\sin\left(z_1+\vartheta(x)\right)\sin z_2\right). \tag*{(184)} α N ( x ) α N ( y ) ( a ( y ) sin z 1 sin ( z 2 + ϑ ( y ) ) − a ( x ) sin ( z 1 + ϑ ( x ) ) sin z 2 ) . (184) Lemma 4.1(iii) implies that
∣ a ( y ) − a ( x ) ∣ ≤ C N − 3 / 2 ∣ x − y ∣ (185) |a(y)-a(x)|\le C N^{-3/2}|x-y| \tag*{(185)} ∣ a ( y ) − a ( x ) ∣ ≤ C N − 3/2 ∣ x − y ∣ (185) and
∣ ϑ ( y ) − ϑ ( x ) ∣ ≤ C N − 1 / 2 ∣ x − y ∣ . (186) |\vartheta(y)-\vartheta(x)|\le C N^{-1/2}|x-y|. \tag*{(186)} ∣ ϑ ( y ) − ϑ ( x ) ∣ ≤ C N − 1/2 ∣ x − y ∣. (186) Replacing a ( y ) a(y) a ( y ) a ( x ) a(x) a ( x ) ϑ ( x ) \vartheta(x) ϑ ( x ) ϑ ( y ) \vartheta(y) ϑ ( y ) C N − 1 ∣ x − y ∣ C N^{-1}|x-y| C N − 1 ∣ x − y ∣ K ( u , v ) \mathcal{K}(u,v) K ( u , v )
C N 1 / 2 N − 1 ∣ x − y ∣ / ∣ u − v ∣ = C N − 1 / 2 ϱ − 1 ≤ C N − 1 . (187) C N^{1/2}N^{-1}|x-y|/|u-v|=C N^{-1/2}\varrho^{-1}\le C N^{-1}. \tag*{(187)} C N 1/2 N − 1 ∣ x − y ∣/∣ u − v ∣ = C N − 1/2 ϱ − 1 ≤ C N − 1 . (187) After these replacements, we use the identity
sin z 1 sin ( z 2 + w ) − sin ( z 1 + w ) sin z 2 = sin w sin ( z 1 − z 2 ) , z 1 , z 2 , w ∈ R , (188) \sin z_1\sin(z_2+w)-\sin(z_1+w)\sin z_2=\sin w\sin(z_1-z_2),\qquad z_1,z_2,w\in\mathbb{R}, \tag*{(188)} sin z 1 sin ( z 2 + w ) − sin ( z 1 + w ) sin z 2 = sin w sin ( z 1 − z 2 ) , z 1 , z 2 , w ∈ R , (188) with w = ϑ ( y ) w=\vartheta(y) w = ϑ ( y )
K ( u , v ) = N 2 α N ( x ) α N ( y ) a ( x ) sin ϑ ( y ) sin ( z 1 − z 2 ) u − v + O ( N − 1 / 2 ) . (189) K(u,v)=\sqrt{\frac{N}{2}}\,\alpha_N(x)\alpha_N(y)\,a(x)\sin\vartheta(y)\frac{\sin(z_1-z_2)}{u-v}+O(N^{-1/2}). \tag*{(189)} K ( u , v ) = 2 N α N ( x ) α N ( y ) a ( x ) sin ϑ ( y ) u − v sin ( z 1 − z 2 ) + O ( N − 1/2 ) . (189) We now evaluate the prefactor. It follows from Lemma 4.1(iii) and (i) that
sin ϑ ( y ) = sin θ N ( y ) + O ( N − 1 ) (190) \sin\vartheta(y)=\sin\theta_N(y)+O(N^{-1}) \tag*{(190)} sin ϑ ( y ) = sin θ N ( y ) + O ( N − 1 ) (190) and
sin θ N ( x ) sin θ N ( y ) = 1 + O ( N − 3 / 4 ) , (191) \frac{\sin\theta_N(x)}{\sin\theta_N(y)}=1+O(N^{-3/4}), \tag*{(191)} sin θ N ( y ) sin θ N ( x ) = 1 + O ( N − 3/4 ) , (191) since ∣ x − y ∣ ≤ C N − 1 / 4 |x-y|\le CN^{-1/4} ∣ x − y ∣ ≤ C N − 1/4 (160) , we have
N 2 α N ( x ) α N ( y ) a ( x ) sin ϑ ( y ) = 1 π ( sin θ N ( y ) sin θ N ( x ) ) 1 / 2 ( 1 + O ( N − 1 ) ) = 1 π ( 1 + O ( N − 3 / 4 ) ) . (192) \begin{aligned}
\sqrt{\frac{N}{2}}\,\alpha_N(x)\alpha_N(y)\,a(x)\sin\vartheta(y)
=\frac{1}{\pi}\left(\frac{\sin\theta_N(y)}{\sin\theta_N(x)}\right)^{1/2}\left(1+O(N^{-1})\right) \\
&=\frac{1}{\pi}\left(1+O(N^{-3/4})\right). \tag*{(192)}
\end{aligned} 2 N α N ( x ) α N ( y ) a ( x ) sin ϑ ( y ) = π 1 ( sin θ N ( x ) sin θ N ( y ) ) 1/2 ( 1 + O ( N − 1 ) ) = π 1 ( 1 + O ( N − 3/4 ) ) . (192) Next, Lemma 4.1(i) and Taylor’s theorem give
z 1 − z 2 = Θ N ′ ( x 0 ) ( x − y ) + O ( ∣ x − y ∣ ( ∣ x − x 0 ∣ + ∣ y − x 0 ∣ ) ) = ( u − v ) ( 1 + O ( N − 3 / 4 ) ) , (193) \begin{aligned}
z_1-z_2=\Theta_N'(x_0)(x-y)+O\left(|x-y|(|x-x_0|+|y-x_0|)\right) \\
&=(u-v)\left(1+O(N^{-3/4})\right), \tag*{(193)}
\end{aligned} z 1 − z 2 = Θ N ′ ( x 0 ) ( x − y ) + O ( ∣ x − y ∣ ( ∣ x − x 0 ∣ + ∣ y − x 0 ∣ ) ) = ( u − v ) ( 1 + O ( N − 3/4 ) ) , (193) where we used the bound ∣ x − x 0 ∣ + ∣ y − x 0 ∣ ≤ C N 1 / 4 / ϱ |x-x_0|+|y-x_0|\le CN^{1/4}/\varrho ∣ x − x 0 ∣ + ∣ y − x 0 ∣ ≤ C N 1/4 / ϱ
Θ N ′ ( x 0 ) = ϱ 2 + 1 = ϱ ( 1 + O ( N − 1 ) ) . (194) \Theta_N'(x_0)=\sqrt{\varrho^2+1}=\varrho\left(1+O(N^{-1})\right). \tag*{(194)} Θ N ′ ( x 0 ) = ϱ 2 + 1 = ϱ ( 1 + O ( N − 1 ) ) . (194) Using the inequality
∣ sin ( w ( 1 + w ′ ) ) − sin w ∣ ≤ ∣ w w ′ ∣ , (195) \left|\sin\left(w(1+w'\right))-\sin w\right|\le|ww'|, \tag*{(195)} ∣ sin ( w ( 1 + w ′ ) ) − sin w ∣ ≤ ∣ w w ′ ∣ , (195) which holds for all w , w ′ ∈ R w,w'\in\mathbb{R} w , w ′ ∈ R
sin ( z 1 − z 2 ) u − v = sin ( u − v ) u − v + O ( N − 3 / 4 ) . (196) \frac{\sin(z_1-z_2)}{u-v}=\frac{\sin(u-v)}{u-v}+O(N^{-3/4}). \tag*{(196)} u − v sin ( z 1 − z 2 ) = u − v sin ( u − v ) + O ( N − 3/4 ) . (196) Inserting this estimate and the evaluation of the prefactor into (189) , we obtain (179) when ∣ u − v ∣ ≥ N − 1 / 2 |u-v|\ge N^{-1/2} ∣ u − v ∣ ≥ N − 1/2
Now suppose that ∣ u − v ∣ < N − 1 / 2 |u-v|<N^{-1/2} ∣ u − v ∣ < N − 1/2 u ′ = v − sgn ( v ) N − 1 / 2 u'=v-\operatorname{sgn}(v)N^{-1/2} u ′ = v − sgn ( v ) N − 1/2 v ≠ 0 v\ne0 v = 0 u ′ = N − 1 / 2 u'=N^{-1/2} u ′ = N − 1/2 v = 0 v=0 v = 0 ∣ u ′ − v ∣ = N − 1 / 2 |u'-v|=N^{-1/2} ∣ u ′ − v ∣ = N − 1/2 ∣ u ′ ∣ ≤ 2 N 1 / 4 |u'|\le2N^{1/4} ∣ u ′ ∣ ≤ 2 N 1/4 ∣ u − u ′ ∣ ≤ 2 N − 1 / 2 |u-u'|\le2N^{-1/2} ∣ u − u ′ ∣ ≤ 2 N − 1/2 Lemma 4.2 (d), we have
∣ ∂ u K ( u , v ) ∣ = ϱ − 2 ∣ ∂ x K N ( x , y ) ∣ ≤ C , (197) |\partial_uK(u,v)|=\varrho^{-2}|\partial_xK_N(x,y)|\le C, \tag*{(197)} ∣ ∂ u K ( u , v ) ∣ = ϱ − 2 ∣ ∂ x K N ( x , y ) ∣ ≤ C , (197) and ∣ S ′ ∣ ≤ 1 |S'|\le1 ∣ S ′ ∣ ≤ 1
∣ K ( u , v ) − S ( u − v ) ∣ ≤ ∣ K ( u ′ , v ) − S ( u ′ − v ) ∣ + ( C + 1 ) ∣ u − u ′ ∣ ≤ C N − 1 / 2 . (198) |K(u,v)-S(u-v)|\le|K(u',v)-S(u'-v)|+(C+1)|u-u'|\le CN^{-1/2}. \tag*{(198)} ∣ K ( u , v ) − S ( u − v ) ∣ ≤ ∣ K ( u ′ , v ) − S ( u ′ − v ) ∣ + ( C + 1 ) ∣ u − u ′ ∣ ≤ C N − 1/2 . (198) This completes the proof.
Oscillatory integrals and global functions
Lemma 4.4. For n ∈ { N − 1 , N } n \in\{N-1,N\} n ∈ { N − 1 , N } a < b a<b a < b B N ( δ ) \mathcal{B}_N(\delta) B N ( δ ) f f f [ a , b ] [a,b] [ a , b ]
∣ ∫ a b h n ( x ) f ( x ) d x ∣ ≤ C N − 3 / 4 ( sup [ a , b ] ∣ f ∣ + ∫ a b ∣ f ′ ( x ) ∣ d x ) + C N − 5 / 4 ∫ a b ∣ f ( x ) ∣ d x . (199) \left|\int_a^b h_n(x)f(x)\,\mathrm{d}x\right| \le CN^{-3/4}\left(\sup_{[a,b]} |f|+\int_a^b |f'(x)|\,\mathrm{d}x\right)+CN^{-5/4}\int_a^b |f(x)|\,\mathrm{d}x. \tag*{(199)} ∫ a b h n ( x ) f ( x ) d x ≤ C N − 3/4 ( [ a , b ] sup ∣ f ∣ + ∫ a b ∣ f ′ ( x ) ∣ d x ) + C N − 5/4 ∫ a b ∣ f ( x ) ∣ d x . (199) In particular, ∣ ∫ a b φ N ∣ + ∣ ∫ a b ψ N ∣ ≤ C N − 1 / 2 \left|\int_a^b \varphi_N\right|+\left|\int_a^b \psi_N\right|\le CN^{-1/2} ∫ a b φ N + ∫ a b ψ N ≤ C N − 1/2
Proof. By (4.5), the contribution of ρ n \rho_n ρ n
C N − 1 / 4 N − 1 ∫ a b ∣ f ∣ . (200) CN^{-1/4}N^{-1}\int_a^b |f|. \tag*{(200)} C N − 1/4 N − 1 ∫ a b ∣ f ∣. (200) To bound the main term, we integrate by parts using sin Θ n = − ( cos Θ n ) ′ / Θ n ′ \sin\Theta_n=-(\cos\Theta_n)'/\Theta_n' sin Θ n = − ( cos Θ n ) ′ / Θ n ′
∫ a b α n f sin Θ n d x = [ − α n f Θ n ′ cos Θ n ] a b + ∫ a b cos Θ n ( α n Θ n ′ ) ′ f d x + ∫ a b cos Θ n α n Θ n ′ f ′ d x . (201) \int_a^b \alpha_n f\sin\Theta_n\,\mathrm{d}x
=
\left[-\frac{\alpha_n f}{\Theta_n'}\cos\Theta_n\right]_a^b
+\int_a^b \cos\Theta_n\left(\frac{\alpha_n}{\Theta_n'}\right)'f\,\mathrm{d}x
+\int_a^b \cos\Theta_n\frac{\alpha_n}{\Theta_n'}f'\,\mathrm{d}x. \tag*{(201)} ∫ a b α n f sin Θ n d x = [ − Θ n ′ α n f cos Θ n ] a b + ∫ a b cos Θ n ( Θ n ′ α n ) ′ f d x + ∫ a b cos Θ n Θ n ′ α n f ′ d x . (201) By Lemma 4.1, we have ∣ α n / Θ n ′ ∣ ≤ C N − 3 / 4 |\alpha_n/\Theta_n'|\le CN^{-3/4} ∣ α n / Θ n ′ ∣ ≤ C N − 3/4 ∣ ( α n / Θ n ′ ) ′ ∣ ≤ C N − 5 / 4 |(\alpha_n/\Theta_n')'|\le CN^{-5/4} ∣ ( α n / Θ n ′ ) ′ ∣ ≤ C N − 5/4 B N ( δ ) \mathcal{B}_N(\delta) B N ( δ ) f = 1 f=1 f = 1 b − a ≤ C N b-a\le C\sqrt{N} b − a ≤ C N ( N / 2 ) 1 / 4 ≤ N 1 / 4 (N/2)^{1/4}\le N^{1/4} ( N /2 ) 1/4 ≤ N 1/4
We now show that the global functions introduced in Section 2.4 are close to their limiting values in the bulk. Recall that p N p_N p N ω N \omega_N ω N c N c_N c N
Lemma 4.5. For all y ∈ B N ( δ ) y\in\mathcal{B}_N(\delta) y ∈ B N ( δ )
(a) We have ∣ p N ( y ) − 1 ∣ ≤ C N − 1 / 2 |p_N(y)-1|\le CN^{-1/2} ∣ p N ( y ) − 1∣ ≤ C N − 1/2
(b) We have ∣ ( ε φ N ) ( y ) ∣ + ∣ ( ε ψ N ) ( y ) ∣ ≤ C |(\varepsilon\varphi_N)(y)|+|(\varepsilon\psi_N)(y)|\le C ∣ ( ε φ N ) ( y ) ∣ + ∣ ( ε ψ N ) ( y ) ∣ ≤ C ∣ ( ε φ N ) ( y ) ( ε ψ N ) ( y ) ∣ ≤ C N − 1 / 2 |(\varepsilon\varphi_N)(y)(\varepsilon\psi_N)(y)|\le CN^{-1/2} ∣ ( ε φ N ) ( y ) ( ε ψ N ) ( y ) ∣ ≤ C N − 1/2 N N N ∣ ( ε ψ N ) ( y ) ∣ ≤ C N − 1 / 2 |(\varepsilon\psi_N)(y)|\le CN^{-1/2} ∣ ( ε ψ N ) ( y ) ∣ ≤ C N − 1/2
(c) We have ∣ ω N ( y ) ∣ ≤ C N − 1 / 2 |\omega_N(y)|\le CN^{-1/2} ∣ ω N ( y ) ∣ ≤ C N − 1/2
Proof. To prove (a), we use (2.30), which gives
p N ( y ) − p N ( 0 ) = − ( ∫ R φ N ) ∫ 0 y ψ N ( x ) d x − ( ∫ R ψ N ) ∫ 0 y φ N ( x ) d x . (202) p_N(y)-p_N(0)
=
-\left(\int_{\mathbb{R}}\varphi_N\right)\int_0^y\psi_N(x)\,\mathrm{d}x
-\left(\int_{\mathbb{R}}\psi_N\right)\int_0^y\varphi_N(x)\,\mathrm{d}x. \tag*{(202)} p N ( y ) − p N ( 0 ) = − ( ∫ R φ N ) ∫ 0 y ψ N ( x ) d x − ( ∫ R ψ N ) ∫ 0 y φ N ( x ) d x . (202) Lemmas 2.4 and 4.4 show that the right side is bounded by C N − 1 / 2 CN^{-1/2} C N − 1/2 [ 0 , y ] ⊂ B N ( δ ) [0,y]\subset\mathcal{B}_N(\delta) [ 0 , y ] ⊂ B N ( δ ) ∣ p N ( 0 ) − 1 ∣ ≤ 2 N − 1 / 2 |p_N(0)-1|\le2N^{-1/2} ∣ p N ( 0 ) − 1∣ ≤ 2 N − 1/2
For (b), observe that ( ε f ) ( 0 ) = 0 (\varepsilon f)(0)=0 ( ε f ) ( 0 ) = 0 f f f
( ε f ) ( y ) = ∫ 0 y f . (203) (\varepsilon f)(y)=\int_0^y f. \tag*{(203)} ( ε f ) ( y ) = ∫ 0 y f . (203) Suppose first that N N N φ N \varphi_N φ N ∣ ( ε φ N ) ( y ) ∣ ≤ C N − 1 / 2 \lvert(\varepsilon\varphi_N)(y)\rvert\le CN^{-1/2} ∣( ε φ N ) ( y )∣ ≤ C N − 1/2 ∫ φ N ≥ 1 \int\varphi_N \ge1 ∫ φ N ≥ 1
∣ ( ε ψ N ) ( y ) ∣ = p N ( y ) / ∫ φ N ≤ 2. (204) \lvert(\varepsilon\psi_N)(y)\rvert= p_N(y)/\int\varphi_N \le2. \tag*{(204)} ∣( ε ψ N ) ( y )∣ = p N ( y ) / ∫ φ N ≤ 2. (204) Now suppose that N N N ψ N \psi_N ψ N ∣ ( ε ψ N ) ( y ) ∣ ≤ C N − 1 / 2 \lvert(\varepsilon\psi_N)(y)\rvert\le CN^{-1/2} ∣( ε ψ N ) ( y )∣ ≤ C N − 1/2
∣ ( ε φ N ) ( y ) ∣ = c N p N ( y ) ≤ 2. (205) \lvert(\varepsilon\varphi_N)(y)\rvert= c_Np_N(y) \le2. \tag*{(205)} ∣( ε φ N ) ( y )∣ = c N p N ( y ) ≤ 2. (205) This proves (b).
It remains to prove (c). If N N N ω N = ε φ N \omega_N=\varepsilon\varphi_N ω N = ε φ N N N N ω N = c N ( 1 − p N ) \omega_N=c_N(1-p_N) ω N = c N ( 1 − p N ) c N ≤ 1 c_N\le1 c N ≤ 1
Integrals of the kernel
Lemma 4.6. For all a , y ∈ B N ( δ ) a,y\in B_N(\delta) a , y ∈ B N ( δ ) a < y a<y a < y
(a) We have
∣ ∫ a y K N ( x , y ) d x − 1 2 ∣ ≤ C 1 + N ( y − a ) + C N − 1 / 4 . (206) \left\lvert\int_a^y K_N(x,y)\,\mathrm{d}x-\frac{1}{2}\right\rvert\le\frac{C}{1+\sqrt{N}(y-a)}+CN^{-1/4}. \tag*{(206)} ∫ a y K N ( x , y ) d x − 2 1 ≤ 1 + N ( y − a ) C + C N − 1/4 . (206) (b) Set ϱ = ϱ N ( y ) \varrho=\varrho_N(y) ϱ = ϱ N ( y ) T = ϱ ( y − a ) T=\varrho(y-a) T = ϱ ( y − a ) T ≤ 2 N 1 / 4 T\le2N^{1/4} T ≤ 2 N 1/4
∣ ∫ a y K N ( x , y ) d x − ∫ 0 T S ( t ) d t ∣ ≤ C ( 1 + T ) N − 1 / 2 . (207) \left\lvert\int_a^y K_N(x,y)\,\mathrm{d}x-\int_0^T S(t)\,\mathrm{d}t\right\rvert\le C(1+T)N^{-1/2}. \tag*{(207)} ∫ a y K N ( x , y ) d x − ∫ 0 T S ( t ) d t ≤ C ( 1 + T ) N − 1/2 . (207) The same estimates hold for ∫ y a K N ( x , y ) d x \int_y^a K_N(x,y)\,\mathrm{d}x ∫ y a K N ( x , y ) d x a > y a>y a > y y − a y-a y − a a − y a-y a − y
Proof. The last sentence of the lemma follows from the first two claims and the symmetry K N ( − x , − y ) = K N ( x , y ) K_N(-x,-y)=K_N(x,y) K N ( − x , − y ) = K N ( x , y ) a < y a<y a < y x = y − t / ϱ x=y-t/\varrho x = y − t / ϱ
∫ a y K N ( x , y ) d x = ∫ 0 T ϱ − 1 K N ( y − t / ϱ , y ) d t , (208) \int_a^y K_N(x,y)\,\mathrm{d}x=\int_0^T \varrho^{-1}K_N(y-t/\varrho,y)\,\mathrm{d}t, \tag*{(208)} ∫ a y K N ( x , y ) d x = ∫ 0 T ϱ − 1 K N ( y − t / ϱ , y ) d t , (208) and Lemma 4.3 with x 0 = y x_0=y x 0 = y u = − t u=-t u = − t v = 0 v=0 v = 0 S ( t ) + O ( N − 1 / 2 ) S(t)+O(N^{-1/2}) S ( t ) + O ( N − 1/2 ) S S S
For (a), suppose first that T ≤ N 1 / 4 T\le N^{1/4} T ≤ N 1/4
∣ 1 2 − ∫ 0 T S ( t ) d t ∣ = ∣ ∫ T ∞ sin t π t d t ∣ ≤ min ( 1 2 , 2 π T ) , (209) \left\lvert\frac{1}{2}-\int_0^T S(t)\,\mathrm{d}t\right\rvert
=\left\lvert\int_T^\infty\frac{\sin t}{\pi t}\,\mathrm{d}t\right\rvert
\le\min\left(\frac{1}{2},\frac{2}{\pi T}\right), \tag*{(209)} 2 1 − ∫ 0 T S ( t ) d t = ∫ T ∞ π t sin t d t ≤ min ( 2 1 , π T 2 ) , (209) where we integrated by parts once for the second bound. Combining (209) with (b), and using T ≥ c N ( y − a ) T\ge c\sqrt{N}(y-a) T ≥ c N ( y − a ) ( 1 + T ) N − 1 / 2 ≤ 2 N − 1 / 4 (1+T)N^{-1/2}\le2N^{-1/4} ( 1 + T ) N − 1/2 ≤ 2 N − 1/4 T > N 1 / 4 T > N^{1/4} T > N 1/4
a ′ = y − N 1 / 4 / ϱ ∈ ( a , y ) . (210) a' = y - N^{1/4}/\varrho\in(a,y). \tag*{(210)} a ′ = y − N 1/4 / ϱ ∈ ( a , y ) . (210) The case already treated gives
∫ a ′ y K N ( x , y ) d x = 1 / 2 + O ( N − 1 / 4 ) , (211) \int_{a'}^y K_N(x,y)\,\mathrm{d}x = 1/2 + O(N^{-1/4}), \tag*{(211)} ∫ a ′ y K N ( x , y ) d x = 1/2 + O ( N − 1/4 ) , (211) and (66) yields
∫ a a ′ K N ( x , y ) d x = N 2 ( h N − 1 ( y ) ∫ a a ′ h N ( x ) x − y d x − h N ( y ) ∫ a a ′ h N − 1 ( x ) x − y d x ) . (212) \int_a^{a'} K_N(x,y)\,\mathrm{d}x = \sqrt{\frac{N}{2}}\left(h_{N-1}(y)\int_a^{a'}\frac{h_N(x)}{x-y}\,\mathrm{d}x-h_N(y)\int_a^{a'}\frac{h_{N-1}(x)}{x-y}\,\mathrm{d}x\right). \tag*{(212)} ∫ a a ′ K N ( x , y ) d x = 2 N ( h N − 1 ( y ) ∫ a a ′ x − y h N ( x ) d x − h N ( y ) ∫ a a ′ x − y h N − 1 ( x ) d x ) . (212) We apply Lemma 4.4 with f ( x ) = ( x − y ) − 1 f(x)=(x-y)^{-1} f ( x ) = ( x − y ) − 1 [ a , a ′ ] [a,a'] [ a , a ′ ] sup ∣ f ∣ = ( y − a ′ ) − 1 \sup|f|=(y-a')^{-1} sup ∣ f ∣ = ( y − a ′ ) − 1
∫ ∣ f ′ ∣ ≤ ( y − a ′ ) − 1 , (213) \int|f'| \le(y-a')^{-1}, \tag*{(213)} ∫ ∣ f ′ ∣ ≤ ( y − a ′ ) − 1 , (213) and
∫ ∣ f ∣ ≤ log ( ( y − a ) / ( y − a ′ ) ) ≤ C log N . (214) \int|f| \le\log((y-a)/(y-a')) \le C\log N. \tag*{(214)} ∫ ∣ f ∣ ≤ log (( y − a ) / ( y − a ′ )) ≤ C log N . (214) We also have
( y − a ′ ) − 1 = ϱ N − 1 / 4 ≤ C N 1 / 4 , (215) (y-a')^{-1}=\varrho N^{-1/4}\le CN^{1/4}, \tag*{(215)} ( y − a ′ ) − 1 = ϱ N − 1/4 ≤ C N 1/4 , (215) so these bounds give
∣ ∫ a a ′ h n ( x ) ( x − y ) − 1 d x ∣ ≤ C N − 1 / 2 (216) \left|\int_a^{a'}h_n(x)(x-y)^{-1}\,\mathrm{d}x\right|\le CN^{-1/2} \tag*{(216)} ∫ a a ′ h n ( x ) ( x − y ) − 1 d x ≤ C N − 1/2 (216) for n ∈ { N − 1 , N } n\in\{N-1,N\} n ∈ { N − 1 , N } Lemma 4.2 (a), this implies that
∣ ∫ a a ′ K N ( x , y ) d x ∣ ≤ C N 1 / 2 N − 1 / 4 N − 1 / 2 = C N − 1 / 4 . (217) \left|\int_a^{a'}K_N(x,y)\,\mathrm{d}x\right|\le CN^{1/2}N^{-1/4}N^{-1/2}=CN^{-1/4}. \tag*{(217)} ∫ a a ′ K N ( x , y ) d x ≤ C N 1/2 N − 1/4 N − 1/2 = C N − 1/4 . (217) In this case, we have N ( y − a ) ≥ c N 1 / 4 \sqrt{N}(y-a)\ge cN^{1/4} N ( y − a ) ≥ c N 1/4 (206) follows.
We now apply these estimates to the projection of a step function. For t ∈ R t\in\mathbb{R} t ∈ R
g t = K N ε t − ε t , (218) g_t=K_N\varepsilon_t-\varepsilon_t, \tag*{(218)} g t = K N ε t − ε t , (218) where ε t \varepsilon_t ε t
g s i n e ( u ) = − sgn ( u ) ∫ ∣ u ∣ ∞ S ( v ) d v , u ∈ R . (219) g_{\mathrm{sine}}(u)=-\operatorname{sgn}(u)\int_{|u|}^{\infty}S(v)\,\mathrm{d}v,\qquad u\in\mathbb{R}. \tag*{(219)} g sine ( u ) = − sgn ( u ) ∫ ∣ u ∣ ∞ S ( v ) d v , u ∈ R . (219) The function g t g_t g t K N K_N K N ε t \varepsilon_t ε t t t t g s i n e g_{\mathrm{sine}} g sine
Corollary 4.7. For all t , y ∈ B N ( δ ) t,y \in B_N(\delta) t , y ∈ B N ( δ ) y ≠ t y \ne t y = t
(a) We have
g t ( y ) = ( ϵ φ N ) ( y ) ( ϵ ψ N ) ( y ) + sgn ( y − t ) ( ∫ min ( t , y ) max ( t , y ) K N ( x , y ) d x − 1 2 ) . (220) g_t(y) = (\epsilon\varphi_N)(y)\,(\epsilon\psi_N)(y) + \operatorname{sgn}(y-t)\left(\int_{\min(t,y)}^{\max(t,y)} K_N(x,y)\,\mathrm{d}x-\frac{1}{2}\right). \tag*{(220)} g t ( y ) = ( ϵ φ N ) ( y ) ( ϵ ψ N ) ( y ) + sgn ( y − t ) ( ∫ m i n ( t , y ) m a x ( t , y ) K N ( x , y ) d x − 2 1 ) . (220) (b) We have ∣ g t ( y ) ∣ ≤ C ( 1 + N ∣ y − t ∣ ) − 1 + C N − 1 / 4 |g_t(y)| \le C(1+\sqrt{N}|y-t|)^{-1}+CN^{-1/4} ∣ g t ( y ) ∣ ≤ C ( 1 + N ∣ y − t ∣ ) − 1 + C N − 1/4
(c) Set ϱ = ϱ N ( t ) \varrho=\varrho_N(t) ϱ = ϱ N ( t ) y = t + u / ϱ y=t+u/\varrho y = t + u / ϱ 0 < ∣ u ∣ ≤ N 1 / 4 0<|u|\le N^{1/4} 0 < ∣ u ∣ ≤ N 1/4 ∣ g t ( y ) − g s i n e ( u ) ∣ ≤ C ( 1 + ∣ u ∣ ) N − 1 / 2 |g_t(y)-g_{\mathrm{sine}}(u)|\le C(1+|u|)N^{-1/2} ∣ g t ( y ) − g sine ( u ) ∣ ≤ C ( 1 + ∣ u ∣ ) N − 1/2
Proof. The identity (a) is a restatement of (103) , since ϵ t ( y ) = sgn ( y − t ) / 2 \epsilon_t(y)=\operatorname{sgn}(y-t)/2 ϵ t ( y ) = sgn ( y − t ) /2 ∫ t y \int_t^y ∫ t y sgn ( y − t ) ∫ min ( t , y ) max ( t , y ) \operatorname{sgn}(y-t)\int_{\min(t,y)}^{\max(t,y)} sgn ( y − t ) ∫ m i n ( t , y ) m a x ( t , y )
To prove (c), we apply Lemma 4.6(b) with the local scale ϱ N ( y ) \varrho_N(y) ϱ N ( y ) ϱ \varrho ϱ
T = ϱ N ( y ) ∣ y − t ∣ = ∣ u ∣ ϱ N ( y ) ϱ . (221) T=\varrho_N(y)|y-t|=|u|\frac{\varrho_N(y)}{\varrho}. \tag*{(221)} T = ϱ N ( y ) ∣ y − t ∣ = ∣ u ∣ ϱ ϱ N ( y ) . (221) Since ϱ ≥ c N \varrho\ge c\sqrt{N} ϱ ≥ c N
∣ ϱ N ( y ) − ϱ ∣ ≤ C ∣ y − t ∣ ≤ C ∣ u ∣ ϱ , (222) |\varrho_N(y)-\varrho|\le C|y-t|\le C\frac{|u|}{\varrho}, \tag*{(222)} ∣ ϱ N ( y ) − ϱ ∣ ≤ C ∣ y − t ∣ ≤ C ϱ ∣ u ∣ , (222) we have
∣ T − ∣ u ∣ ∣ ≤ C ∣ u ∣ 2 N ≤ C N − 1 / 2 , (223) |T-|u||\le C\frac{|u|^2}{N}\le CN^{-1/2}, \tag*{(223)} ∣ T − ∣ u ∣∣ ≤ C N ∣ u ∣ 2 ≤ C N − 1/2 , (223) and in particular T ≤ 2 N 1 / 4 T\le2N^{1/4} T ≤ 2 N 1/4 ∣ S ∣ ≤ 1 / π |S|\le1/\pi ∣ S ∣ ≤ 1/ π
∫ min ( t , y ) max ( t , y ) K N ( x , y ) d x = ∫ 0 ∣ u ∣ S ( v ) d v + O ( ( 1 + ∣ u ∣ ) N − 1 / 2 ) . (224) \int_{\min(t,y)}^{\max(t,y)}K_N(x,y)\,\mathrm{d}x
=\int_0^{|u|}S(v)\,\mathrm{d}v+O\left((1+|u|)N^{-1/2}\right). \tag*{(224)} ∫ m i n ( t , y ) m a x ( t , y ) K N ( x , y ) d x = ∫ 0 ∣ u ∣ S ( v ) d v + O ( ( 1 + ∣ u ∣ ) N − 1/2 ) . (224) Inserting this into (4.64), and using Lemma 4.5(b) and the identity
∫ 0 ∣ u ∣ ( S − 1 2 ) = − ∫ ∣ u ∣ ∞ S , (225) \int_0^{|u|}\left(S-\frac{1}{2}\right)=-\int_{|u|}^{\infty}S, \tag*{(225)} ∫ 0 ∣ u ∣ ( S − 2 1 ) = − ∫ ∣ u ∣ ∞ S , (225) we obtain (c).
An inverse bound from the uncertainty principle
The entries of the matrix M N ♯ ( J ) M_N^{\sharp}(J) M N ♯ ( J ) Corollary 3.3 involve the inverse of Id − K J \operatorname{Id}-K_J Id − K J J J J J ⊂ R J\subset\mathbb{R} J ⊂ R microscopic length
L ( J ) = 2 N ∣ J ∣ . (226) L(J)=\sqrt{2N}\lvert J\rvert. \tag*{(226)} L ( J ) = 2 N ∣ J ∣ . (226) In eigenvalue coordinates, we have L ( J ) = N ∣ A ∣ L(J)=N\lvert A\rvert L ( J ) = N ∣ A ∣ J = N / 2 A J=\sqrt{N/2}\,A J = N /2 A L ( J ) L(J) L ( J ) J J J J J J
Since every eigenvalue of K J K_J K J [ 0 , 1 ) [0,1) [ 0 , 1 ) ∥ ( Id − K J ) − 1 ∥ ≤ det ( Id − K J ) − 1 \lVert(\operatorname{Id}-K_J)^{-1}\rVert\leq\det(\operatorname{Id}-K_J)^{-1} ∥( Id − K J ) − 1 ∥ ≤ det ( Id − K J ) − 1 det ( Id − K J ) \det(\operatorname{Id}-K_J) det ( Id − K J ) e − c L ( J ) 2 e^{-cL(J)^2} e − c L ( J ) 2 ∥ ( Id − K J ) − 1 ∥ ≤ e c L ( J ) 2 \lVert(\operatorname{Id}-K_J)^{-1}\rVert\leq e^{cL(J)^2} ∥( Id − K J ) − 1 ∥ ≤ e c L ( J ) 2 L ( J ) L(J) L ( J ) log N \sqrt{\log N} log N e c L ( J ) 2 e^{cL(J)^2} e c L ( J ) 2 N N N e C L ( J ) e^{CL(J)} e C L ( J ) N N N [39 ] , in the form given by Jaming [29 ] .
Proposition 5.1. There exist absolute constants C , N 0 > 0 C,N_0>0 C , N 0 > 0 N ≥ N 0 N\geq N_0 N ≥ N 0 J ⊂ R J\subset\mathbb{R} J ⊂ R ∣ J ∣ ≤ 1 \lvert J\rvert\leq1 ∣ J ∣ ≤ 1
∥ ( Id − K J ) − 1 ∥ L 2 ( J ) → L 2 ( J ) ≤ C e C L ( J ) . (227) \left\lVert(\operatorname{Id}-K_J)^{-1}\right\rVert_{L^2(J)\to L^2(J)}\leq Ce^{CL(J)}. \tag*{(227)} ( Id − K J ) − 1 L 2 ( J ) → L 2 ( J ) ≤ C e C L ( J ) . (227) Further, for every measurable set J ⊂ R J\subset\mathbb{R} J ⊂ R S J S_J S J L 2 ( J ) L^2(J) L 2 ( J ) S ( u − v ) S(u-v) S ( u − v ) u , v ∈ J u,v\in J u , v ∈ J ∥ ( Id − S J ) − 1 ∥ ≤ C e C ∣ J ∣ \lVert(\operatorname{Id}-S_J)^{-1}\rVert\leq Ce^{C\lvert J\rvert} ∥( Id − S J ) − 1 ∥ ≤ C e C ∣ J ∣
We will use the following one-dimensional form of Nazarov’s uncertainty principle [39 ] , as stated in [29 Theorem] . Recall that F \mathcal{F} F (61) .
Lemma 5.2. There exists an absolute constant C 0 > 0 C_0>0 C 0 > 0 A , Σ ⊂ R A,\Sigma\subset\mathbb{R} A , Σ ⊂ R f ∈ L 2 ( R ) f\in L^2(\mathbb{R}) f ∈ L 2 ( R )
∥ f ∥ 2 2 ≤ C 0 e C 0 ∣ A ∣ ∣ Σ ∣ ( ∫ A c ∣ f ( x ) ∣ 2 d x + ∫ Σ c ∣ ( F f ) ( ξ ) ∣ 2 d ξ ) . (228) \lVert f\rVert_2^2\leq C_0e^{C_0\lvert A\rvert\lvert\Sigma\rvert}\left(\int_{A^c}\lvert f(x)\rvert^2\,\mathrm{d}x+\int_{\Sigma^c}\lvert(\mathcal{F}f)(\xi)\rvert^2\,\mathrm{d}\xi\right). \tag*{(228)} ∥ f ∥ 2 2 ≤ C 0 e C 0 ∣ A ∣ ∣ Σ ∣ ( ∫ A c ∣ f ( x ) ∣ 2 d x + ∫ Σ c ∣( F f ) ( ξ ) ∣ 2 d ξ ) . (228) Proof. We define
f ^ ( η ) = ∫ R f ( x ) e 2 π i x η d x . (229) \widehat{f}(\eta)=\int_{\mathbb{R}}f(x)e^{2\pi ix\eta}\,\mathrm{d}x. \tag*{(229)} f ( η ) = ∫ R f ( x ) e 2 π i x η d x . (229) In [29 ] , the inequality is stated with a Fourier transform of this form, normalized so that it is unitary on L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) Σ \Sigma Σ − Σ -\Sigma − Σ ∣ Σ ∣ |\Sigma| ∣Σ∣ e C min ( ⋅ ) e^{C\min(\cdot)} e C m i n ( ⋅ ) e C ∣ A ∣ ∣ Σ ∣ e^{C|A||\Sigma|} e C ∣ A ∣∣Σ∣ [39 ] . We set
Σ ′ = − Σ / ( 2 π ) . (230) \Sigma'=-\Sigma/(2\pi). \tag*{(230)} Σ ′ = − Σ/ ( 2 π ) . (230) Using
( F f ) ( ζ ) = ( 2 π ) − 1 / 2 f ^ ( − ζ / ( 2 π ) ) (231) (\mathcal{F}f)(\zeta)=(2\pi)^{-1/2}\widehat{f}(-\zeta/(2\pi)) \tag*{(231)} ( F f ) ( ζ ) = ( 2 π ) − 1/2 f ( − ζ / ( 2 π )) (231) and changing variables, we have
∫ Σ c ∣ F f ∣ 2 = ∫ Σ ′ c ∣ f ^ ∣ 2 . (232) \int_{\Sigma^c}|\mathcal{F}f|^2=\int_{\Sigma'^c}|\widehat{f}|^2. \tag*{(232)} ∫ Σ c ∣ F f ∣ 2 = ∫ Σ ′ c ∣ f ∣ 2 . (232) Together with ∣ Σ ′ ∣ = ∣ Σ ∣ / ( 2 π ) |\Sigma'|=|\Sigma|/(2\pi) ∣ Σ ′ ∣ = ∣Σ∣/ ( 2 π ) ∥ f ^ ∥ 2 = ∥ f ∥ 2 \|\widehat{f}\|_2=\|f\|_2 ∥ f ∥ 2 = ∥ f ∥ 2 (228) .
Proof of Proposition 5.1. Set
Σ N = [ − 8 N , 8 N ] . (233) \Sigma_N=[-8\sqrt{N},8\sqrt{N}]. \tag*{(233)} Σ N = [ − 8 N , 8 N ] . (233) We first show that every f ∈ V N f\in V_N f ∈ V N
∫ Σ N c ∣ f ( x ) ∣ 2 d x ≤ e − N ∥ f ∥ 2 2 . (234) \int_{\Sigma_N^c}|f(x)|^2\,\mathrm{d}x\le e^{-N}\|f\|_2^2. \tag*{(234)} ∫ Σ N c ∣ f ( x ) ∣ 2 d x ≤ e − N ∥ f ∥ 2 2 . (234) Indeed, (67) and Lemma 2.1 imply that
∣ f ( x ) ∣ 2 ≤ K N ( x , x ) ∥ f ∥ 2 2 ≤ N e − x 2 / 4 ∥ f ∥ 2 2 (235) |f(x)|^2\le K_N(x,x)\|f\|_2^2\le Ne^{-x^2/4}\|f\|_2^2 \tag*{(235)} ∣ f ( x ) ∣ 2 ≤ K N ( x , x ) ∥ f ∥ 2 2 ≤ N e − x 2 /4 ∥ f ∥ 2 2 (235) for ∣ x ∣ ≥ 8 N |x|\ge8\sqrt{N} ∣ x ∣ ≥ 8 N
∫ ∣ x ∣ ≥ 8 N N e − x 2 / 4 d x ≤ 2 N ∫ 8 N ∞ e − 2 N x d x ≤ e − N , (236) \int_{|x|\ge8\sqrt{N}}Ne^{-x^2/4}\,\mathrm{d}x\le2N\int_{8\sqrt{N}}^\infty e^{-2\sqrt{N}x}\,\mathrm{d}x\le e^{-N}, \tag*{(236)} ∫ ∣ x ∣ ≥ 8 N N e − x 2 /4 d x ≤ 2 N ∫ 8 N ∞ e − 2 N x d x ≤ e − N , (236) where we used x 2 / 4 ≥ 2 N x x^2/4\ge2\sqrt{N}x x 2 /4 ≥ 2 N x x ≥ 8 N x\ge8\sqrt{N} x ≥ 8 N N ≥ N 0 N\ge N_0 N ≥ N 0
Now let f ∈ V N f\in V_N f ∈ V N (62) that the function F f \mathcal{F}f F f h 0 , … , h N − 1 h_0,\ldots,h_{N-1} h 0 , … , h N − 1 ∥ F f ∥ 2 = ∥ f ∥ 2 \|\mathcal{F}f\|_2=\|f\|_2 ∥ F f ∥ 2 = ∥ f ∥ 2 (234) applies verbatim to this complex span. Applying Lemma 5.2 with A = J A=J A = J Σ = Σ N \Sigma=\Sigma_N Σ = Σ N (234) to F f \mathcal{F}f F f
∥ f ∥ 2 2 ≤ C 0 e 16 C 0 N ∣ J ∣ ( ∫ J c ∣ f ∣ 2 + e − N ∥ f ∥ 2 2 ) . (237) \|f\|_2^2\le C_0e^{16C_0\sqrt{N}|J|}\left(\int_{J^c}|f|^2+e^{-N}\|f\|_2^2\right). \tag*{(237)} ∥ f ∥ 2 2 ≤ C 0 e 16 C 0 N ∣ J ∣ ( ∫ J c ∣ f ∣ 2 + e − N ∥ f ∥ 2 2 ) . (237) Since 16 N ∣ J ∣ ≤ 12 L ( J ) 16\sqrt{N}|J|\le12L(J) 16 N ∣ J ∣ ≤ 12 L ( J ) L ( J ) ≤ 2 N L(J)\le\sqrt{2N} L ( J ) ≤ 2 N
C 0 e 16 C 0 N ∣ J ∣ − N ≤ 1 / 2 (238) C_0e^{16C_0\sqrt{N}|J|-N}\le1/2 \tag*{(238)} C 0 e 16 C 0 N ∣ J ∣ − N ≤ 1/2 (238) for N ≥ N 0 N \ge N_0 N ≥ N 0
∥ 1 J f ∥ 2 2 ≤ ( 1 − c e − C L ( J ) ) ∥ f ∥ 2 2 , f ∈ V N , (239) \|1_Jf\|_2^2 \le\left(1-ce^{-CL(J)}\right)\|f\|_2^2,\qquad f\in V_N, \tag*{(239)} ∥ 1 J f ∥ 2 2 ≤ ( 1 − c e − C L ( J ) ) ∥ f ∥ 2 2 , f ∈ V N , (239) with c = ( 2 C 0 ) − 1 c=(2C_0)^{-1} c = ( 2 C 0 ) − 1 C = 12 C 0 C=12C_0 C = 12 C 0 K N 1 J K N K_N1_JK_N K N 1 J K N L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) 1 − c e − C L ( J ) 1-ce^{-CL(J)} 1 − c e − C L ( J ) K J K_J K J L 2 ( J ) L^2(J) L 2 ( J ) ( 1 J K N ) ( 1 J K N ) ∗ (1_JK_N)(1_JK_N)^* ( 1 J K N ) ( 1 J K N ) ∗ K N 1 J K N = ( 1 J K N ) ∗ ( 1 J K N ) K_N1_JK_N=(1_JK_N)^*(1_JK_N) K N 1 J K N = ( 1 J K N ) ∗ ( 1 J K N ) K J K_J K J
∥ ( Id − K J ) − 1 ∥ ≤ c − 1 e C L ( J ) . (240) \|( \operatorname{Id}-K_J)^{-1}\|\le c^{-1}e^{CL(J)}. \tag*{(240)} ∥ ( Id − K J ) − 1 ∥ ≤ c − 1 e C L ( J ) . (240) Finally, we consider the sine kernel. Let P P P L 2 ( R ) L^2(\mathbb{R}) L 2 ( R ) f f f F f \mathcal{F}f F f [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] S J = 1 J P 1 J S_J=1_JP1_J S J = 1 J P 1 J Lemma 5.2 with A = J A=J A = J Σ = [ − 1 , 1 ] \Sigma=[-1,1] Σ = [ − 1 , 1 ] f f f
∥ 1 J f ∥ 2 2 ≤ ( 1 − C 0 − 1 e − 2 C 0 ∣ J ∣ ) ∥ f ∥ 2 2 , (241) \|1_Jf\|_2^2\le\left(1-C_0^{-1}e^{-2C_0|J|}\right)\|f\|_2^2, \tag*{(241)} ∥ 1 J f ∥ 2 2 ≤ ( 1 − C 0 − 1 e − 2 C 0 ∣ J ∣ ) ∥ f ∥ 2 2 , (241) and the rest of the argument is the same as above.
We will use Proposition 5.1 through the following consequence for values at single points. For a bounded measurable set J J J f f f J J J ( Id − K J ) − 1 f (\operatorname{Id}-K_J)^{-1}f ( Id − K J ) − 1 f L 2 ( J ) L^2(J) L 2 ( J )
( Id − K J ) − 1 f = f + K J ( Id − K J ) − 1 f , (242) (\operatorname{Id}-K_J)^{-1}f=f+K_J(\operatorname{Id}-K_J)^{-1}f, \tag*{(242)} ( Id − K J ) − 1 f = f + K J ( Id − K J ) − 1 f , (242) and the second term on the right is the restriction to J J J V N V_N V N (242) to define ( Id − K J ) − 1 f (\operatorname{Id}-K_J)^{-1}f ( Id − K J ) − 1 f J J J f f f J J J f f f ( Id − K J ) − 1 f (\operatorname{Id}-K_J)^{-1}f ( Id − K J ) − 1 f
Lemma 5.3. Let δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ] C , N 0 > 0 C,N_0>0 C , N 0 > 0 δ \delta δ N ≥ N 0 N\ge N_0 N ≥ N 0 J ⊂ B N ( δ ) J\subset B_N(\delta) J ⊂ B N ( δ ) ∣ J ∣ ≤ 1 |J|\le1 ∣ J ∣ ≤ 1 f f f J J J G = ( Id − K J ) − 1 f G=(\operatorname{Id}-K_J)^{-1}f G = ( Id − K J ) − 1 f
∣ ∫ J K N ( t , z ) G ( z ) d z ∣ ≤ C e C L ( J ) sup J ∣ f ∣ , t ∈ B N ( δ ) . (243) \left|\int_J K_N(t,z)G(z)\,\mathrm{d}z\right|\le Ce^{CL(J)}\sup_J|f|,\qquad t\in B_N(\delta). \tag*{(243)} ∫ J K N ( t , z ) G ( z ) d z ≤ C e C L ( J ) J sup ∣ f ∣ , t ∈ B N ( δ ) . (243) Consequently, ∣ G ( t ) ∣ ≤ C e C L ( J ) sup J ∣ f ∣ |G(t)|\le Ce^{CL(J)}\sup_J|f| ∣ G ( t ) ∣ ≤ C e C L ( J ) sup J ∣ f ∣ t ∈ J t\in J t ∈ J G ( t ) G(t) G ( t ) (242) . The same bound holds for boundary values obtained as limits from within J J J
Proof. Using the Cauchy–Schwarz inequality, Lemma 4.2 (b), and Proposition 5.1 , we find that the left side of (243) is bounded by
K N ( t , t ) 1 / 2 ∥ G ∥ L 2 ( J ) ≤ C N 1 / 4 e C L ( J ) / 2 sup J ∣ f ∣ ≤ C e C L ( J ) L ( J ) 1 / 2 sup J ∣ f ∣ . (244) K_N(t,t)^{1/2}\|G\|_{L^2(J)}\le CN^{1/4}e^{CL(J)/2}\sup_J|f|\le Ce^{CL(J)}L(J)^{1/2}\sup_J|f|. \tag*{(244)} K N ( t , t ) 1/2 ∥ G ∥ L 2 ( J ) ≤ C N 1/4 e C L ( J ) /2 J sup ∣ f ∣ ≤ C e C L ( J ) L ( J ) 1/2 J sup ∣ f ∣. (244) Since L ( J ) 1 / 2 ≤ e L ( J ) L(J)^{1/2}\le e^{L(J)} L ( J ) 1/2 ≤ e L ( J ) (243) , and the bound for G G G J J J (242) . Passing to limits gives the boundary assertion.
The endpoint matrix
In this section, we show that, up to a small error, the determinant of M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J ) Corollary 3.3 is the determinant of an r × r r \times r r × r A N ( J ) A_N(J) A N ( J ) ( Id − K J ) − 1 g a i (\operatorname{Id}-K_J)^{-1}g_{a_i} ( Id − K J ) − 1 g a i J J J g a i g_{a_i} g a i (218) . We then show that A N ( J ) A_N(J) A N ( J ) J J J
Setting
Throughout this section, we fix δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ] r 0 ≥ 1 r_0 \ge1 r 0 ≥ 1 C C C c c c N 0 N_0 N 0 δ \delta δ r 0 r_0 r 0 N ≥ N 0 N \ge N_0 N ≥ N 0
J = ⋃ j = 1 r ( a j , b j ) J=\bigcup_{j=1}^{r}(a_j,b_j) J = j = 1 ⋃ r ( a j , b j ) as in (109) , with r ≤ r 0 r \le r_0 r ≤ r 0
J ‾ ⊂ B N ( δ ) , L ( J ) ≤ N 1 / 8 , (245) \overline{J} \subset B_N(\delta), \qquad L(J) \le N^{1/8}, \tag*{(245)} J ⊂ B N ( δ ) , L ( J ) ≤ N 1/8 , (245) where L ( J ) L(J) L ( J ) (226) . In particular, ∣ J ∣ ≤ 1 |J| \le1 ∣ J ∣ ≤ 1 Proposition 5.1 and Lemma 5.3 apply. We also define the microscopic separation of the intervals of J J J
sep ( J ) = 2 N min i ≠ j dist ( J i , J j ) (246) \operatorname{sep}(J)=\sqrt{2N}\min_{i\ne j}\operatorname{dist}(J_i,J_j) \tag*{(246)} sep ( J ) = 2 N i = j min dist ( J i , J j ) (246) when r ≥ 2 r \ge2 r ≥ 2 sep ( J ) = ∞ \operatorname{sep}(J)=\infty sep ( J ) = ∞ r = 1 r=1 r = 1 G G G J J J J J J
Δ j G = G ( b j − ) − G ( a j + ) , 1 ≤ j ≤ r . (247) \Delta_jG=G(b_j^-)-G(a_j^+), \qquad1 \le j \le r. \tag*{(247)} Δ j G = G ( b j − ) − G ( a j + ) , 1 ≤ j ≤ r . (247) Definition 6.1. We define the endpoint matrix of J J J r × r r \times r r × r A N ( J ) A_N(J) A N ( J )
A N ( J ) i j = 1 i = j − Δ j [ ( Id − K J ) − 1 ( 1 J g a i ) ] , 1 ≤ i , j ≤ r . (248) A_N(J)_{ij}=\mathbf{1}_{i=j}-\Delta_j\left[(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_Jg_{a_i})\right], \qquad1 \le i,j \le r. \tag*{(248)} A N ( J ) ij = 1 i = j − Δ j [ ( Id − K J ) − 1 ( 1 J g a i ) ] , 1 ≤ i , j ≤ r . (248) Note that the restriction of g a i g_{a_i} g a i J k J_k J k J k ‾ \overline{J_k} J k a i a_i a i lim y ↓ a i g a i ( y ) \lim_{y\downarrow a_i}g_{a_i}(y) lim y ↓ a i g a i ( y ) (248) are defined through (242) .
The main result of this section is the following proposition.
Proposition 6.2. Under the assumptions (6.1), we have
∣ det M N ♯ ( J ) − det A N ( J ) ∣ ≤ C e C L ( J ) N − 1 / 2 , ∣ det A N ( J ) ∣ ≤ C e C L ( J ) . (249) \left|\det M_N^\sharp(J)-\det A_N(J)\right|\le Ce^{CL(J)}N^{-1/2}, \qquad\left|\det A_N(J)\right|\le Ce^{CL(J)}. \tag*{(249)} det M N ♯ ( J ) − det A N ( J ) ≤ C e C L ( J ) N − 1/2 , ∣ det A N ( J ) ∣ ≤ C e C L ( J ) . (249) Consequently, P N ( J ) 2 ≤ C e C L ( J ) D N ( J ) P_N(J)^2\le Ce^{CL(J)}D_N(J) P N ( J ) 2 ≤ C e C L ( J ) D N ( J )
Reduction to local quantities
The entries of M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J ) Y Y Y R \mathbb{R} R Q N ( J ) − 1 X Q_N(J)^{-1}X Q N ( J ) − 1 X X ∈ V N X \in V_N X ∈ V N Y Y Y ⟨ Y , X ⟩ \langle Y,X\rangle ⟨ Y , X ⟩ K N Y K_NY K N Y J J J X = k j X=k_j X = k j
Lemma 6.3. Let Y Y Y R \mathbb{R} R X ∈ V N X \in V_N X ∈ V N
⟨ Y , Q N ( J ) − 1 X ⟩ = ⟨ Y , X ⟩ + ⟨ 1 J K N Y , ( Id − K J ) − 1 ( 1 J X ) ⟩ . (250) \langle Y,Q_N(J)^{-1}X\rangle=\langle Y,X\rangle+\left\langle\mathbf{1}_J K_NY,(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_JX)\right\rangle. \tag*{(250)} ⟨ Y , Q N ( J ) − 1 X ⟩ = ⟨ Y , X ⟩ + ⟨ 1 J K N Y , ( Id − K J ) − 1 ( 1 J X ) ⟩ . (250) Moreover, for 1 ≤ j ≤ r 1 \le j \le r 1 ≤ j ≤ r ⟨ Y , Q N ( J ) − 1 k j ⟩ = Δ j [ ( Id − K J ) − 1 ( 1 J K N Y ) ] \langle Y,Q_N(J)^{-1}k_j\rangle=\Delta_j\left[(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_JK_NY)\right] ⟨ Y , Q N ( J ) − 1 k j ⟩ = Δ j [ ( Id − K J ) − 1 ( 1 J K N Y ) ]
Proof. Let f ∈ V N f \in V_N f ∈ V N
G = ( Id − K J ) − 1 ( 1 J f ) ∈ L 2 ( J ) . (251) G=(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_Jf)\in L^2(J). \tag*{(251)} G = ( Id − K J ) − 1 ( 1 J f ) ∈ L 2 ( J ) . (251) Then f + K N ( 1 J G ) ∈ V N f+K_N(\mathbf{1}_JG)\in V_N f + K N ( 1 J G ) ∈ V N
Q N ( J ) ( f + K N ( 1 J G ) ) = f + K N ( 1 J G ) − K N ( 1 J f + 1 J K N ( 1 J G ) ) = f − K N ( 1 J f ) + K N ( 1 J ( Id − K J ) G ) = f , (252) \begin{aligned}
Q_N(J)(f+K_N(\mathbf{1}_JG))
&=f+K_N(\mathbf{1}_JG)-K_N(\mathbf{1}_Jf+\mathbf{1}_JK_N(\mathbf{1}_JG))\\
&=f-K_N(\mathbf{1}_Jf)+K_N(\mathbf{1}_J(\operatorname{Id}-K_J)G)=f, \tag*{(252)}
\end{aligned} Q N ( J ) ( f + K N ( 1 J G )) = f + K N ( 1 J G ) − K N ( 1 J f + 1 J K N ( 1 J G )) = f − K N ( 1 J f ) + K N ( 1 J ( Id − K J ) G ) = f , (252) where we used 1 J K N ( 1 J G ) = K J G \mathbf{1}_JK_N(\mathbf{1}_JG)=K_JG 1 J K N ( 1 J G ) = K J G J J J Q N ( J ) − 1 f = f + K N ( 1 J G ) Q_N(J)^{-1}f=f+K_N(\mathbf{1}_JG) Q N ( J ) − 1 f = f + K N ( 1 J G ) Y Y Y K N K_N K N f = X f=X f = X
To prove the second claim, we take X = k j X=k_j X = k j K N Y ∈ V N K_NY\in V_N K N Y ∈ V N ⟨ Y , k j ⟩ = Δ j ( K N Y ) \langle Y,k_j\rangle=\Delta_j(K_NY) ⟨ Y , k j ⟩ = Δ j ( K N Y ) ( Id − K J ) − 1 (\operatorname{Id}-K_J)^{-1} ( Id − K J ) − 1
⟨ 1 J K N Y , ( Id − K J ) − 1 ( 1 J k j ) ⟩ = ⟨ ( Id − K J ) − 1 ( 1 J K N Y ) , 1 J k j ⟩ = Δ j [ K N ( 1 J ( Id − K J ) − 1 ( 1 J K N Y ) ) ] . (253) \begin{aligned}
\left\langle\mathbf{1}_JK_NY,(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_Jk_j)\right\rangle
&=\left\langle(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_JK_NY),\mathbf{1}_Jk_j\right\rangle\\
&=\Delta_j\left[K_N\left(\mathbf{1}_J(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_JK_NY)\right)\right]. \tag*{(253)}
\end{aligned} ⟨ 1 J K N Y , ( Id − K J ) − 1 ( 1 J k j ) ⟩ = ⟨ ( Id − K J ) − 1 ( 1 J K N Y ) , 1 J k j ⟩ = Δ j [ K N ( 1 J ( Id − K J ) − 1 ( 1 J K N Y ) ) ] . (253) Adding the two terms and using (5.17), we obtain the claim.
We now compute 1 J K N w i \mathbf{1}_JK_Nw_i 1 J K N w i w i = 1 J c ε a i w_i=\mathbf{1}_{J^c}\varepsilon_{a_i} w i = 1 J c ε a i
Lemma 6.4. For 1 ≤ i ≤ r 1 \le i \le r 1 ≤ i ≤ r
1 J K N w i = ( Id − K J ) ( 1 J ε a i ) + 1 J g a i . (254) \mathbf{1}_JK_Nw_i=(\operatorname{Id}-K_J)(\mathbf{1}_J\varepsilon_{a_i})+\mathbf{1}_Jg_{a_i}. \tag*{(254)} 1 J K N w i = ( Id − K J ) ( 1 J ε a i ) + 1 J g a i . (254) Consequently, M N ( J ) i j = A N ( J ) i j M_N(J)_{ij}=A_N(J)_{ij} M N ( J ) ij = A N ( J ) ij 1 ≤ i , j ≤ r 1 \le i,j \le r 1 ≤ i , j ≤ r
Proof. We have K N w i = K N ε a i − K N ( 1 J ε a i ) K_Nw_i=K_N\varepsilon_{a_i}-K_N(\mathbf{1}_J\varepsilon_{a_i}) K N w i = K N ε a i − K N ( 1 J ε a i ) 1 J K N ( 1 J ε a i ) = K J ( 1 J ε a i ) \mathbf{1}_JK_N(\mathbf{1}_J\varepsilon_{a_i})=K_J(\mathbf{1}_J\varepsilon_{a_i}) 1 J K N ( 1 J ε a i ) = K J ( 1 J ε a i ) 1 J K N ε a i = 1 J ε a i + 1 J g a i \mathbf{1}_JK_N\varepsilon_{a_i}=\mathbf{1}_J\varepsilon_{a_i}+\mathbf{1}_Jg_{a_i} 1 J K N ε a i = 1 J ε a i + 1 J g a i
Applying Lemma 6.3 with Y = w i Y=w_i Y = w i X = k j X=k_j X = k j
⟨ w i , Q N ( J ) − 1 k j ⟩ = Δ j [ 1 J ε a i ] + Δ j [ ( Id − K J ) − 1 ( 1 J g a i ) ] . (255) \langle w_i,Q_N(J)^{-1}k_j\rangle=\Delta_j[\mathbf{1}_J\varepsilon_{a_i}]+\Delta_j\left[(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_Jg_{a_i})\right]. \tag*{(255)} ⟨ w i , Q N ( J ) − 1 k j ⟩ = Δ j [ 1 J ε a i ] + Δ j [ ( Id − K J ) − 1 ( 1 J g a i ) ] . (255) The function ε a i \varepsilon_{a_i} ε a i J j J_j J j 1 / 2 1/2 1/2 J i J_i J i J i J_i J i − 1 / 2 -1/2 − 1/2 J i J_i J i Δ j [ 1 J ε a i ] = 0 \Delta_j[\mathbf{1}_J\varepsilon_{a_i}]=0 Δ j [ 1 J ε a i ] = 0 j j j Y i = w i Y_i=w_i Y i = w i X j = k j X_j=k_j X j = k j (117) . The claim now follows from (118) and (248) .
Remark 6.5. Lemma 6.4 is where the nondecaying sign function in the GOE kernel is removed. The function w i w_i w i ⟨ w i , Q N ( J ) − 1 k j ⟩ \langle w_i,Q_N(J)^{-1}k_j\rangle ⟨ w i , Q N ( J ) − 1 k j ⟩ K N K_N K N w i w_i w i J J J (254) shows that after discarding the term ( Id − K J ) ( 1 J ε a i ) (\operatorname{Id}-K_J)(\mathbf{1}_J\varepsilon_{a_i}) ( Id − K J ) ( 1 J ε a i ) g a i g_{a_i} g a i Corollary 4.7 (b), we have
∣ g a i ( y ) ∣ ≤ C ( 1 + N ∣ y − a i ∣ ) − 1 + C N − 1 / 4 , (256) |g_{a_i}(y)| \le C(1+\sqrt{N}|y-a_i|)^{-1}+CN^{-1/4}, \tag*{(256)} ∣ g a i ( y ) ∣ ≤ C ( 1 + N ∣ y − a i ∣ ) − 1 + C N − 1/4 , (256) so this function is small away from a i a_i a i
Bounds on the remaining entries
We now bound the entries of M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J )
∫ J ∣ K N ( y , z ) ∣ d z ≤ C log ( 2 + L ( J ) ) , y ∈ B N ( δ ) . (257) \int_J |K_N(y,z)|\,\mathrm{d}z \le C\log(2+L(J)), \qquad y\in B_N(\delta). \tag*{(257)} ∫ J ∣ K N ( y , z ) ∣ d z ≤ C log ( 2 + L ( J )) , y ∈ B N ( δ ) . (257) To see this, note that the bound C N ( 1 + N ∣ y − z ∣ ) − 1 C\sqrt{N}(1+\sqrt{N}|y-z|)^{-1} C N ( 1 + N ∣ y − z ∣ ) − 1 Lemma 4.2 (c) is a decreasing function of ∣ y − z ∣ |y-z| ∣ y − z ∣ J J J { z : ∣ z − y ∣ ≤ ∣ J ∣ / 2 } \{z:|z-y|\le|J|/2\} { z : ∣ z − y ∣ ≤ ∣ J ∣/2 } 2 C log ( 1 + N ∣ J ∣ / 2 ) 2C\log(1+\sqrt{N}|J|/2) 2 C log ( 1 + N ∣ J ∣/2 ) N ∣ J ∣ ≤ L ( J ) \sqrt{N}|J|\le L(J) N ∣ J ∣ ≤ L ( J )
Lemma 6.6. For 1 ≤ i ≤ r 1\le i\le r 1 ≤ i ≤ r sup J ∣ K N w i ∣ ≤ C log ( 2 + L ( J ) ) \sup_J|K_Nw_i|\le C\log(2+L(J)) sup J ∣ K N w i ∣ ≤ C log ( 2 + L ( J )) Y 0 = Y N ( J ) Y_0=Y_N(J) Y 0 = Y N ( J ) Y N ♯ ( J ) Y_N^\sharp(J) Y N ♯ ( J ) (111) and (113) ,
sup J ∣ K N Y 0 ∣ + sup J ∣ K N Y N ♯ ( J ) ∣ ≤ C ( 1 + L ( J ) ) 2 N − 1 / 2 . (258) \sup_J|K_NY_0|+\sup_J|K_NY_N^\sharp(J)|\le C(1+L(J))^2N^{-1/2}. \tag*{(258)} J sup ∣ K N Y 0 ∣ + J sup ∣ K N Y N ♯ ( J ) ∣ ≤ C ( 1 + L ( J ) ) 2 N − 1/2 . (258) Proof. The first claim follows from (254) , (257) , and Corollary 4.7 (b). Next, we observe that for all integrable f f f
1 J c E ( 1 J f ) = ∑ k = 1 r ( ∫ J k f ) w k (259) \mathbf{1}_{J^c}\mathcal{E}(\mathbf{1}_Jf)=\sum_{k=1}^{r}\left(\int_{J_k}f\right)w_k \tag*{(259)} 1 J c E ( 1 J f ) = k = 1 ∑ r ( ∫ J k f ) w k (259) almost everywhere. Indeed, sgn ( x − z ) = sgn ( x − a k ) \operatorname{sgn}(x-z)=\operatorname{sgn}(x-a_k) sgn ( x − z ) = sgn ( x − a k ) x ∉ J k ‾ x \notin\overline{J_k} x ∈ / J k z ∈ J k z \in J_k z ∈ J k
Y 0 = 1 J c ω N − ∑ k = 1 r ( ∫ J k φ N ) w k , Y N ♯ ( J ) = − 2 c N ( 1 J ε ψ N + ∑ k = 1 r ( ∫ J k ψ N ) w k ) . (260) Y_0=\mathbf{1}_{J^c}\omega_N-\sum_{k=1}^{r}\left(\int_{J_k}\varphi_N\right)w_k,\qquad Y_N^\sharp(J)=-2c_N\left(\mathbf{1}_{J}\varepsilon\psi_N+\sum_{k=1}^{r}\left(\int_{J_k}\psi_N\right)w_k\right). \tag*{(260)} Y 0 = 1 J c ω N − k = 1 ∑ r ( ∫ J k φ N ) w k , Y N ♯ ( J ) = − 2 c N ( 1 J ε ψ N + k = 1 ∑ r ( ∫ J k ψ N ) w k ) . (260) To bound sup J ∣ K N Y 0 ∣ \sup_J|K_NY_0| sup J ∣ K N Y 0 ∣ K N ω N = 0 K_N\omega_N=0 K N ω N = 0 K N ( 1 J c ω N ) = − K N ( 1 J ω N ) K_N(\mathbf{1}_{J^c}\omega_N)=-K_N(\mathbf{1}_{J}\omega_N) K N ( 1 J c ω N ) = − K N ( 1 J ω N ) (257) imply that
sup J ∣ K N ( 1 J ω N ) ∣ ≤ C N − 1 / 2 log ( 2 + L ( J ) ) . (261) \sup_J |K_N(\mathbf{1}_{J}\omega_N)|\le CN^{-1/2}\log(2+L(J)). \tag*{(261)} J sup ∣ K N ( 1 J ω N ) ∣ ≤ C N − 1/2 log ( 2 + L ( J )) . (261) Further, Lemma 4.2(a) gives
∣ ∫ J k φ N ∣ ≤ C ∣ J ∣ ≤ C L ( J ) N − 1 / 2 . (262) \left|\int_{J_k}\varphi_N\right|\le C|J|\le CL(J)N^{-1/2}. \tag*{(262)} ∫ J k φ N ≤ C ∣ J ∣ ≤ C L ( J ) N − 1/2 . (262) Combining these bounds with the first claim, we obtain the desired bound. For Y N ♯ ( J ) Y_N^\sharp(J) Y N ♯ ( J ) N N N Y N ♯ ( J ) = 0 Y_N^\sharp(J)=0 Y N ♯ ( J ) = 0 ∣ ε ψ N ∣ ≤ C N − 1 / 2 |\varepsilon\psi_N|\le CN^{-1/2} ∣ ε ψ N ∣ ≤ C N − 1/2 J J J c N ≤ 1 c_N\le1 c N ≤ 1
Lemma 6.7. The entries of M N ♯ ( J ) \mathbf{M}_N^\sharp(J) M N ♯ ( J ) 1 ≤ i , j ≤ r 1\le i,j\le r 1 ≤ i , j ≤ r 0 ≤ β ≤ r 0\le\beta\le r 0 ≤ β ≤ r
∣ M N ( J ) i 0 ∣ + ∣ M N ( J ) i j ∣ ≤ C e C L ( J ) , |\mathbf{M}_N(J)_{i0}|+|\mathbf{M}_N(J)_{ij}|\le Ce^{CL(J)}, ∣ M N ( J ) i 0 ∣ + ∣ M N ( J ) ij ∣ ≤ C e C L ( J ) , ∣ M N ( J ) 00 − 1 ∣ + ∣ M N ( J ) 0 j ∣ + ∣ z N ( J ) β ∣ + ∣ d N ( J ) − 1 ∣ + ∣ v N ( J ) 0 ∣ ≤ C e C L ( J ) N − 1 / 2 . (263) |\mathbf{M}_N(J)_{00}-1|+|\mathbf{M}_N(J)_{0j}|+|z_N(J)_\beta|+|d_N(J)-1|+|v_N(J)_0|\le Ce^{CL(J)}N^{-1/2}. \tag*{(263)} ∣ M N ( J ) 00 − 1∣ + ∣ M N ( J ) 0 j ∣ + ∣ z N ( J ) β ∣ + ∣ d N ( J ) − 1∣ + ∣ v N ( J ) 0 ∣ ≤ C e C L ( J ) N − 1/2 . (263) Proof. We first consider the entries in the columns j ≥ 1 j\ge1 j ≥ 1 Y Y Y
⟨ Y , Q N ( J ) − 1 k j ⟩ = Δ j [ ( Id − K J ) − 1 ( 1 J K N Y ) ] , (264) \langle Y,Q_N(J)^{-1}k_j\rangle=\Delta_j\left[(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_JK_NY)\right], \tag*{(264)} ⟨ Y , Q N ( J ) − 1 k j ⟩ = Δ j [ ( Id − K J ) − 1 ( 1 J K N Y ) ] , (264) and Lemma 5.3 bounds this by C e C L ( J ) sup J ∣ K N Y ∣ Ce^{CL(J)}\sup_J|K_NY| C e C L ( J ) sup J ∣ K N Y ∣ Y = w i Y=w_i Y = w i Y = Y 0 Y=Y_0 Y = Y 0 Y = Y N ♯ ( J ) Y=Y_N^\sharp(J) Y = Y N ♯ ( J ) M N ( J ) i j \mathbf{M}_N(J)_{ij} M N ( J ) ij M N ( J ) 0 j \mathbf{M}_N(J)_{0j} M N ( J ) 0 j z N ( J ) j z_N(J)_j z N ( J ) j
Next, we consider the column β = 0 \beta=0 β = 0 X 0 = − ψ N X_0=-\psi_N X 0 = − ψ N Y Y Y
⟨ Y , Q N ( J ) − 1 ψ N ⟩ = ⟨ Y , ψ N ⟩ + ⟨ 1 J K N Y , ( Id − K J ) − 1 ( 1 J ψ N ) ⟩ . (265) \langle Y,Q_N(J)^{-1}\psi_N\rangle=\langle Y,\psi_N\rangle+\langle\mathbf{1}_JK_NY,(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_J\psi_N)\rangle. \tag*{(265)} ⟨ Y , Q N ( J ) − 1 ψ N ⟩ = ⟨ Y , ψ N ⟩ + ⟨ 1 J K N Y , ( Id − K J ) − 1 ( 1 J ψ N )⟩ . (265) Using Proposition 5.1 and Lemma 4.2(a), we bound the second term in (265) by
∣ J ∣ 1 / 2 sup J ∣ K N Y ∣ ⋅ C e C L ( J ) ∥ 1 J ψ N ∥ 2 ≤ C e C L ( J ) sup J ∣ K N Y ∣ . (266) |J|^{1/2}\sup_J|K_NY|\cdot Ce^{CL(J)}\|\mathbf{1}_J\psi_N\|_2\le Ce^{CL(J)}\sup_J|K_NY|. \tag*{(266)} ∣ J ∣ 1/2 J sup ∣ K N Y ∣ ⋅ C e C L ( J ) ∥ 1 J ψ N ∥ 2 ≤ C e C L ( J ) J sup ∣ K N Y ∣. (266) It remains to bound ⟨ Y , ψ N ⟩ \langle Y,\psi_N\rangle ⟨ Y , ψ N ⟩ Y Y Y Y = w i Y=w_i Y = w i
⟨ w i , ψ N ⟩ = ⟨ ε a i , ψ N ⟩ − ⟨ 1 J ε a i , ψ N ⟩ = − ( ε ψ N ) ( a i ) − ⟨ 1 J ε a i , ψ N ⟩ . (267) \langle w_i,\psi_N\rangle=\langle\varepsilon a_i,\psi_N\rangle-\langle\mathbf{1}_J\varepsilon a_i,\psi_N\rangle=-(\varepsilon\psi_N)(a_i)-\langle\mathbf{1}_J\varepsilon a_i,\psi_N\rangle. \tag*{(267)} ⟨ w i , ψ N ⟩ = ⟨ ε a i , ψ N ⟩ − ⟨ 1 J ε a i , ψ N ⟩ = − ( ε ψ N ) ( a i ) − ⟨ 1 J ε a i , ψ N ⟩ . (267) which Lemma 4.5 (b) and Lemma 4.2 (a) bound by C C C Y = Y 0 Y=Y_0 Y = Y 0 ⟨ ω N , ψ N ⟩ = 0 \langle\omega_N,\psi_N\rangle=0 ⟨ ω N , ψ N ⟩ = 0 Lemma 2.6 (c) and the formula for Y 0 Y_0 Y 0 Lemma 6.6 to write
⟨ Y 0 , ψ N ⟩ = − ⟨ 1 J ω N , ψ N ⟩ − ∑ k = 1 r ( ∫ J k φ N ) ⟨ w k , ψ N ⟩ (268) \langle Y_0,\psi_N\rangle=-\langle\mathbf{1}_J\omega_N,\psi_N\rangle-\sum_{k=1}^{r}\left(\int_{J_k}\varphi_N\right)\langle w_k,\psi_N\rangle \tag*{(268)} ⟨ Y 0 , ψ N ⟩ = − ⟨ 1 J ω N , ψ N ⟩ − k = 1 ∑ r ( ∫ J k φ N ) ⟨ w k , ψ N ⟩ (268) which is bounded by C N − 1 / 2 ∣ J ∣ + C L ( J ) N − 1 / 2 CN^{-1/2}|J|+CL(J)N^{-1/2} C N − 1/2 ∣ J ∣ + C L ( J ) N − 1/2 Y = Y N ♯ ( J ) Y=Y_N^\sharp(J) Y = Y N ♯ ( J ) N N N
∣ ⟨ Y N ♯ ( J ) , ψ N ⟩ ∣ ≤ C N − 1 / 2 ∣ J ∣ + C ∣ J ∣ max k ∣ ⟨ w k , ψ N ⟩ ∣ (269) \left|\langle Y_N^\sharp(J),\psi_N\rangle\right|\le CN^{-1/2}|J|+C|J|\max_k\left|\langle w_k,\psi_N\rangle\right| \tag*{(269)} ⟨ Y N ♯ ( J ) , ψ N ⟩ ≤ C N − 1/2 ∣ J ∣ + C ∣ J ∣ k max ∣ ⟨ w k , ψ N ⟩ ∣ (269) and the right side can be bounded by C L ( J ) N − 1 / 2 CL(J)N^{-1/2} C L ( J ) N − 1/2 Lemma 6.6 , and recalling that M N ( J ) α 0 = 1 α = 0 + ⟨ Y α , Q N ( J ) − 1 ψ N ⟩ M_N(J)_{\alpha0}=\mathbf{1}_{\alpha=0}+\langle Y_\alpha,Q_N(J)^{-1}\psi_N\rangle M N ( J ) α 0 = 1 α = 0 + ⟨ Y α , Q N ( J ) − 1 ψ N ⟩ z N ( J ) 0 = − ⟨ Y N ♯ ( J ) , Q N ( J ) − 1 ψ N ⟩ z_N(J)_0=-\langle Y_N^\sharp(J),Q_N(J)^{-1}\psi_N\rangle z N ( J ) 0 = − ⟨ Y N ♯ ( J ) , Q N ( J ) − 1 ψ N ⟩ β = 0 \beta=0 β = 0
Finally, Lemma 4.2 (a) gives
∣ d N ( J ) − 1 ∣ = c N ∣ ∫ J ψ N ∣ ≤ C ∣ J ∣ (270) |d_N(J)-1|=c_N\left|\int_J\psi_N\right|\le C|J| \tag*{(270)} ∣ d N ( J ) − 1∣ = c N ∫ J ψ N ≤ C ∣ J ∣ (270) and ∣ v N ( J ) 0 ∣ ≤ C ∣ J ∣ |v_N(J)_0|\le C|J| ∣ v N ( J ) 0 ∣ ≤ C ∣ J ∣ ∣ J ∣ ≤ C L ( J ) N − 1 / 2 |J|\le CL(J)N^{-1/2} ∣ J ∣ ≤ C L ( J ) N − 1/2 L ( J ) ≤ e L ( J ) L(J)\le e^{L(J)} L ( J ) ≤ e L ( J )
Proof of Proposition 6.2 . We order the rows and columns of M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J ) 0 , 1 , … , r , r + 1 0,1,\ldots,r,r+1 0 , 1 , … , r , r + 1 r + 1 r+1 r + 1 (119) . Let E \mathbf{E} E M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J ) 0 0 0 e 0 e_0 e 0 r + 1 r+1 r + 1 e r + 1 e_{r+1} e r + 1 Lemma 6.7 implies that every entry of M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J ) C e C L ( J ) Ce^{CL(J)} C e C L ( J ) 0 0 0 r + 1 r+1 r + 1 e 0 e_0 e 0 e r + 1 e_{r+1} e r + 1 C e C L ( J ) N − 1 / 2 Ce^{CL(J)}N^{-1/2} C e C L ( J ) N − 1/2 r ≤ r 0 r\le r_0 r ≤ r 0
∣ det M N ♯ ( J ) − det E ∣ ≤ C e C L ( J ) N − 1 / 2 . (271) \left|\det M_N^\sharp(J)-\det E\right|\le Ce^{CL(J)}N^{-1/2}. \tag*{(271)} det M N ♯ ( J ) − det E ≤ C e C L ( J ) N − 1/2 . (271) Expanding det E \det E det E det E \det E det E M N ♯ ( J ) M_N^\sharp(J) M N ♯ ( J ) 1 , … , r 1,\ldots,r 1 , … , r A N ( J ) A_N(J) A N ( J ) (249) . The second bound follows from Lemma 6.7 and r ≤ r 0 r\le r_0 r ≤ r 0 Corollary 3.3 .
Separated intervals
We now show that the endpoint matrix of a union of well-separated intervals is close to the diagonal matrix formed by the endpoint matrices of the individual intervals.
Lemma 6.8. Under the assumptions (6.1), suppose that r ≥ 2 r \ge2 r ≥ 2 sep ( J ) ≥ 1 \operatorname{sep}(J) \ge1 sep ( J ) ≥ 1 1 ≤ i , j ≤ r 1 \le i,j \le r 1 ≤ i , j ≤ r i ≠ j i \ne j i = j
∣ A N ( J ) i j ∣ ≤ C e C L ( J ) ( sep ( J ) − 1 + N − 1 / 4 ) , \left|\mathcal{A}_N(J)_{ij}\right| \le C e^{CL(J)}\left(\operatorname{sep}(J)^{-1}+N^{-1/4}\right), ∣ A N ( J ) ij ∣ ≤ C e C L ( J ) ( sep ( J ) − 1 + N − 1/4 ) , ∣ A N ( J ) j j − A N ( J j ) ∣ ≤ C e C L ( J ) sep ( J ) − 1 , (272) \left|\mathcal{A}_N(J)_{jj}-\mathcal{A}_N(J_j)\right| \le C e^{CL(J)}\operatorname{sep}(J)^{-1}, \tag*{(272)} ∣ A N ( J ) j j − A N ( J j ) ∣ ≤ C e C L ( J ) sep ( J ) − 1 , (272) where A N ( J j ) \mathcal{A}_N(J_j) A N ( J j ) J j J_j J j
Proof. Throughout the proof, we write L = L ( J ) L=L(J) L = L ( J ) d = sep ( J ) \mathrm{d}=\operatorname{sep}(J) d = sep ( J ) K 0 K_0 K 0 L 2 ( J ) = ⨁ k L 2 ( J k ) L^2(J)=\bigoplus_k L^2(J_k) L 2 ( J ) = ⨁ k L 2 ( J k ) K J k K_{J_k} K J k L 2 ( J k ) L^2(J_k) L 2 ( J k ) R = K J − K 0 R=K_J-K_0 R = K J − K 0 K N ( x , y ) K_N(x,y) K N ( x , y ) x x x y y y J J J 0 0 0 ∣ K N ( x , y ) ∣ ≤ C N d − 1 \left|K_N(x,y)\right|\le C\sqrt{N}\mathrm{d}^{-1} ∣ K N ( x , y ) ∣ ≤ C N d − 1 x x x y y y R R R
∥ R ∥ 2 2 ≤ ∑ k ≠ l ∣ J k ∣ ∣ J l ∣ C N d − 2 ≤ C L 2 d − 2 . (273) \|R\|_2^2 \le\sum_{k\ne l}|J_k||J_l|CN\mathrm{d}^{-2}\le CL^2\mathrm{d}^{-2}. \tag*{(273)} ∥ R ∥ 2 2 ≤ k = l ∑ ∣ J k ∣∣ J l ∣ C N d − 2 ≤ C L 2 d − 2 . (273) By the resolvent identity, we have
( Id − K J ) − 1 = ( Id − K 0 ) − 1 + ( Id − K J ) − 1 R ( Id − K 0 ) − 1 . (274) (\operatorname{Id}-K_J)^{-1}=(\operatorname{Id}-K_0)^{-1}+(\operatorname{Id}-K_J)^{-1}R(\operatorname{Id}-K_0)^{-1}. \tag*{(274)} ( Id − K J ) − 1 = ( Id − K 0 ) − 1 + ( Id − K J ) − 1 R ( Id − K 0 ) − 1 . (274) Fix i i i
G = ( Id − K J ) − 1 ( 1 J g a i ) (275) G=(\operatorname{Id}-K_J)^{-1}(\mathbf{1}_{J}g_{a_i}) \tag*{(275)} G = ( Id − K J ) − 1 ( 1 J g a i ) (275) and
G 0 = ( Id − K 0 ) − 1 ( 1 J g a i ) . (276) G_0=(\operatorname{Id}-K_0)^{-1}(\mathbf{1}_{J}g_{a_i}). \tag*{(276)} G 0 = ( Id − K 0 ) − 1 ( 1 J g a i ) . (276) By Corollary 4.7(b), we have ∣ g a i ∣ ≤ C |g_{a_i}|\le C ∣ g a i ∣ ≤ C J J J ∣ g a i ( y ) ∣ ≤ C ( d − 1 + N − 1 / 4 ) |g_{a_i}(y)|\le C(\mathrm{d}^{-1}+N^{-1/4}) ∣ g a i ( y ) ∣ ≤ C ( d − 1 + N − 1/4 ) y ∈ J k y\in J_k y ∈ J k k ≠ i k\ne i k = i N ∣ y − a i ∣ ≥ d / 2 \sqrt{N}|y-a_i|\ge\mathrm{d}/\sqrt{2} N ∣ y − a i ∣ ≥ d / 2 J k J_k J k
∥ 1 J k G 0 ∥ 2 ≤ C e C L ∣ J ∣ 1 / 2 ( d − 1 + N − 1 / 4 ) , k ≠ i , ∥ G 0 ∥ 2 ≤ C e C L ∣ J ∣ 1 / 2 . (277) \|\mathbf{1}_{J_k}G_0\|_2\le Ce^{CL}|J|^{1/2}(\mathrm{d}^{-1}+N^{-1/4}),\qquad k\ne i,\qquad\|G_0\|_2\le Ce^{CL}|J|^{1/2}. \tag*{(277)} ∥ 1 J k G 0 ∥ 2 ≤ C e C L ∣ J ∣ 1/2 ( d − 1 + N − 1/4 ) , k = i , ∥ G 0 ∥ 2 ≤ C e C L ∣ J ∣ 1/2 . (277) Combining (6.30) and (6.29) with Proposition 5.1, we also have
∥ G − G 0 ∥ 2 ≤ C e C L ∥ R ∥ 2 ∥ G 0 ∥ 2 ≤ C e C L d − 1 ∣ J ∣ 1 / 2 , (278) \|G-G_0\|_2\le Ce^{CL}\|R\|_2\|G_0\|_2\le Ce^{CL}\mathrm{d}^{-1}|J|^{1/2}, \tag*{(278)} ∥ G − G 0 ∥ 2 ≤ C e C L ∥ R ∥ 2 ∥ G 0 ∥ 2 ≤ C e C L d − 1 ∣ J ∣ 1/2 , (278) where in the last inequality we absorbed the factor L L L e C L e^{CL} e C L
We now evaluate G G G t ∈ { a j , b j } t\in\{a_j,b_j\} t ∈ { a j , b j } j ≠ i j\ne i j = i
G ( t ) = g a i ( t ) + ∑ k = 1 r ∫ J k K N ( t , z ) G ( z ) d z . (279) G(t)=g_{a_i}(t)+\sum_{k=1}^{r}\int_{J_k}K_N(t,z)G(z)\,\mathrm{d}z. \tag*{(279)} G ( t ) = g a i ( t ) + k = 1 ∑ r ∫ J k K N ( t , z ) G ( z ) d z . (279) The first term is bounded by C ( d − 1 + N − 1 / 4 ) C(\mathrm{d}^{-1}+N^{-1/4}) C ( d − 1 + N − 1/4 ) N ∣ t − a i ∣ ≥ d / 2 \sqrt{N}|t-a_i|\ge\mathrm{d}/\sqrt{2} N ∣ t − a i ∣ ≥ d / 2 k ≠ j k\ne j k = j ∣ K N ( t , z ) ∣ ≤ C N d − 1 |K_N(t,z)|\le C\sqrt{N}\mathrm{d}^{-1} ∣ K N ( t , z ) ∣ ≤ C N d − 1 z ∈ J k z\in J_k z ∈ J k k k k
C N d − 1 ∣ J ∣ 1 / 2 ∥ G ∥ 2 ≤ C e C L L d − 1 . (280) C\sqrt{N}\mathrm{d}^{-1}|J|^{1/2}\|G\|_2\le Ce^{CL}L\mathrm{d}^{-1}. \tag*{(280)} C N d − 1 ∣ J ∣ 1/2 ∥ G ∥ 2 ≤ C e C L L d − 1 . (280) For k = j k=j k = j Lemma 4.2 (b) give
∣ ∫ J j K N ( t , z ) G ( z ) d z ∣ ≤ C N 1 / 4 ( ∥ 1 J j G 0 ∥ 2 + ∥ G − G 0 ∥ 2 ) ≤ C e C L d − 1 + N − 1 / 4 , (281) \left|\int_{J_j} K_N(t,z)G(z)\,\mathrm{d}z\right| \le CN^{1/4}\left(\|\mathbf{1}_{J_j}G_0\|_2+\|G-G_0\|_2\right) \le Ce^{CL}d^{-1}+N^{-1/4}, \tag*{(281)} ∫ J j K N ( t , z ) G ( z ) d z ≤ C N 1/4 ( ∥ 1 J j G 0 ∥ 2 + ∥ G − G 0 ∥ 2 ) ≤ C e C L d − 1 + N − 1/4 , (281) where we used N 1 / 4 ∣ J ∣ 1 / 2 ≤ L 1 / 2 N^{1/4}|J|^{1/2}\le L^{1/2} N 1/4 ∣ J ∣ 1/2 ≤ L 1/2 j ≠ i j\ne i j = i t = b j t=b_j t = b j t = a j t=a_j t = a j (272) .
To prove the second bound, we define
G ( i ) = ( Id − K J i ) − 1 ( 1 J i g a i ) . (282) G^{(i)}=(\operatorname{Id}-K_{J_i})^{-1}(\mathbf{1}_{J_i}g_{a_i}). \tag*{(282)} G ( i ) = ( Id − K J i ) − 1 ( 1 J i g a i ) . (282) By the definition of the endpoint matrix, A N ( J i ) = 1 − Δ i G ( i ) A_N(J_i)=1-\Delta_iG^{(i)} A N ( J i ) = 1 − Δ i G ( i ) 1 J i G 0 = G ( i ) \mathbf{1}_{J_i}G_0=G^{(i)} 1 J i G 0 = G ( i ) t ∈ { a i , b i } t\in\{a_i,b_i\} t ∈ { a i , b i }
G ( t ) − G ( i ) ( t ) = ∑ k ≠ i ∫ J k K N ( t , z ) G ( z ) d z + ∫ J i K N ( t , z ) ( G − G 0 ) ( z ) d z . (283) G(t)-G^{(i)}(t)=\sum_{k\ne i}\int_{J_k}K_N(t,z)G(z)\,\mathrm{d}z+\int_{J_i}K_N(t,z)(G-G_0)(z)\,\mathrm{d}z. \tag*{(283)} G ( t ) − G ( i ) ( t ) = k = i ∑ ∫ J k K N ( t , z ) G ( z ) d z + ∫ J i K N ( t , z ) ( G − G 0 ) ( z ) d z . (283) As above, the first sum is bounded by C e C L L d − 1 Ce^{CL}Ld^{-1} C e C L L d − 1 (278) , we bound the last term by C N 1 / 4 ∥ G − G 0 ∥ 2 ≤ C e C L d − 1 CN^{1/4}\|G-G_0\|_2\le Ce^{CL}d^{-1} C N 1/4 ∥ G − G 0 ∥ 2 ≤ C e C L d − 1 (272) , after renaming i i i j j j
The hole probability of one interval
In this section, we prove that the GOE hole probability of an interval ( x 0 , x 0 + s / ( 2 ϱ N ( x 0 ) ) ) (x_0,x_0+s/(2\varrho_N(x_0))) ( x 0 , x 0 + s / ( 2 ϱ N ( x 0 ))) s ≤ γ log N s \le\gamma\sqrt{\log N} s ≤ γ log N q 1 q_1 q 1 N N N
Sine-kernel determinants
Recall from (13) that D ± ( t ) = det ( Id − S ± , t ) D_\pm(t)=\det(\operatorname{Id}-S_{\pm,t}) D ± ( t ) = det ( Id − S ± , t ) t > 0 t>0 t > 0 S ± , t S_{\pm,t} S ± , t L 2 ( 0 , t ) L^2(0,t) L 2 ( 0 , t ) S ( u − v ) ± S ( u + v ) S(u-v)\pm S(u+v) S ( u − v ) ± S ( u + v ) S S S (159) . We set D ± ( 0 ) = 1 D_\pm(0)=1 D ± ( 0 ) = 1 S ± , t S_{\pm,t} S ± , t S ( u − v ) S(u-v) S ( u − v ) L 2 ( − t , t ) L^2(-t,t) L 2 ( − t , t ) [ 0 , 1 ) [0,1) [ 0 , 1 )
D + ( t ) D − ( t ) = det ( Id − S ( − t , t ) ) = det ( Id − S ( 0 , 2 t ) ) , (284) D_+(t)D_-(t)=\det(\operatorname{Id}-S_{(-t,t)})=\det(\operatorname{Id}-S_{(0,2t)}), \tag*{(284)} D + ( t ) D − ( t ) = det ( Id − S ( − t , t ) ) = det ( Id − S ( 0 , 2 t ) ) , (284) where S A S_A S A L 2 ( A ) L^2(A) L 2 ( A ) S ( u − v ) S(u-v) S ( u − v ) [48 Section IV] or [16 (4)] . The functions D ± D_\pm D ± [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) S + , t S_{+,t} S + , t S + , t ′ S_{+,t'} S + , t ′ t < t ′ t<t' t < t ′ t t t q 1 ( s ) = D + ( s / 4 ) q_1(s)=D_+(s/4) q 1 ( s ) = D + ( s /4 )
μ ( s ) = D + ( s / 4 ) D − ( s / 4 ) , s ≥ 0 , (285) \mu(s)=\frac{D_+(s/4)}{D_-(s/4)},\qquad s\geq0, \tag*{(285)} μ ( s ) = D − ( s /4 ) D + ( s /4 ) , s ≥ 0 , (285) and we set q 1 ( 0 ) = 1 q_1(0)=1 q 1 ( 0 ) = 1 q 1 q_1 q 1 § 2.2 , is not used in our proofs.) Recall from (14) the asymptotics of D ± ( t ) D_\pm(t) D ± ( t ) t → ∞ t\to\infty t → ∞ [16 Theorem 4.8] .
Lemma 7.1. Recall the function F ( s ) = s 2 / 64 + s / 8 F(s)=s^2/64+s/8 F ( s ) = s 2 /64 + s /8 and the constant κ 1 \kappa_1 κ 1 from (3) . Then the following statements hold.
(a) We have q 1 ( s ) = κ 1 s − 1 / 8 e − F ( s ) ( 1 + o ( 1 ) ) q_1(s)=\kappa_1s^{-1/8}e^{-F(s)}(1+o(1)) q 1 ( s ) = κ 1 s − 1/8 e − F ( s ) ( 1 + o ( 1 )) s → ∞ s\to\infty s → ∞ q 1 q_1 q 1
(b) We have μ ( s ) = 2 e − s / 4 ( 1 + o ( 1 ) ) \mu(s)=\sqrt{2}e^{-s/4}(1+o(1)) μ ( s ) = 2 e − s /4 ( 1 + o ( 1 )) s → ∞ s\to\infty s → ∞ c > 0 c>0 c > 0 μ ( s ) ≥ c e − s / 4 \mu(s)\geq ce^{-s/4} μ ( s ) ≥ c e − s /4 s ≥ 0 s\geq0 s ≥ 0
Proof. Setting t = s / 4 t=s/4 t = s /4 (14) with the upper sign, we find
log q 1 ( s ) = − s 2 64 − s 8 − 1 8 log s + 1 4 log 2 + 7 24 log 2 + 3 2 ζ ′ ( − 1 ) + o ( 1 ) , (286) \log q_1(s)=-\frac{s^2}{64}-\frac{s}{8}-\frac{1}{8}\log s+\frac{1}{4}\log2+\frac{7}{24}\log2+\frac{3}{2}\zeta'(-1)+o(1), \tag*{(286)} log q 1 ( s ) = − 64 s 2 − 8 s − 8 1 log s + 4 1 log 2 + 24 7 log 2 + 2 3 ζ ′ ( − 1 ) + o ( 1 ) , (286) which is the asymptotic formula in (a), since 1 / 4 + 7 / 24 = 13 / 24 1/4 + 7/24 = 13/24 1/4 + 7/24 = 13/24 (14) , we find
log μ ( s ) = − s / 4 + ( log 2 ) / 2 + o ( 1 ) . (287) \log\mu(s) = -s/4 + (\log2)/2 + o(1). \tag*{(287)} log μ ( s ) = − s /4 + ( log 2 ) /2 + o ( 1 ) . (287) Since μ \mu μ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
Next, we identify the limit of the endpoint matrix of one interval. For ℓ > 0 \ell> 0 ℓ > 0
A s i n e ( ℓ ) = 1 − ( ( Id − S ( 0 , ℓ ) ) − 1 g s i n e ) ( ℓ − ) − ( ( Id − S ( 0 , ℓ ) ) − 1 g s i n e ) ( 0 + ) , (288) A_{\mathrm{sine}}(\ell) = 1 - \left((\operatorname{Id} - S_{(0,\ell)})^{-1}g_{\mathrm{sine}}\right)(\ell^-) - \left((\operatorname{Id} - S_{(0,\ell)})^{-1}g_{\mathrm{sine}}\right)(0^+), \tag*{(288)} A sine ( ℓ ) = 1 − ( ( Id − S ( 0 , ℓ ) ) − 1 g sine ) ( ℓ − ) − ( ( Id − S ( 0 , ℓ ) ) − 1 g sine ) ( 0 + ) , (288) where g s i n e g_{\mathrm{sine}} g sine (219) , and the values at the endpoints are defined through the analogue of (242) for S S S g s i n e ( 0 + ) = − 1 / 2 g_{\mathrm{sine}}(0^+)=-1/2 g sine ( 0 + ) = − 1/2
Lemma 7.2. For all ℓ > 0 \ell> 0 ℓ > 0 A s i n e ( ℓ ) = D + ( ℓ / 2 ) / D − ( ℓ / 2 ) A_{\mathrm{sine}}(\ell) = D_+(\ell/2)/D_-(\ell/2) A sine ( ℓ ) = D + ( ℓ /2 ) / D − ( ℓ /2 ) A s i n e ( s / 2 ) = μ ( s ) A_{\mathrm{sine}}(s/2)=\mu(s) A sine ( s /2 ) = μ ( s ) s > 0 s>0 s > 0
Proof. Set t = ℓ / 2 t=\ell/2 t = ℓ /2 J = ( − t , t ) J=(-t,t) J = ( − t , t )
Δ G = G ( t − ) − G ( ( − t ) + ) . (289) \Delta G = G(t^-) - G((-t)^+). \tag*{(289)} Δ G = G ( t − ) − G (( − t ) + ) . (289) Since the sine kernel is translation invariant and g s i n e g_{\mathrm{sine}} g sine
A s i n e ( ℓ ) = 1 − Δ [ ( Id − S J ) − 1 ( 1 J g s i n e ( ⋅ + t ) ) ] . (290) A_{\mathrm{sine}}(\ell) = 1 - \Delta\left[(\operatorname{Id}-S_J)^{-1}(1_Jg_{\mathrm{sine}}(\cdot+t))\right]. \tag*{(290)} A sine ( ℓ ) = 1 − Δ [ ( Id − S J ) − 1 ( 1 J g sine ( ⋅ + t )) ] . (290) Let R ( x , y ) R(x,y) R ( x , y ) x ∈ R x\in\mathbb{R} x ∈ R y ∈ J ‾ y\in\overline{J} y ∈ J ( Id − S 1 J ) − 1 S 1 J (\operatorname{Id}-S1_J)^{-1}S1_J ( Id − S 1 J ) − 1 S 1 J
R ( x , y ) = S ( x − y ) + ∫ J S ( x − z ) R ( z , y ) d z , (291) R(x,y)=S(x-y)+\int_J S(x-z)R(z,y)\,\mathrm{d}z, \tag*{(291)} R ( x , y ) = S ( x − y ) + ∫ J S ( x − z ) R ( z , y ) d z , (291) and R R R J × J J\times J J × J
Set w = 1 J c ε − t w=1_{J^c}\varepsilon_{-t} w = 1 J c ε − t [48 (19), (25), (30)] that
1 − ∫ J c w ( x ) ( R ( x , t ) − R ( x , − t ) ) d x = D + ( t ) D − ( t ) , (292) 1-\int_{J^c}w(x)\left(R(x,t)-R(x,-t)\right)\,\mathrm{d}x=\frac{D_+(t)}{D_-(t)}, \tag*{(292)} 1 − ∫ J c w ( x ) ( R ( x , t ) − R ( x , − t ) ) d x = D − ( t ) D + ( t ) , (292) where the integral over J c J^c J c M → ∞ M\to\infty M → ∞ J c ∩ [ − M , M ] J^c\cap[-M,M] J c ∩ [ − M , M ]
a ± = ∫ J c w ( x ) R ( x , ± t ) d x . (293) a_\pm=\int_{J^c}w(x)R(x,\pm t)\,\mathrm{d}x. \tag*{(293)} a ± = ∫ J c w ( x ) R ( x , ± t ) d x . (293) Since 1 J c ε − t = 1 J c ε t 1_{J^c}\varepsilon_{-t}=1_{J^c}\varepsilon_t 1 J c ε − t = 1 J c ε t [48 (19)] with m = 1 m=1 m = 1
det ( 1 + a − − a + a − 1 − a + ) = 1 − ( a + − a − ) , (294) \det\begin{pmatrix}1+a_- & -a_+ \\ a_- & 1-a_+\end{pmatrix}=1-(a_+-a_-), \tag*{(294)} det ( 1 + a − a − − a + 1 − a + ) = 1 − ( a + − a − ) , (294) which is the left side of (7.9). This determinant is computed in [48 (25)] and is equal to D + ( t ) / D − ( t ) D_{+}(t)/D_{-}(t) D + ( t ) / D − ( t ) [48 (30)] $.
It remains to identify the left side of (7.9) with A s i n e ( ℓ ) \mathcal{A}_{\mathrm{sine}}(\ell) A sine ( ℓ ) z ∈ J ‾ z \in\overline{J} z ∈ J
Φ ( z ) = lim M → ∞ ∫ J c ∩ [ − M , M ] w ( x ) S ( x − z ) d x . (295) \Phi(z)=\lim_{M\to\infty}\int_{J^c\cap[-M,M]}w(x)S(x-z)\,\mathrm{d}x. \tag*{(295)} Φ ( z ) = M → ∞ lim ∫ J c ∩ [ − M , M ] w ( x ) S ( x − z ) d x . (295) A direct computation using ∫ 0 ∞ S = 1 / 2 \int_{0}^{\infty}S=1/2 ∫ 0 ∞ S = 1/2
lim M → ∞ ∫ − M M ε − t ( x ) S ( x − z ) d x = ∫ 0 z + t S ( v ) d v = ε − t ( z ) + g s i n e ( z + t ) , (296) \lim_{M\to\infty}\int_{-M}^{M}\varepsilon_{-t}(x)S(x-z)\,\mathrm{d}x=\int_{0}^{z+t}S(v)\,\mathrm{d}v=\varepsilon_{-t}(z)+g_{\mathrm{sine}}(z+t), \tag*{(296)} M → ∞ lim ∫ − M M ε − t ( x ) S ( x − z ) d x = ∫ 0 z + t S ( v ) d v = ε − t ( z ) + g sine ( z + t ) , (296) and the convergence is uniform in z ∈ J ‾ z \in\overline{J} z ∈ J ∣ ∫ M ∞ S ∣ ≤ 2 / M \left|\int_{M}^{\infty}S\right|\le2/M ∫ M ∞ S ≤ 2/ M
Φ = ( Id − S J ) ( 1 J ε − t ) + 1 J g s i n e ( ⋅ + t ) . (297) \Phi=(\operatorname{Id}-S_J)(\mathbf{1}_J\varepsilon_{-t})+\mathbf{1}_Jg_{\mathrm{sine}}(\mathord{\cdot}+t). \tag*{(297)} Φ = ( Id − S J ) ( 1 J ε − t ) + 1 J g sine ( ⋅ + t ) . (297) Inserting the equation for R R R [ − M , M ] [-M,M] [ − M , M ] J J J
∫ J c w ( x ) R ( x , y ) d x = Φ ( y ) + ∫ J Φ ( z ) R ( z , y ) d z = ( ( Id − S J ) − 1 Φ ) ( y ) , y = ± t . (298) \int_{J^c}w(x)R(x,y)\,\mathrm{d}x=\Phi(y)+\int_J\Phi(z)R(z,y)\,\mathrm{d}z=((\operatorname{Id}-S_J)^{-1}\Phi)(y),\qquad y=\pm t. \tag*{(298)} ∫ J c w ( x ) R ( x , y ) d x = Φ ( y ) + ∫ J Φ ( z ) R ( z , y ) d z = (( Id − S J ) − 1 Φ ) ( y ) , y = ± t . (298) Since ε − t \varepsilon_{-t} ε − t 1 / 2 1/2 1/2 J J J
Δ [ ( Id − S J ) − 1 Φ ] = Δ [ ( Id − S J ) − 1 ( 1 J g s i n e ( ⋅ + t ) ) ] , (299) \Delta[(\operatorname{Id}-S_J)^{-1}\Phi]=\Delta[(\operatorname{Id}-S_J)^{-1}(\mathbf{1}_Jg_{\mathrm{sine}}(\mathord{\cdot}+t))], \tag*{(299)} Δ [( Id − S J ) − 1 Φ ] = Δ [( Id − S J ) − 1 ( 1 J g sine ( ⋅ + t ))] , (299) which completes the proof.
The comparison
Proposition 7.3. Let δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ] γ > 0 \gamma>0 γ > 0 C , N 0 > 0 C,N_0>0 C , N 0 > 0 δ \delta δ γ \gamma γ N ≥ N 0 N\ge N_0 N ≥ N 0 x 0 ∈ B N ( δ ) x_0\in B_N(\delta) x 0 ∈ B N ( δ ) ϱ = ϱ N ( x 0 ) \varrho=\varrho_N(x_0) ϱ = ϱ N ( x 0 ) 0 < s ≤ γ log N 0<s\le\gamma\sqrt{\log N} 0 < s ≤ γ log N J = ( x 0 , x 0 + s / ( 2 ϱ ) ) J=(x_0,x_0+s/(2\varrho)) J = ( x 0 , x 0 + s / ( 2 ϱ ))
∣ P N ( J ) q 1 ( s ) − 1 ∣ ≤ C e C s N − 1 / 2 , ∣ D N ( J ) D + ( s / 4 ) D − ( s / 4 ) − 1 ∣ ≤ C e C s N − 1 / 2 . (300) \left|\frac{P_N(J)}{q_1(s)}-1\right|\le Ce^{Cs}N^{-1/2},\qquad\left|\frac{D_N(J)}{D_+(s/4)D_-(s/4)}-1\right|\le Ce^{Cs}N^{-1/2}. \tag*{(300)} q 1 ( s ) P N ( J ) − 1 ≤ C e C s N − 1/2 , D + ( s /4 ) D − ( s /4 ) D N ( J ) − 1 ≤ C e C s N − 1/2 . (300) Before giving the proof, we record a standard bound for determinants. We let ∥ Z ∥ 1 \lVert Z\rVert_1 ∥ Z ∥ 1 ∥ Z ∥ 2 \lVert Z\rVert_2 ∥ Z ∥ 2 Z Z Z
Lemma 7.4. Let Z Z Z ∥ Z ∥ 2 ≤ 1 / 2 \lVert Z\rVert_2\le1/2 ∥ Z ∥ 2 ≤ 1/2 ( y k ) (y_k) ( y k ) log det ( Id − Z ) = ∑ k log ( 1 − y k ) \log\det(\operatorname{Id}-Z)=\sum_k\log(1-y_k) log det ( Id − Z ) = ∑ k log ( 1 − y k ) log \log log
∣ log det ( Id − Z ) + Tr Z ∣ ≤ ∥ Z ∥ 2 2 . (301) \left|\log\det(\operatorname{Id}-Z)+\operatorname{Tr}Z\right|\le\lVert Z\rVert_2^2. \tag*{(301)} ∣ log det ( Id − Z ) + Tr Z ∣ ≤ ∥ Z ∥ 2 2 . (301) Proof. Recall that, by Lidskii’s theorem,
Tr Z = ∑ k y k (302) \operatorname{Tr} Z = \sum_k y_k \tag*{(302)} Tr Z = k ∑ y k (302) and
det ( Id − Z ) = ∏ k ( 1 − y k ) , (303) \det(\operatorname{Id} - Z) = \prod_k (1-y_k), \tag*{(303)} det ( Id − Z ) = k ∏ ( 1 − y k ) , (303) and that, by Weyl’s inequality (see [43 ] ),
∑ k ∣ y k ∣ 2 ≤ ∥ Z ∥ 2 2 . (304) \sum_k \lvert y_k\rvert^2 \leq\lVert Z\rVert_2^2. \tag*{(304)} k ∑ ∣ y k ∣ 2 ≤ ∥ Z ∥ 2 2 . (304) In particular, ∣ y k ∣ ≤ 1 / 2 \lvert y_k\rvert\leq1/2 ∣ y k ∣ ≤ 1/2 k k k ∣ log ( 1 − y ) + y ∣ ≤ ∣ y ∣ 2 \lvert\log(1-y)+y\rvert\leq\lvert y\rvert^2 ∣ log ( 1 − y ) + y ∣ ≤ ∣ y ∣ 2 y y y ∣ y ∣ ≤ 1 / 2 \lvert y\rvert\leq1/2 ∣ y ∣ ≤ 1/2
Proof of Proposition 7.3. Set ℓ = s / 2 \ell=s/2 ℓ = s /2 J = ( x 0 , x 0 + ℓ / ϱ ) J=(x_0,x_0+\ell/\varrho) J = ( x 0 , x 0 + ℓ / ϱ )
L ( J ) = 2 N ℓ / ϱ ≤ C s . (305) L(J)=\sqrt{2N}\ell/\varrho\leq Cs. \tag*{(305)} L ( J ) = 2 N ℓ / ϱ ≤ C s . (305) Since s ≤ γ log N s\leq\gamma\sqrt{\log N} s ≤ γ log N J ⊂ B N ( δ / 2 ) J\subset B_N(\delta/2) J ⊂ B N ( δ /2 ) L ( J ) ≤ N 1 / 8 L(J)\leq N^{1/8} L ( J ) ≤ N 1/8 C e C s N − 1 / 2 ≤ N − 1 / 4 Ce^{Cs}N^{-1/2}\leq N^{-1/4} C e C s N − 1/2 ≤ N − 1/4 C C C N 0 N_0 N 0 δ / 2 \delta/2 δ /2 δ \delta δ
We define a map from L 2 ( J ) L^2(J) L 2 ( J ) L 2 ( 0 , ℓ ) L^2(0,\ell) L 2 ( 0 , ℓ )
( U f ) ( u ) = ϱ − 1 / 2 f ( x 0 + u / ϱ ) . (306) (Uf)(u)=\varrho^{-1/2}f(x_0+u/\varrho). \tag*{(306)} ( U f ) ( u ) = ϱ − 1/2 f ( x 0 + u / ϱ ) . (306) This map is unitary. Let K ℓ K_\ell K ℓ L 2 ( 0 , ℓ ) L^2(0,\ell) L 2 ( 0 , ℓ )
K ( u , v ) = ϱ − 1 K N ( x 0 + u / ϱ , x 0 + v / ϱ ) . (307) K(u,v)=\varrho^{-1}K_N(x_0+u/\varrho,x_0+v/\varrho). \tag*{(307)} K ( u , v ) = ϱ − 1 K N ( x 0 + u / ϱ , x 0 + v / ϱ ) . (307) Note that U K J U − 1 UK_JU^{-1} U K J U − 1 S ℓ = S ( 0 , ℓ ) S_\ell=S_{(0,\ell)} S ℓ = S ( 0 , ℓ ) Ξ = K ℓ − S ℓ \Xi=K_\ell-S_\ell Ξ = K ℓ − S ℓ ∣ Ξ ( u , v ) ∣ ≤ C N − 1 / 2 \lvert\Xi(u,v)\rvert\leq CN^{-1/2} ∣ Ξ ( u , v )∣ ≤ C N − 1/2 u , v ∈ [ 0 , ℓ ] u,v\in[0,\ell] u , v ∈ [ 0 , ℓ ] ∥ ( Id − K ℓ ) − 1 ∥ \lVert(\operatorname{Id}-K_\ell)^{-1}\rVert ∥( Id − K ℓ ) − 1 ∥ ∥ ( Id − S ℓ ) − 1 ∥ \lVert(\operatorname{Id}-S_\ell)^{-1}\rVert ∥( Id − S ℓ ) − 1 ∥ C e C s Ce^{Cs} C e C s
We first prove the second bound in (7.17). By (7.1), we have D + ( s / 4 ) D − ( s / 4 ) = det ( Id − S ℓ ) D_+(s/4)D_-(s/4)=\det(\operatorname{Id}-S_\ell) D + ( s /4 ) D − ( s /4 ) = det ( Id − S ℓ ) D N ( J ) = det ( Id − K ℓ ) D_N(J)=\det(\operatorname{Id}-K_\ell) D N ( J ) = det ( Id − K ℓ ) Z = ( Id − S ℓ ) − 1 Ξ Z=(\operatorname{Id}-S_\ell)^{-1}\Xi Z = ( Id − S ℓ ) − 1 Ξ
det ( Id − K ℓ ) = det ( Id − S ℓ ) det ( Id − Z ) . (308) \det(\operatorname{Id}-K_\ell)=\det(\operatorname{Id}-S_\ell)\det(\operatorname{Id}-Z). \tag*{(308)} det ( Id − K ℓ ) = det ( Id − S ℓ ) det ( Id − Z ) . (308) We have ∥ Ξ ∥ 2 ≤ C ℓ N − 1 / 2 \lVert\Xi\rVert_2\leq C\ell N^{-1/2} ∥ Ξ ∥ 2 ≤ C ℓ N − 1/2 ∥ Z ∥ 2 ≤ C e C s N − 1 / 2 \lVert Z\rVert_2\leq Ce^{Cs}N^{-1/2} ∥ Z ∥ 2 ≤ C e C s N − 1/2 K ℓ K_\ell K ℓ S ℓ S_\ell S ℓ ∣ Tr Ξ ∣ ≤ C ℓ N − 1 / 2 \lvert\operatorname{Tr}\Xi\rvert\leq C\ell N^{-1/2} ∣ Tr Ξ ∣ ≤ C ℓ N − 1/2 ( Id − S ℓ ) − 1 = Id + S ℓ ( Id − S ℓ ) − 1 (\operatorname{Id}-S_\ell)^{-1}=\operatorname{Id}+S_\ell(\operatorname{Id}-S_\ell)^{-1} ( Id − S ℓ ) − 1 = Id + S ℓ ( Id − S ℓ ) − 1
∣ Tr Z ∣ ≤ ∣ Tr Ξ ∣ + ∣ Tr ( S ℓ ( Id − S ℓ ) − 1 Ξ ) ∣ ≤ ∣ Tr Ξ ∣ + ∥ S ℓ ∥ 2 ∥ ( Id − S ℓ ) − 1 ∥ ∥ Ξ ∥ 2 ≤ C e C s N − 1 / 2 , (309) \begin{aligned}
\lvert\operatorname{Tr}Z\rvert
&\leq\lvert\operatorname{Tr}\Xi\rvert+\lvert\operatorname{Tr}(S_\ell(\operatorname{Id}-S_\ell)^{-1}\Xi)\rvert\\
&\leq\lvert\operatorname{Tr}\Xi\rvert+\lVert S_\ell\rVert_2\lVert(\operatorname{Id}-S_\ell)^{-1}\rVert\lVert\Xi\rVert_2
\leq Ce^{Cs}N^{-1/2}, \tag*{(309)}
\end{aligned} ∣ Tr Z ∣ ≤ ∣ Tr Ξ ∣ + ∣ Tr ( S ℓ ( Id − S ℓ ) − 1 Ξ )∣ ≤ ∣ Tr Ξ ∣ + ∥ S ℓ ∥ 2 ∥( Id − S ℓ ) − 1 ∥ ∥ Ξ ∥ 2 ≤ C e C s N − 1/2 , (309) where we used ∥ S ℓ ∥ 2 ≤ ℓ / π \lVert S_{\ell}\rVert_{2} \le\ell/\pi ∥ S ℓ ∥ 2 ≤ ℓ / π ∣ log det ( Id − Z ) ∣ ≤ C e C s N − 1 / 2 \lvert\log\det(\operatorname{Id}-Z)\rvert\le Ce^{C s}N^{-1/2} ∣ log det ( Id − Z )∣ ≤ C e C s N − 1/2 (300) .
Next, we compare the endpoint matrices. We define
G = ( Id − K J ) − 1 ( 1 J g x 0 ) . (310) G=(\operatorname{Id}-K_J)^{-1}(1_Jg_{x_0}). \tag*{(310)} G = ( Id − K J ) − 1 ( 1 J g x 0 ) . (310) The endpoint matrix of J J J A N ( J ) = 1 − Δ 1 G A_N(J)=1-\Delta_1G A N ( J ) = 1 − Δ 1 G
g ~ ( u ) = g x 0 ( x 0 + u / ϱ ) (311) \widetilde{g}(u)=g_{x_0}(x_0+u/\varrho) \tag*{(311)} g ( u ) = g x 0 ( x 0 + u / ϱ ) (311) and
G ~ ( u ) = G ( x 0 + u / ϱ ) (312) \widetilde{G}(u)=G(x_0+u/\varrho) \tag*{(312)} G ( u ) = G ( x 0 + u / ϱ ) (312) for u ∈ ( 0 , ℓ ) u\in(0,\ell) u ∈ ( 0 , ℓ ) (242) and the change of variables z = x 0 + v / ϱ z=x_0+v/\varrho z = x 0 + v / ϱ
G ~ = g ~ + K ℓ G ~ , (313) \widetilde{G}=\widetilde{g}+K_{\ell}\widetilde{G}, \tag*{(313)} G = g + K ℓ G , (313) and this identity extends to u ∈ { 0 , ℓ } u\in\{0,\ell\} u ∈ { 0 , ℓ }
Let G s i n e = ( Id − S ℓ ) − 1 g s i n e G_{\mathrm{sine}}=(\operatorname{Id}-S_{\ell})^{-1}g_{\mathrm{sine}} G sine = ( Id − S ℓ ) − 1 g sine ∣ g ~ ( u ) − g s i n e ( u ) ∣ ≤ C ( 1 + ℓ ) N − 1 / 2 \lvert\widetilde{g}(u)-g_{\mathrm{sine}}(u)\rvert\le C(1+\ell)N^{-1/2} ∣ g ( u ) − g sine ( u )∣ ≤ C ( 1 + ℓ ) N − 1/2 u ∈ ( 0 , ℓ ) u\in(0,\ell) u ∈ ( 0 , ℓ )
G ~ − G s i n e = ( Id − K ℓ ) − 1 ( ( g ~ − g s i n e ) + Ξ G s i n e ) (314) \widetilde{G}-G_{\mathrm{sine}}=(\operatorname{Id}-K_{\ell})^{-1}\left((\widetilde{g}-g_{\mathrm{sine}})+\Xi G_{\mathrm{sine}}\right) \tag*{(314)} G − G sine = ( Id − K ℓ ) − 1 ( ( g − g sine ) + Ξ G sine ) (314) and that ∥ G s i n e ∥ 2 ≤ C e C s ℓ 1 / 2 \lVert G_{\mathrm{sine}}\rVert_2\le Ce^{Cs}\ell^{1/2} ∥ G sine ∥ 2 ≤ C e C s ℓ 1/2
∥ G ~ − G s i n e ∥ 2 ≤ C e C s N − 1 / 2 . (315) \lVert\widetilde{G}-G_{\mathrm{sine}}\rVert_2\le Ce^{Cs}N^{-1/2}. \tag*{(315)} ∥ G − G sine ∥ 2 ≤ C e C s N − 1/2 . (315) For u ∈ { 0 , ℓ } u\in\{0,\ell\} u ∈ { 0 , ℓ }
∣ G ~ ( u ) − G s i n e ( u ) ∣ ≤ ∣ g ~ ( u ) − g s i n e ( u ) ∣ + ∫ 0 ℓ ∣ Ξ ( u , v ) ∣ ∣ G ~ ( v ) ∣ d v + ∫ 0 ℓ ∣ S ( u − v ) ∣ ∣ G ~ ( v ) − G s i n e ( v ) ∣ d v ≤ C e C s N − 1 / 2 , (316) \begin{aligned}
\lvert\widetilde{G}(u)-G_{\mathrm{sine}}(u)\rvert
&\le\lvert\widetilde{g}(u)-g_{\mathrm{sine}}(u)\rvert+\int_0^\ell\lvert\Xi(u,v)\rvert\lvert\widetilde{G}(v)\rvert\,\mathrm{d}v\\
&\quad+\int_0^\ell\lvert S(u-v)\rvert\lvert\widetilde{G}(v)-G_{\mathrm{sine}}(v)\rvert\,\mathrm{d}v\\
&\le Ce^{Cs}N^{-1/2}, \tag*{(316)}
\end{aligned} ∣ G ( u ) − G sine ( u )∣ ≤ ∣ g ( u ) − g sine ( u )∣ + ∫ 0 ℓ ∣ Ξ ( u , v )∣ ∣ G ( v )∣ d v + ∫ 0 ℓ ∣ S ( u − v )∣ ∣ G ( v ) − G sine ( v )∣ d v ≤ C e C s N − 1/2 , (316) by the Cauchy–Schwarz inequality. Therefore, by Lemma 7.2,
∣ A N ( J ) − μ ( s ) ∣ = ∣ A N ( J ) − A s i n e ( ℓ ) ∣ ≤ C e C s N − 1 / 2 . (317) \lvert A_N(J)-\mu(s)\rvert=\lvert A_N(J)-A_{\mathrm{sine}}(\ell)\rvert\le Ce^{Cs}N^{-1/2}. \tag*{(317)} ∣ A N ( J ) − μ ( s )∣ = ∣ A N ( J ) − A sine ( ℓ )∣ ≤ C e C s N − 1/2 . (317) We now conclude. By Corollary 3.3, Proposition 6.2, (317) , and Lemma 7.1(b), we have
P N ( J ) 2 D N ( J ) = det M N ♯ ( J ) = μ ( s ) + O ( e C s N − 1 / 2 ) = μ ( s ) ( 1 + O ( e C s N − 1 / 2 ) ) . (318) \frac{\mathcal{P}_N(J)^2}{D_N(J)}=\det M_N^{\sharp}(J)=\mu(s)+O(e^{Cs}N^{-1/2})=\mu(s)\left(1+O(e^{Cs}N^{-1/2})\right). \tag*{(318)} D N ( J ) P N ( J ) 2 = det M N ♯ ( J ) = μ ( s ) + O ( e C s N − 1/2 ) = μ ( s ) ( 1 + O ( e C s N − 1/2 ) ) . (318) Combining this with the second bound in (300) and D + ( s / 4 ) D − ( s / 4 ) μ ( s ) = q 1 ( s ) 2 D_+(s/4)D_-(s/4)\mu(s)=q_1(s)^2 D + ( s /4 ) D − ( s /4 ) μ ( s ) = q 1 ( s ) 2 P N ( J ) 2 = q 1 ( s ) 2 ( 1 + O ( e C s N − 1 / 2 ) ) \mathcal{P}_N(J)^2=q_1(s)^2(1+O(e^{Cs}N^{-1/2})) P N ( J ) 2 = q 1 ( s ) 2 ( 1 + O ( e C s N − 1/2 )) (300) . ▫
We now translate Proposition 7.3 to eigenvalue coordinates. Recall from the introduction that E 1 , N ( A ) E_{1,N}(A) E 1 , N ( A ) H N H_N H N A A A σ ( y ) = 4 − y 2 \sigma(y)=\sqrt{4-y^2} σ ( y ) = 4 − y 2
Corollary 7.5. Let δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) γ > 0 \gamma>0 γ > 0 C , N 0 > 0 C,N_0>0 C , N 0 > 0 δ \delta δ γ \gamma γ N ≥ N 0 N\ge N_0 N ≥ N 0 y ∈ [ − 2 + δ , 2 − δ ] y\in[-2+\delta,2-\delta] y ∈ [ − 2 + δ , 2 − δ ] 0 < s ≤ γ log N 0<s\le\gamma\sqrt{\log N} 0 < s ≤ γ log N
∣ E 1 , N ( ( y , y + s N σ ( y ) ) ) − q 1 ( s ) ∣ ≤ C e C s N − 1 / 2 q 1 ( s ) . (319) \left|E_{1,N}\left(\left(y,y+\frac{s}{N\sigma(y)}\right)\right)-q_1(s)\right|\le Ce^{Cs}N^{-1/2}q_1(s). \tag*{(319)} E 1 , N ( ( y , y + N σ ( y ) s ) ) − q 1 ( s ) ≤ C e C s N − 1/2 q 1 ( s ) . (319) Proof. Set x 0 = N / 2 y x_0=\sqrt{N}/2\,y x 0 = N /2 y x 0 ∈ B N ( δ / 2 ) x_0\in B_N(\delta/2) x 0 ∈ B N ( δ /2 ) ϱ N ( x 0 ) = N / 2 σ ( y ) \varrho_N(x_0)=\sqrt{N}/2\,\sigma(y) ϱ N ( x 0 ) = N /2 σ ( y ) ( y , y + s / ( N σ ( y ) ) ) (y,y+s/(N\sigma(y))) ( y , y + s / ( N σ ( y ))) λ ↦ N / 2 λ \lambda\mapsto\sqrt{N}/2\,\lambda λ ↦ N /2 λ ( x 0 , x 0 + s / ( 2 ϱ N ( x 0 ) ) ) (x_0,x_0+s/(2\varrho_N(x_0))) ( x 0 , x 0 + s / ( 2 ϱ N ( x 0 ))) Proposition 7.3 .
Several intervals
In this section, we prove two estimates for unions of intervals, which are used in the proof of Theorem 1.1 . The first states that the hole probability of a union of well-separated intervals factorizes. The second states that the hole probability of an arbitrary union of disjoint intervals is bounded by e C L e^{CL} e C L e C L e^{CL} e C L [4 Lemma 3.8] . For the GOE, the factor e C L e^{CL} e C L L L L log N \sqrt{\log N} log N
Throughout this section, we fix δ ∈ ( 0 , 1 / 2 ] \delta\in(0,1/2] δ ∈ ( 0 , 1/2 ] r 0 ≥ 1 r_0 \ge1 r 0 ≥ 1 γ > 0 \gamma> 0 γ > 0 J = ⋃ j = 1 r J j J = \bigcup_{j=1}^{r} J_j J = ⋃ j = 1 r J j
r ≤ r 0 , J ‾ ⊂ B N ( δ ) , L ( J ) ≤ γ log N . (320) r \le r_0,\qquad\overline{J} \subset\mathcal{B}_N(\delta),\qquad L(J) \le\gamma\sqrt{\log N}. \tag*{(320)} r ≤ r 0 , J ⊂ B N ( δ ) , L ( J ) ≤ γ log N . (320) The constants C C C c c c N 0 N_0 N 0 δ \delta δ r 0 r_0 r 0 γ \gamma γ N ≥ N 0 N \ge N_0 N ≥ N 0
The determinantal part
Lemma 8.1. Let J ′ J' J ′ J ′ ′ ⊂ R J'' \subset\mathbb{R} J ′′ ⊂ R
D N ( J ′ ∪ J ′ ′ ) ≤ D N ( J ′ ) D N ( J ′ ′ ) . (321) D_N(J' \cup J'') \le D_N(J')D_N(J''). \tag*{(321)} D N ( J ′ ∪ J ′′ ) ≤ D N ( J ′ ) D N ( J ′′ ) . (321) Moreover, if J ′ J' J ′ J ′ ′ J'' J ′′ Proposition 5.1 and ∥ K J ′ J ′ ′ ∥ 2 ≤ c e − C ( L ( J ′ ) + L ( J ′ ′ ) ) \lVert K_{J'J''}\rVert_2 \le c e^{-C(L(J')+L(J''))} ∥ K J ′ J ′′ ∥ 2 ≤ c e − C ( L ( J ′ ) + L ( J ′′ )) K J ′ J ′ ′ K_{J'J''} K J ′ J ′′ L 2 ( J ′ ′ ) L^2(J'') L 2 ( J ′′ ) L 2 ( J ′ ) L^2(J') L 2 ( J ′ ) K N ( x , y ) K_N(x,y) K N ( x , y )
∣ log D N ( J ′ ∪ J ′ ′ ) D N ( J ′ ) D N ( J ′ ′ ) ∣ ≤ C e C ( L ( J ′ ) + L ( J ′ ′ ) ) ∥ K J ′ J ′ ′ ∥ 2 2 . (322) \left|\log\frac{D_N(J'\cup J'')}{D_N(J')D_N(J'')}\right| \le C e^{C(L(J')+L(J''))}\lVert K_{J'J''}\rVert_2^2. \tag*{(322)} log D N ( J ′ ) D N ( J ′′ ) D N ( J ′ ∪ J ′′ ) ≤ C e C ( L ( J ′ ) + L ( J ′′ )) ∥ K J ′ J ′′ ∥ 2 2 . (322) Proof. With respect to the decomposition L 2 ( J ′ ∪ J ′ ′ ) = L 2 ( J ′ ) ⊕ L 2 ( J ′ ′ ) L^2(J'\cup J'')=L^2(J')\oplus L^2(J'') L 2 ( J ′ ∪ J ′′ ) = L 2 ( J ′ ) ⊕ L 2 ( J ′′ )
Id − K J ′ ∪ J ′ ′ = ( Id − K J ′ − K J ′ J ′ ′ − K J ′ ′ J ′ Id − K J ′ ′ ) . (323) \operatorname{Id}-K_{J'\cup J''}=
\begin{pmatrix}
\operatorname{Id}-K_{J'} & -K_{J'J''}\\
-K_{J''J'} & \operatorname{Id}-K_{J''}
\end{pmatrix}. \tag*{(323)} Id − K J ′ ∪ J ′′ = ( Id − K J ′ − K J ′′ J ′ − K J ′ J ′′ Id − K J ′′ ) . (323) Factoring this operator into block triangular operators whose diagonal blocks are Id − K J ′ \operatorname{Id}-K_{J'} Id − K J ′
D N ( J ′ ∪ J ′ ′ ) D N ( J ′ ) D N ( J ′ ′ ) = det ( Id − Z ) , (324) \frac{D_N(J'\cup J'')}{D_N(J')D_N(J'')}=\det(\operatorname{Id}-Z), \tag*{(324)} D N ( J ′ ) D N ( J ′′ ) D N ( J ′ ∪ J ′′ ) = det ( Id − Z ) , (324) where we define
Z = ( Id − K J ′ ′ ) − 1 / 2 K J ′ ′ J ′ ( Id − K J ′ ) − 1 K J ′ J ′ ′ ( Id − K J ′ ′ ) − 1 / 2 . (325) Z = (\operatorname{Id} - K_{J''})^{-1/2}K_{J''J'}(\operatorname{Id} - K_{J'})^{-1}K_{J'J''}(\operatorname{Id} - K_{J''})^{-1/2}. \tag*{(325)} Z = ( Id − K J ′′ ) − 1/2 K J ′′ J ′ ( Id − K J ′ ) − 1 K J ′ J ′′ ( Id − K J ′′ ) − 1/2 . (325) Note that the operator Z Z Z Id − K J ′ ∪ J ′ ′ \operatorname{Id} - K_{J'\cup J''} Id − K J ′ ∪ J ′′ Id − K J ′ ′ − K J ′ ′ J ′ ( Id − K J ′ ) − 1 K J ′ J ′ ′ \operatorname{Id} - K_{J''} - K_{J''J'}(\operatorname{Id} - K_{J'})^{-1}K_{J'J''} Id − K J ′′ − K J ′′ J ′ ( Id − K J ′ ) − 1 K J ′ J ′′ Z ≤ Id Z \le\operatorname{Id} Z ≤ Id Z Z Z [ 0 , 1 ] [0,1] [ 0 , 1 ] det ( Id − Z ) ≤ 1 \det(\operatorname{Id} - Z) \le1 det ( Id − Z ) ≤ 1 (321) .
We now prove the second claim. Under the additional assumptions, Proposition 5.1 gives
∥ Z ∥ 1 ≤ C e C ( L ( J ′ ) + L ( J ′ ′ ) ) ∥ K J ′ J ′ ′ ∥ 2 2 ≤ 1 / 2. (326) \lVert Z\rVert_1 \le Ce^{C(L(J')+L(J''))}\lVert K_{J'J''}\rVert_2^2 \le1/2. \tag*{(326)} ∥ Z ∥ 1 ≤ C e C ( L ( J ′ ) + L ( J ′′ )) ∥ K J ′ J ′′ ∥ 2 2 ≤ 1/2. (326) Therefore, (322) follows from
∣ log det ( Id − Z ) ∣ ≤ ∑ k ∣ log ( 1 − z k ) ∣ ≤ 2 ∥ Z ∥ 1 , (327) \left|\log\det(\operatorname{Id} - Z)\right| \le\sum_k \left|\log(1-z_k)\right| \le2\lVert Z\rVert_1, \tag*{(327)} ∣ log det ( Id − Z ) ∣ ≤ k ∑ ∣ log ( 1 − z k ) ∣ ≤ 2 ∥ Z ∥ 1 , (327) where we let z k ∈ [ 0 , 1 / 2 ] z_k \in[0,1/2] z k ∈ [ 0 , 1/2 ]
Factorization and the quasi-product bound
We first record a lower bound for the endpoint matrix of a single interval.
Lemma 8.2. Let J ′ J' J ′ det M N ♯ ( J ′ ) ≥ c e − C L ( J ′ ) \det M_N^\sharp(J') \ge c e^{-CL(J')} det M N ♯ ( J ′ ) ≥ c e − C L ( J ′ ) A N ( J ′ ) ≥ c e − C L ( J ′ ) \mathcal{A}_N(J') \ge c e^{-CL(J')} A N ( J ′ ) ≥ c e − C L ( J ′ )
Proof. We write
J ′ = ( x 0 , x 0 + s / ( 2 ϱ ) ) (328) J' = (x_0,x_0+s/(2\varrho)) \tag*{(328)} J ′ = ( x 0 , x 0 + s / ( 2 ϱ )) (328) with ϱ = ϱ N ( x 0 ) \varrho=\varrho_N(x_0) ϱ = ϱ N ( x 0 ) s ≤ 2 L ( J ′ ) s \le2L(J') s ≤ 2 L ( J ′ ) Corollary 3.3 and Proposition 7.3 , we have
det M N ♯ ( J ′ ) = P N ( J ′ ) 2 D N ( J ′ ) = μ ( s ) ( 1 + O ( e C s N − 1 / 2 ) ) (329) \det M_N^\sharp(J') = \frac{\mathcal{P}_N(J')^2}{\mathcal{D}_N(J')} = \mu(s)(1+O(e^{Cs}N^{-1/2})) \tag*{(329)} det M N ♯ ( J ′ ) = D N ( J ′ ) P N ( J ′ ) 2 = μ ( s ) ( 1 + O ( e C s N − 1/2 )) (329) and Lemma 7.1 (b) gives μ ( s ) ≥ c e − s / 4 \mu(s) \ge ce^{-s/4} μ ( s ) ≥ c e − s /4 N ≥ N 0 N \ge N_0 N ≥ N 0 e C s N − 1 / 2 ≤ N − 1 / 4 e^{Cs}N^{-1/2} \le N^{-1/4} e C s N − 1/2 ≤ N − 1/4 Proposition 6.2 .
Proposition 8.3. Assume (8.1), and suppose that sep ( J ) ≥ N \operatorname{sep}(J) \ge\sqrt{N} sep ( J ) ≥ N
∣ P N ( J ) ∏ j = 1 k P N ( J j ) − 1 ∣ ≤ C e C L ( J ) N − 1 / 4 . (330) \left|\frac{\mathcal{P}_N(J)}{\prod_{j=1}^{k}\mathcal{P}_N(J_j)}-1\right| \le Ce^{CL(J)}N^{-1/4}. \tag*{(330)} ∏ j = 1 k P N ( J j ) P N ( J ) − 1 ≤ C e C L ( J ) N − 1/4 . (330) Proof. We write L = L ( J ) L=L(J) L = L ( J ) d = sep ( J ) d=\operatorname{sep}(J) d = sep ( J )
J k ′ = J 1 ∪ ⋯ ∪ J k − 1 (331) J'_k = J_1 \cup\cdots\cup J_{k-1} \tag*{(331)} J k ′ = J 1 ∪ ⋯ ∪ J k − 1 (331) :::`
for 2 ≤ k ≤ r 2 \le k \le r 2 ≤ k ≤ r ∣ K N ( x , y ) ∣ ≤ C N d − 1 |K_N(x,y)| \le C\sqrt{N}d^{-1} ∣ K N ( x , y ) ∣ ≤ C N d − 1 x ∈ J k ′ x \in J_k' x ∈ J k ′ y ∈ J k y \in J_k y ∈ J k
∥ K J k ′ J k ∥ 2 2 ≤ C N d − 2 ∣ J ∣ 2 ≤ C L 2 d − 2 ≤ C L 2 N − 1 . (332) \left\|K_{J_k'J_k}\right\|_2^2 \le CNd^{-2}|J|^2 \le CL^2d^{-2} \le CL^2N^{-1}. \tag*{(332)} K J k ′ J k 2 2 ≤ C N d − 2 ∣ J ∣ 2 ≤ C L 2 d − 2 ≤ C L 2 N − 1 . (332) Applying (8.3) for k = r , r − 1 , … , 2 k=r,r-1,\ldots,2 k = r , r − 1 , … , 2
D N ( J ) = ( ∏ j = 1 r D N ( J j ) ) ( 1 + O ( e c C L N − 1 ) ) . (333) \mathcal{D}_N(J)=\left(\prod_{j=1}^{r}\mathcal{D}_N(J_j)\right)\left(1+O(e^{cCL}N^{-1})\right). \tag*{(333)} D N ( J ) = ( j = 1 ∏ r D N ( J j ) ) ( 1 + O ( e c C L N − 1 ) ) . (333) Next, we consider the endpoint matrices. Let A 0 \mathcal{A}_0 A 0 A N ( J j ) \mathcal{A}_N(J_j) A N ( J j ) A N ( J ) = A 0 + E \mathcal{A}_N(J)=\mathcal{A}_0+E A N ( J ) = A 0 + E d ≥ N d\ge\sqrt{N} d ≥ N E E E C e C L N − 1 / 4 Ce^{CL}N^{-1/4} C e C L N − 1/4 A 0 \mathcal{A}_0 A 0 c e − C L ce^{-CL} c e − C L
det A N ( J ) = det A 0 det ( Id + A 0 − 1 E ) = ( ∏ j = 1 r A N ( J j ) ) ( 1 + O ( e C L N − 1 / 4 ) ) . (334) \det\mathcal{A}_N(J)=\det\mathcal{A}_0\det(\operatorname{Id}+\mathcal{A}_0^{-1}E)=\left(\prod_{j=1}^{r}\mathcal{A}_N(J_j)\right)\left(1+O(e^{CL}N^{-1/4})\right). \tag*{(334)} det A N ( J ) = det A 0 det ( Id + A 0 − 1 E ) = ( j = 1 ∏ r A N ( J j ) ) ( 1 + O ( e C L N − 1/4 ) ) . (334) In particular, det A N ( J ) ≥ c e − C L \det\mathcal{A}_N(J)\ge ce^{-CL} det A N ( J ) ≥ c e − C L
We now apply Proposition 6.2 to J J J J j J_j J j
det M N ♯ ( J ) = det A N ( J ) ( 1 + O ( e C L N − 1 / 2 ) ) (335) \det\mathsf{M}^{\sharp}_N(J)=\det\mathcal{A}_N(J)\left(1+O(e^{CL}N^{-1/2})\right) \tag*{(335)} det M N ♯ ( J ) = det A N ( J ) ( 1 + O ( e C L N − 1/2 ) ) (335) and
det M N ♯ ( J j ) = A N ( J j ) ( 1 + O ( e C L N − 1 / 2 ) ) . (336) \det\mathsf{M}^{\sharp}_N(J_j)=\mathcal{A}_N(J_j)\left(1+O(e^{CL}N^{-1/2})\right). \tag*{(336)} det M N ♯ ( J j ) = A N ( J j ) ( 1 + O ( e C L N − 1/2 ) ) . (336) Hence
det M N ♯ ( J ) = ( ∏ j = 1 r det M N ♯ ( J j ) ) ( 1 + O ( e C L N − 1 / 4 ) ) . (337) \det\mathsf{M}^{\sharp}_N(J)=\left(\prod_{j=1}^{r}\det\mathsf{M}^{\sharp}_N(J_j)\right)\left(1+O(e^{CL}N^{-1/4})\right). \tag*{(337)} det M N ♯ ( J ) = ( j = 1 ∏ r det M N ♯ ( J j ) ) ( 1 + O ( e C L N − 1/4 ) ) . (337) Combining (8.14) and (8.18) with Corollary 3.3, we find
P N ( J ) 2 = ∏ j P N ( J j ) 2 ( 1 + O ( e C L N − 1 / 4 ) ) . (338) \mathcal{P}_N(J)^2=\prod_j\mathcal{P}_N(J_j)^2\left(1+O(e^{CL}N^{-1/4})\right). \tag*{(338)} P N ( J ) 2 = j ∏ P N ( J j ) 2 ( 1 + O ( e C L N − 1/4 ) ) . (338) Taking square roots completes the proof.
Proposition 8.4. Assume (8.1). Then
P N ( J ) ≤ C e C L ( J ) ∏ j = 1 r P N ( J j ) . (339) \mathcal{P}_N(J)\le Ce^{CL(J)}\prod_{j=1}^{r}\mathcal{P}_N(J_j). \tag*{(339)} P N ( J ) ≤ C e C L ( J ) j = 1 ∏ r P N ( J j ) . (339) Proof. Combining Proposition 6.2, Lemma 8.1, Corollary 3.3, and Lemma 8.2, we have
P N ( J ) 2 ≤ C e C L ( J ) D N ( J ) ≤ C e C L ( J ) ∏ j = 1 r D N ( J j ) = C e C L ( J ) ∏ j = 1 r P N ( J j ) 2 det M N ♯ ( J j ) ≤ C e C L ( J ) ∏ j = 1 r P N ( J j ) 2 . (340) \begin{aligned}
\mathcal{P}_N(J)^2\le Ce^{CL(J)}\mathcal{D}_N(J)\le Ce^{CL(J)}\prod_{j=1}^{r}\mathcal{D}_N(J_j) \\
&=Ce^{CL(J)}\prod_{j=1}^{r}\frac{\mathcal{P}_N(J_j)^2}{\det\mathsf{M}^{\sharp}_N(J_j)}
\le Ce^{CL(J)}\prod_{j=1}^{r}\mathcal{P}_N(J_j)^2. \tag*{(340)}
\end{aligned} P N ( J ) 2 ≤ C e C L ( J ) D N ( J ) ≤ C e C L ( J ) j = 1 ∏ r D N ( J j ) = C e C L ( J ) j = 1 ∏ r det M N ♯ ( J j ) P N ( J j ) 2 ≤ C e C L ( J ) j = 1 ∏ r P N ( J j ) 2 . (340) Taking square roots, we obtain (8.20).
We now translate these results to eigenvalue coordinates.
Corollary 8.5. Let δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) r 0 ≥ 1 r_0 \ge1 r 0 ≥ 1 γ > 0 \gamma> 0 γ > 0 C , N 0 > 0 C, N_0 > 0 C , N 0 > 0 δ \delta δ r 0 r_0 r 0 γ \gamma γ N ≥ N 0 N \ge N_0 N ≥ N 0 A 1 , … , A r ⊂ [ − 2 + δ , 2 − δ ] A_1,\ldots,A_r \subset[-2+\delta,2-\delta] A 1 , … , A r ⊂ [ − 2 + δ , 2 − δ ] r ≤ r 0 r \le r_0 r ≤ r 0 L ≤ γ log N L \le\gamma\sqrt{\log N} L ≤ γ log N L = N ∑ j ∣ A j ∣ L=N\sum_j |A_j| L = N ∑ j ∣ A j ∣
E 1 , N ( ⋃ j = 1 r A j ) ≤ C e C L ∏ j = 1 r E 1 , N ( A j ) . (341) E_{1,N}\left(\bigcup_{j=1}^{r} A_j\right) \le C e^{CL} \prod_{j=1}^{r} E_{1,N}(A_j). \tag*{(341)} E 1 , N ( j = 1 ⋃ r A j ) ≤ C e C L j = 1 ∏ r E 1 , N ( A j ) . (341) If, in addition, dist ( A i , A j ) ≥ N − 1 / 2 \operatorname{dist}(A_i,A_j) \ge N^{-1/2} dist ( A i , A j ) ≥ N − 1/2 i ≠ j i \ne j i = j
∣ E 1 , N ( ⋃ j = 1 r A j ) − ∏ j = 1 r E 1 , N ( A j ) ∣ ≤ C e C L N − 1 / 4 ∏ j = 1 r E 1 , N ( A j ) . (342) \left|E_{1,N}\left(\bigcup_{j=1}^{r} A_j\right)-\prod_{j=1}^{r} E_{1,N}(A_j)\right| \le C e^{CL}N^{-1/4}\prod_{j=1}^{r} E_{1,N}(A_j). \tag*{(342)} E 1 , N ( j = 1 ⋃ r A j ) − j = 1 ∏ r E 1 , N ( A j ) ≤ C e C L N − 1/4 j = 1 ∏ r E 1 , N ( A j ) . (342) Proof. Let J j J_j J j A j A_j A j λ ↦ N / 2 λ \lambda\mapsto\sqrt{N/2}\lambda λ ↦ N /2 λ J = ⋃ j J j J=\bigcup_j J_j J = ⋃ j J j δ / 2 \delta/2 δ /2 δ \delta δ L ( J ) = L L(J)=L L ( J ) = L
sep ( J ) = N min i ≠ j dist ( A i , A j ) . (343) \operatorname{sep}(J)=N\min_{i\ne j}\operatorname{dist}(A_i,A_j). \tag*{(343)} sep ( J ) = N i = j min dist ( A i , A j ) . (343) So (8.22) follows from Proposition 8.4. Under the additional assumption that dist ( A i , A j ) ≥ N − 1 / 2 \operatorname{dist}(A_i,A_j) \ge N^{-1/2} dist ( A i , A j ) ≥ N − 1/2 i ≠ j i \ne j i = j sep ( J ) ≥ N \operatorname{sep}(J) \ge\sqrt{N} sep ( J ) ≥ N
The first moment
In this section, we compute the integral of the one-interval hole probability over the starting point of the interval, at the threshold relevant for the largest gaps. This calculation determines the centering, the scaling, and the constant Λ I \Lambda_{\mathrm{I}} Λ I Theorem 1.1 . We work in eigenvalue coordinates, with the notation of § 2.2 . Recall the quantities ℓ N ( x ) \ell_N(x) ℓ N ( x ) χ N \chi_N χ N (20) . For x ∈ R x \in\mathbb{R} x ∈ R s N ( x ) = N σ I ℓ N ( x ) s_N(x) = N\sigma_{\mathrm{I}}\ell_N(x) s N ( x ) = N σ I ℓ N ( x )
s N ( x ) = b N + x − 9 8 log b N a N , ℓ N ( x ) = s N ( x ) N σ I , χ N = a N N σ I . (344) s_N(x) = b_N + \frac{x-\frac{9}{8}\log b_N}{a_N}, \qquad\ell_N(x) = \frac{s_N(x)}{N\sigma_{\mathrm{I}}}, \qquad\chi_N = a_NN\sigma_{\mathrm{I}}. \tag*{(344)} s N ( x ) = b N + a N x − 8 9 log b N , ℓ N ( x ) = N σ I s N ( x ) , χ N = a N N σ I . (344) These definitions are chosen so that, for i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I )
τ i , N − x = χ N ( λ i + 1 − λ i − ℓ N ( x ) ) . (345) \tau_{i,N} - x = \chi_N\left(\lambda_{i+1}-\lambda_i-\ell_N(x)\right). \tag*{(345)} τ i , N − x = χ N ( λ i + 1 − λ i − ℓ N ( x ) ) . (345) Note that s N ( x ) = b N + O x ( log log N / log N ) s_N(x) = b_N + O_x(\log\log N/\sqrt{\log N}) s N ( x ) = b N + O x ( log log N / log N ) O x O_x O x x x x
Proposition 9.1. For every x ∈ R x \in\mathbb{R} x ∈ R
lim N → ∞ χ N ∫ y − y + − ℓ N ( x ) E 1 , N ( y , y + ℓ N ( x ) ) d y = Λ I e − x . (346) \lim_{N\to\infty}\chi_N\int_{y_-}^{y_+-\ell_N(x)}E_{1,N}(y,y+\ell_N(x))\,\mathrm{d}y=\Lambda_{\mathrm{I}}e^{-x}. \tag*{(346)} N → ∞ lim χ N ∫ y − y + − ℓ N ( x ) E 1 , N ( y , y + ℓ N ( x )) d y = Λ I e − x . (346) Moreover, there exist C x , N x > 0 C_x,N_x>0 C x , N x > 0 N ≥ N x N\geq N_x N ≥ N x
sup y ∈ [ y − , y + − ℓ N ( x ) ] χ N E 1 , N ( y , y + ℓ N ( x ) ) ≤ C x log N . (347) \sup_{y\in[y_-,y_+-\ell_N(x)]}\chi_NE_{1,N}(y,y+\ell_N(x))\leq C_x\log N. \tag*{(347)} y ∈ [ y − , y + − ℓ N ( x )] sup χ N E 1 , N ( y , y + ℓ N ( x )) ≤ C x log N . (347) Proof. Throughout the proof, x x x s N = s N ( x ) s_N=s_N(x) s N = s N ( x ) ℓ N = ℓ N ( x ) \ell_N=\ell_N(x) ℓ N = ℓ N ( x ) q 1 q_1 q 1 s N s_N s N F ( b N ) = log N F(b_N)=\log N F ( b N ) = log N F ′ ( b N ) = a N F'(b_N)=a_N F ′ ( b N ) = a N F ′ ′ = 1 / 32 F''=1/32 F ′′ = 1/32
F ( s N ) = log N + x − 9 8 log b N + 1 64 ( x − 9 8 log b N a N ) 2 = log N + x − 9 8 log b N + o ( 1 ) , (348) F(s_N)=\log N+x-\frac{9}{8}\log b_N+\frac{1}{64}\left(\frac{x-\frac{9}{8}\log b_N}{a_N}\right)^2=\log N+x-\frac{9}{8}\log b_N+o(1), \tag*{(348)} F ( s N ) = log N + x − 8 9 log b N + 64 1 ( a N x − 8 9 log b N ) 2 = log N + x − 8 9 log b N + o ( 1 ) , (348) where we used a N ≥ c log N a_N\geq c\sqrt{\log N} a N ≥ c log N s N / b N → 1 s_N/b_N\to1 s N / b N → 1
q 1 ( s N ) = κ 1 s N − 1 / 8 e − F ( s N ) ( 1 + o ( 1 ) ) = κ 1 b N e − x N ( 1 + o ( 1 ) ) . (349) q_1(s_N)=\kappa_1s_N^{-1/8}e^{-F(s_N)}(1+o(1))=\frac{\kappa_1b_Ne^{-x}}{N}(1+o(1)). \tag*{(349)} q 1 ( s N ) = κ 1 s N − 1/8 e − F ( s N ) ( 1 + o ( 1 )) = N κ 1 b N e − x ( 1 + o ( 1 )) . (349) For y ∈ I y \in I y ∈ I
Ψ ( y ) = σ ( y ) / σ I ≥ 1 , (350) \Psi(y)=\sigma(y)/\sigma_I \ge1, \tag*{(350)} Ψ ( y ) = σ ( y ) / σ I ≥ 1 , (350) and for y ∈ [ y − , y + − ℓ N ] y \in[y_-,y_+ - \ell_N] y ∈ [ y − , y + − ℓ N ]
s ( y ) = N σ ( y ) ℓ N = s N Ψ ( y ) , (351) s(y)=N\sigma(y)\ell_N=s_N\Psi(y), \tag*{(351)} s ( y ) = N σ ( y ) ℓ N = s N Ψ ( y ) , (351) so that
( y , y + ℓ N ) = ( y , y + s ( y ) / ( N σ ( y ) ) ) . (352) (y,y+\ell_N)=(y,y+s(y)/(N\sigma(y))). \tag*{(352)} ( y , y + ℓ N ) = ( y , y + s ( y ) / ( N σ ( y ))) . (352) Since I ⊂ ( − 2 , 2 ) I \subset(-2,2) I ⊂ ( − 2 , 2 ) s ( y ) ≤ C log N s(y) \le C\sqrt{\log N} s ( y ) ≤ C log N
E 1 , N ( ( y , y + ℓ N ) ) = q 1 ( s N Ψ ( y ) ) ( 1 + o ( 1 ) ) (353) E_{1,N}((y,y+\ell_N))=q_1(s_N\Psi(y))(1+o(1)) \tag*{(353)} E 1 , N (( y , y + ℓ N )) = q 1 ( s N Ψ ( y )) ( 1 + o ( 1 )) (353) uniformly in y ∈ [ y − , y + − ℓ N ] y \in[y_-,y_+ - \ell_N] y ∈ [ y − , y + − ℓ N ] (353) and (349) imply that
χ N E 1 , N ( ( y , y + ℓ N ) ) ≤ C χ N q 1 ( s N ) ≤ C x N a N b N , (354) \chi_N E_{1,N}((y,y+\ell_N)) \le C\chi_N q_1(s_N) \le Cx_Na_Nb_N, \tag*{(354)} χ N E 1 , N (( y , y + ℓ N )) ≤ C χ N q 1 ( s N ) ≤ C x N a N b N , (354) where we used that q 1 q_1 q 1 Ψ ≥ 1 \Psi\ge1 Ψ ≥ 1 (347) , since a N b N ≤ 2 log N a_Nb_N \le2\log N a N b N ≤ 2 log N (4) .
We now prove (346) . Let d 0 ∈ ( 0 , ∣ I ∣ / 2 ) d_0 \in(0,|I|/2) d 0 ∈ ( 0 , ∣ I ∣/2 ) I I I U \mathcal{U} U y ∈ I y \in I y ∈ I dist ( y , ∂ ∗ I ) < d 0 \operatorname{dist}(y,\partial_* I)<d_0 dist ( y , ∂ ∗ I ) < d 0 Ψ \Psi Ψ I I I 1 1 1 ∂ ∗ I \partial_* I ∂ ∗ I c 0 > 0 c_0>0 c 0 > 0 Ψ ≥ 1 + c 0 \Psi\ge1+c_0 Ψ ≥ 1 + c 0 I ∖ U I\setminus\mathcal{U} I ∖ U (348) that
q 1 ( ( 1 + c 0 ) s N ) ≤ C e − ( 1 + c 0 ) 2 s N 2 / 64 ≤ N − ( 1 + c 0 ) 2 + o ( 1 ) , (355) q_1((1+c_0)s_N)\le Ce^{-(1+c_0)^2s_N^2/64}\le N^{-(1+c_0)^2+o(1)}, \tag*{(355)} q 1 (( 1 + c 0 ) s N ) ≤ C e − ( 1 + c 0 ) 2 s N 2 /64 ≤ N − ( 1 + c 0 ) 2 + o ( 1 ) , (355) so (353) and the monotonicity of q 1 q_1 q 1 I ∖ U I\setminus\mathcal{U} I ∖ U (346) is bounded by C χ N N − ( 1 + c 0 ) 2 + o ( 1 ) → 0 C\chi_NN^{-(1+c_0)^2+o(1)}\to0 C χ N N − ( 1 + c 0 ) 2 + o ( 1 ) → 0
Fix ξ ∈ ∂ ∗ I \xi\in\partial_* I ξ ∈ ∂ ∗ I
∣ ξ ∣ = max ( ∣ y − ∣ , ∣ y + ∣ ) > 0 , (356) |\xi|=\max(|y_-|,|y_+|)>0, \tag*{(356)} ∣ ξ ∣ = max ( ∣ y − ∣ , ∣ y + ∣ ) > 0 , (356) since I I I U \mathcal{U} U ξ \xi ξ y = ξ − sgn ( ξ ) t y=\xi-\operatorname{sgn}(\xi)t y = ξ − sgn ( ξ ) t t ∈ [ 0 , d 0 ) t\in[0,d_0) t ∈ [ 0 , d 0 ) t t t y y y ξ \xi ξ y y y I I I t t t σ ′ ( ξ ) = − ξ / σ I \sigma'(\xi)=-\xi/\sigma_I σ ′ ( ξ ) = − ξ / σ I
Ψ ( ξ − sgn ( ξ ) t ) = 1 + ∣ ξ ∣ σ I 2 t + O ( t 2 ) . (357) \Psi(\xi-\operatorname{sgn}(\xi)t)=1+\frac{|\xi|}{\sigma_I^2}t+O(t^2). \tag*{(357)} Ψ ( ξ − sgn ( ξ ) t ) = 1 + σ I 2 ∣ ξ ∣ t + O ( t 2 ) . (357) We choose d 0 d_0 d 0 t ↦ Ψ ( ξ − sgn ( ξ ) t ) t\mapsto\Psi(\xi-\operatorname{sgn}(\xi)t) t ↦ Ψ ( ξ − sgn ( ξ ) t ) [ 0 , d 0 ) [0,d_0) [ 0 , d 0 ) ∣ ξ ∣ / ( 2 σ I 2 ) |\xi|/(2\sigma_I^2) ∣ ξ ∣/ ( 2 σ I 2 ) 2 ∣ ξ ∣ / σ I 2 2|\xi|/\sigma_I^2 2∣ ξ ∣/ σ I 2
z = ( log N ) ( Ψ ( y ) − 1 ) , (358) z=(\log N)(\Psi(y)-1), \tag*{(358)} z = ( log N ) ( Ψ ( y ) − 1 ) , (358) so that s ( y ) = s N ( 1 + z / log N ) s(y)=s_N(1+z/\log N) s ( y ) = s N ( 1 + z / log N ) d t = σ I 2 ( ∣ ξ ∣ log N ) − 1 ( 1 + O ( t ) ) d z \mathrm{d}t=\sigma_I^2(|\xi|\log N)^{-1}(1+O(t))\,\mathrm{d}z d t = σ I 2 ( ∣ ξ ∣ log N ) − 1 ( 1 + O ( t )) d z t t t [ 0 , d 0 ) [0,d_0) [ 0 , d 0 ) ξ = y − \xi=y_- ξ = y − [ ℓ N , d 0 ) [\ell_N,d_0) [ ℓ N , d 0 ) ξ = y + \xi=y_+ ξ = y + [ z ‾ N , z ‾ N ) [\underline{z}_N,\overline{z}_N) [ z N , z N ) z z z
0 ≤ z ‾ N ≤ C ℓ N log N → 0 (359) 0\leq\underline{z}_N\leq C\ell_N\log N\to0 \tag*{(359)} 0 ≤ z N ≤ C ℓ N log N → 0 (359) and z ‾ N ≥ c log N \overline{z}_N\geq c\log N z N ≥ c log N w = z / log N ≥ 0 w=z/\log N\geq0 w = z / log N ≥ 0
F ( s N ( 1 + w ) ) − F ( s N ) = s N 2 64 ( 2 w + w 2 ) + s N 8 w . (360) F(s_N(1+w))-F(s_N)=\frac{s_N^2}{64}(2w+w^2)+\frac{s_N}{8}w. \tag*{(360)} F ( s N ( 1 + w )) − F ( s N ) = 64 s N 2 ( 2 w + w 2 ) + 8 s N w . (360) By (348) , we have s N 2 / 64 = log N + O ( log N ) s_N^2/64=\log N+O(\sqrt{\log N}) s N 2 /64 = log N + O ( log N ) (360) equals 2 z + o ( 1 ) 2z+o(1) 2 z + o ( 1 ) z z z z z z z ≥ 0 z\geq0 z ≥ 0 N N N Lemma 7.1 (a), which shows that the ratio q 1 ( s ) / ( κ 1 s − 1 / 8 e − F ( s ) ) q_1(s)/(\kappa_1s^{-1/8}e^{-F(s)}) q 1 ( s ) / ( κ 1 s − 1/8 e − F ( s ) ) [ 1 / 2 , 2 ] [1/2,2] [ 1/2 , 2 ] s ≥ s N s\geq s_N s ≥ s N N N N
lim N → ∞ q 1 ( s N ( 1 + z / log N ) ) q 1 ( s N ) = e − 2 z , q 1 ( s N ( 1 + z / log N ) ) q 1 ( s N ) ≤ 4 e − z . (361) \lim_{N\to\infty}\frac{q_1(s_N(1+z/\log N))}{q_1(s_N)}=e^{-2z},\qquad\frac{q_1(s_N(1+z/\log N))}{q_1(s_N)}\leq4e^{-z}. \tag*{(361)} N → ∞ lim q 1 ( s N ) q 1 ( s N ( 1 + z / log N )) = e − 2 z , q 1 ( s N ) q 1 ( s N ( 1 + z / log N )) ≤ 4 e − z . (361) where the limit is locally uniform in z ≥ 0 z\geq0 z ≥ 0 z ≥ 0 z\geq0 z ≥ 0 N N N (353) that the contribution of the part of U \mathcal{U} U ξ \xi ξ (346) is
( 1 + o ( 1 ) ) χ N q 1 ( s N ) σ I 2 ∣ ξ ∣ log N ∫ z ‾ N z ‾ N q 1 ( s N ( 1 + z / log N ) ) q 1 ( s N ) ( 1 + O ( t ) ) d z . (362) (1+o(1))\chi_Nq_1(s_N)\frac{\sigma_I^2}{|\xi|\log N}\int_{\underline{z}_N}^{\overline{z}_N}\frac{q_1(s_N(1+z/\log N))}{q_1(s_N)}(1+O(t))\,\mathrm{d}z. \tag*{(362)} ( 1 + o ( 1 )) χ N q 1 ( s N ) ∣ ξ ∣ log N σ I 2 ∫ z N z N q 1 ( s N ) q 1 ( s N ( 1 + z / log N )) ( 1 + O ( t )) d z . (362) By (361) and dominated convergence, the integral converges to
∫ 0 ∞ e − 2 z d z = 1 / 2. (363) \int_0^\infty e^{-2z}\,\mathrm{d}z=1/2. \tag*{(363)} ∫ 0 ∞ e − 2 z d z = 1/2. (363) Using (349) and χ N = a N N σ I \chi_N=a_NN\sigma_I χ N = a N N σ I
lim N → ∞ a N b N 2 log N ⋅ κ 1 σ I 3 ∣ ξ ∣ e − x = κ 1 σ I 3 ∣ ξ ∣ e − x , (364) \lim_{N\to\infty}\frac{a_Nb_N}{2\log N}\cdot\kappa_1\frac{\sigma_I^3}{|\xi|}e^{-x}=\kappa_1\frac{\sigma_I^3}{|\xi|}e^{-x}, \tag*{(364)} N → ∞ lim 2 log N a N b N ⋅ κ 1 ∣ ξ ∣ σ I 3 e − x = κ 1 ∣ ξ ∣ σ I 3 e − x , (364) where we used that a N b N / ( 2 log N ) → 1 a_Nb_N/(2\log N)\to1 a N b N / ( 2 log N ) → 1 ξ ∈ ∂ ∗ I \xi\in\partial_*I ξ ∈ ∂ ∗ I (346) . □
Remark 9.2. The proof of Proposition 9.1 explains the form of the centering in Theorem 1.1 . Heuristically, the integral of E 1 , N ( ( y , y + ℓ ) ) E_{1,N}((y,y+\ell)) E 1 , N (( y , y + ℓ )) y y y ∑ i ( λ i + 1 − λ i − ℓ ) + \sum_i(\lambda_{i+1}-\lambda_i-\ell)_+ ∑ i ( λ i + 1 − λ i − ℓ ) + ℓ \ell ℓ ℓ \ell ℓ ℓ = s / ( N σ I ) \ell=s/(N\sigma_I) ℓ = s / ( N σ I ) q 1 ( s ) σ I 2 / ( 2 ∣ ξ ∣ log N ) q_1(s)\sigma_I^2/(2|\xi|\log N) q 1 ( s ) σ I 2 / ( 2∣ ξ ∣ log N ) ξ ∈ ∂ ∗ I \xi\in\partial_*I ξ ∈ ∂ ∗ I q 1 ( s ) ≈ κ 1 s − 1 / 8 e − F ( s ) q_1(s)\approx\kappa_1s^{-1/8}e^{-F(s)} q 1 ( s ) ≈ κ 1 s − 1/8 e − F ( s ) F ′ ( s ) ≈ s / 32 F'(s)\approx s/32 F ′ ( s ) ≈ s /32 s / ( N σ I ) s/(N\sigma_I) s / ( N σ I ) N s q 1 ( s ) / log N N s q_1(s)/\log N N s q 1 ( s ) / log N N q 1 ( s ) / s N q_1(s)/s N q 1 ( s ) / s s 2 ≈ 64 log N s^2 \approx64\log N s 2 ≈ 64 log N N q 1 ( s ) ≈ s ≈ b N N q_1(s) \approx s \approx b_N N q 1 ( s ) ≈ s ≈ b N F ( s ) ≈ log N − ( 9 / 8 ) log b N F(s) \approx\log N - (9/8)\log b_N F ( s ) ≈ log N − ( 9/8 ) log b N s / 8 s/8 s /8 F F F ∓ t / 2 \mp t/2 ∓ t /2 (14) , produces the constant − 4 -4 − 4 (10) . By contrast, for the GUE the corresponding exponent s 2 / 32 s^2/32 s 2 /32
Factorial moments and the proof of Theorem 1.1
In this section, we complete the proofs of Theorem 1.1 and Corollary 1.2. We follow the approach of Feng and Wei [21 ] , who reduced the Poisson convergence to the convergence of the moments in (365) below; we state and prove this reduction as Lemma A.1 in Appendix § 12 . The new input is the use of the estimates for several intervals from Section 8 in place of the negative association of determinantal processes used in [21 ] .
The positive-part moments
Fix k ≥ 1 k \ge1 k ≥ 1 x = ( x 1 , … , x k ) ∈ R k \mathbf{x}=(x_1,\ldots,x_k)\in\mathbb{R}^k x = ( x 1 , … , x k ) ∈ R k ℓ j = ℓ N ( x j ) \ell_j=\ell_N(x_j) ℓ j = ℓ N ( x j ) ℓ N ( x ) \ell_N(x) ℓ N ( x ) (344) . We work with N N N 0 < ℓ j < ∣ I ∣ 0<\ell_j<|I| 0 < ℓ j < ∣ I ∣ j j j
M N , k ( x ) = E ∑ i 1 , … , i k ∈ G N ( I ) d i s t i n c t ∏ j = 1 k ( τ i j , N − x j ) + . (365) M_{N,k}(\mathbf{x})=\mathbb{E}\sum_{\substack{i_1,\ldots,i_k\in\mathcal{G}_N(I)\\ \mathrm{distinct}}}\prod_{j=1}^{k}(\tau_{i_j,N}-x_j)_+. \tag*{(365)} M N , k ( x ) = E i 1 , … , i k ∈ G N ( I ) distinct ∑ j = 1 ∏ k ( τ i j , N − x j ) + . (365) where we write ( t ) + = max ( t , 0 ) (t)_+=\max(t,0) ( t ) + = max ( t , 0 ) Y N \mathcal{Y}_N Y N y = ( y 1 , … , y k ) ∈ ∏ j = 1 k [ y − , y + − ℓ j ] \mathbf{y}=(y_1,\ldots,y_k)\in\prod_{j=1}^{k}[y_-,y_+-\ell_j] y = ( y 1 , … , y k ) ∈ ∏ j = 1 k [ y − , y + − ℓ j ] i 1 , … , i k ∈ G N ( I ) i_1,\ldots,i_k\in\mathcal{G}_N(I) i 1 , … , i k ∈ G N ( I )
[ y j , y j + ℓ j ] ⊂ ( λ i j , λ i j + 1 ) , 1 ≤ j ≤ k . (366) [y_j,y_j+\ell_j]\subset(\lambda_{i_j},\lambda_{i_j+1}),\qquad1\le j\le k. \tag*{(366)} [ y j , y j + ℓ j ] ⊂ ( λ i j , λ i j + 1 ) , 1 ≤ j ≤ k . (366) Lemma 10.1. We have
M N , k ( x ) = χ N k ∫ ∏ j = 1 k [ y − , y + − ℓ j ] P ( y ∈ Y N ) d y . (367) M_{N,k}(\mathbf{x})=\chi_N^k\int_{\prod_{j=1}^{k}[y_-,y_+-\ell_j]}\mathbb{P}(\mathbf{y}\in\mathcal{Y}_N)\,d\mathbf{y}. \tag*{(367)} M N , k ( x ) = χ N k ∫ ∏ j = 1 k [ y − , y + − ℓ j ] P ( y ∈ Y N ) d y . (367) Proof. Consider a gap ( λ i , λ i + 1 ) (\lambda_i,\lambda_{i+1}) ( λ i , λ i + 1 ) ℓ > 0 \ell>0 ℓ > 0 y y y [ y , y + ℓ ] ⊂ ( λ i , λ i + 1 ) [y,y+\ell]\subset(\lambda_i,\lambda_{i+1}) [ y , y + ℓ ] ⊂ ( λ i , λ i + 1 ) ( λ i + 1 − λ i − ℓ ) + (\lambda_{i+1}-\lambda_i-\ell)_+ ( λ i + 1 − λ i − ℓ ) + i ∈ G N ( I ) i\in\mathcal{G}_N(I) i ∈ G N ( I ) [ y − , y + − ℓ ] [y_-,y_+-\ell] [ y − , y + − ℓ ] y ∈ Y N \mathbf{y}\in\mathcal{Y}_N y ∈ Y N i 1 , … , i k i_1,\ldots,i_k i 1 , … , i k (366) are determined by y \mathbf{y} y Y N \mathcal{Y}_N Y N k k k G N ( I ) \mathcal{G}_N(I) G N ( I )
∣ Y N ∣ = ∑ i 1 , … , i k ∈ G N ( I ) d i s t i n c t ∏ j = 1 k ( λ i j + 1 − λ i j − ℓ j ) + . (368) |\mathcal{Y}_N|=\sum_{\substack{i_1,\ldots,i_k\in\mathcal{G}_N(I)\\ \mathrm{distinct}}}\prod_{j=1}^{k}(\lambda_{i_j+1}-\lambda_{i_j}-\ell_j)_+. \tag*{(368)} ∣ Y N ∣ = i 1 , … , i k ∈ G N ( I ) distinct ∑ j = 1 ∏ k ( λ i j + 1 − λ i j − ℓ j ) + . (368) By (345) , the right side equals χ N − k \chi_N^{-k} χ N − k (365) . Taking expectations and using Fubini’s theorem, we obtain (367) . ∕
Note that the event { y ∈ Y N } \{y \in\mathcal{Y}_N\} { y ∈ Y N } ⋃ j ( y j , y j + ℓ j ) \bigcup_{j}(y_j,y_j+\ell_j) ⋃ j ( y j , y j + ℓ j ) I I I
η N = N − 1 / 3 , (369) \eta_N=N^{-1/3}, \tag*{(369)} η N = N − 1/3 , (369) and we let R N \mathcal{R}_N R N B ⊂ I B\subset I B ⊂ I ∣ B ∣ ≥ η N |B|\geq\eta_N ∣ B ∣ ≥ η N H N H_N H N
Lemma 10.2. For every D > 0 D>0 D > 0 N 0 > 0 N_0>0 N 0 > 0 P ( R N c ) ≤ N − D \mathbb{P}(\mathcal{R}_N^c)\leq N^{-D} P ( R N c ) ≤ N − D N ≥ N 0 N\geq N_0 N ≥ N 0
Proof. Let γ > 0 \gamma>0 γ > 0 δ > 0 \delta>0 δ > 0 I ⊂ [ − 2 + δ , 2 − δ ] I\subset[-2+\delta,2-\delta] I ⊂ [ − 2 + δ , 2 − δ ] s = γ log N s=\gamma\sqrt{\log N} s = γ log N
B m = [ y − + m η N 2 , y − + ( m + 1 ) η N 2 ] (370) B_m=\left[y_-+\frac{m\eta_N}{2},y_-+\frac{(m+1)\eta_N}{2}\right] \tag*{(370)} B m = [ y − + 2 m η N , y − + 2 ( m + 1 ) η N ] (370) for the integers m ≥ 0 m\geq0 m ≥ 0 B m ⊂ I B_m\subset I B m ⊂ I 2 ∣ I ∣ N 1 / 3 2|I|N^{1/3} 2∣ I ∣ N 1/3 B ⊂ I B\subset I B ⊂ I ∣ B ∣ ≥ η N |B|\geq\eta_N ∣ B ∣ ≥ η N B m B_m B m
Let y y y B m B_m B m s / ( N σ ( y ) ) ≤ η N / 2 s/(N\sigma(y))\leq\eta_N/2 s / ( N σ ( y )) ≤ η N /2 N ≥ N 0 N\geq N_0 N ≥ N 0 ( y , y + s / ( N σ ( y ) ) ) (y,y+s/(N\sigma(y))) ( y , y + s / ( N σ ( y ))) B m B_m B m
2 q 1 ( s ) ≤ C e − F ( s ) ≤ C N − γ 2 / 64 . (371) 2q_1(s)\leq Ce^{-F(s)}\leq CN^{-\gamma^2/64}. \tag*{(371)} 2 q 1 ( s ) ≤ C e − F ( s ) ≤ C N − γ 2 /64 . (371) A union bound over m m m P ( R N c ) ≤ C N 1 / 3 − γ 2 / 64 \mathbb{P}(\mathcal{R}_N^c)\leq CN^{1/3-\gamma^2/64} P ( R N c ) ≤ C N 1/3 − γ 2 /64 γ \gamma γ γ 2 / 64 > D + 1 \gamma^2/64>D+1 γ 2 /64 > D + 1
Remark 10.3. Lemma 10.2 also follows from the rigidity of eigenvalues proved in [18 Theorem 2.2] , after a rescaling of the GOE by the factor ( 1 + 1 / N ) − 1 / 2 (1+1/N)^{-1/2} ( 1 + 1/ N ) − 1/2
Proposition 10.4. For every k ≥ 1 k\geq1 k ≥ 1 x ∈ R k x\in\mathbb{R}^k x ∈ R k
lim N → ∞ M N , k ( x ) = ∏ j = 1 k Λ I e − x j . (372) \lim_{N\to\infty}M_{N,k}(\mathbf{x})=\prod_{j=1}^{k}\Lambda_I e^{-x_j}. \tag*{(372)} N → ∞ lim M N , k ( x ) = j = 1 ∏ k Λ I e − x j . (372) Proof. We define
H j ( y ) = E 1 , N ( ( y , y + ℓ j ) ) , H ( y ) = E 1 , N ( ⋃ j = 1 k ( y j , y j + ℓ j ) ) . (373) \mathcal{H}_j(y)=E_{1,N}\bigl((y,y+\ell_j)\bigr),\qquad\mathcal{H}(y)=E_{1,N}\left(\bigcup_{j=1}^{k}(y_j,y_j+\ell_j)\right). \tag*{(373)} H j ( y ) = E 1 , N ( ( y , y + ℓ j ) ) , H ( y ) = E 1 , N ( j = 1 ⋃ k ( y j , y j + ℓ j ) ) . (373) By Proposition 9.1, we have
lim N → ∞ χ N ∫ y − y + − ℓ j H j ( y ) d y = Λ I e − x j , sup y ∈ [ y − , y + − ℓ j ] χ N H j ( y ) ≤ C log N . (374) \lim_{N\to\infty}\chi_N\int_{y_-}^{y_+-\ell_j}\mathcal{H}_j(y)\,\mathrm{d}y=\Lambda_I e^{-x_j},\qquad\sup_{y\in[y_-,y_+-\ell_j]}\chi_N\mathcal{H}_j(y)\leq C\log N. \tag*{(374)} N → ∞ lim χ N ∫ y − y + − ℓ j H j ( y ) d y = Λ I e − x j , y ∈ [ y − , y + − ℓ j ] sup χ N H j ( y ) ≤ C log N . (374) Here, and for the rest of this proof, the constants C C C k k k x x x I I I S N S_N S N y ∈ ∏ j [ y − , y + − ℓ j ] \mathbf{y} \in\prod_j [y_-,y_+ - \ell_j] y ∈ ∏ j [ y − , y + − ℓ j ]
∣ y i − y j ∣ ≥ 4 η N ( i ≠ j ) , y j − y − ≥ 4 η N , y + − ( y j + ℓ j ) ≥ 4 η N , (375) |y_i-y_j| \ge4\eta_N \quad(i \ne j), \qquad y_j-y_- \ge4\eta_N, \qquad y_+-(y_j+\ell_j) \ge4\eta_N, \tag*{(375)} ∣ y i − y j ∣ ≥ 4 η N ( i = j ) , y j − y − ≥ 4 η N , y + − ( y j + ℓ j ) ≥ 4 η N , (375) and let S N c S_N^c S N c ∏ j [ y − , y + − ℓ j ] \prod_j [y_-,y_+ - \ell_j] ∏ j [ y − , y + − ℓ j ] ℓ j ≤ η N \ell_j \le\eta_N ℓ j ≤ η N j j j ℓ j ≤ C log N / N \ell_j \le C\sqrt{\log N}/N ℓ j ≤ C log N / N
We first bound the integral of the product of the H j \mathcal{H}_j H j S N c S_N^c S N c S N c S_N^c S N c ∣ y i − y j ∣ < 4 η N |y_i-y_j| < 4\eta_N ∣ y i − y j ∣ < 4 η N i ≠ j i \ne j i = j y j y_j y j 4 η N 4\eta_N 4 η N y − y_- y − y + − ℓ j y_+ - \ell_j y + − ℓ j j j j ∣ y i − y j ∣ < 4 η N |y_i-y_j| < 4\eta_N ∣ y i − y j ∣ < 4 η N y i y_i y i H i \mathcal{H}_i H i (374) , we see that the contribution of this set to λ N k ∫ S N c ∏ j H j ( y j ) d y \lambda_N^k \int_{S_N^c} \prod_j \mathcal{H}_j(y_j)\,\mathrm{d}\mathbf{y} λ N k ∫ S N c ∏ j H j ( y j ) d y C η N log N C\eta_N \log N C η N log N
λ N k ∫ S N c ∏ j = 1 k H j ( y j ) d y ≤ C η N log N . (376) \lambda_N^k \int_{S_N^c} \prod_{j=1}^{k}\mathcal{H}_j(y_j)\,\mathrm{d}\mathbf{y} \le C\eta_N \log N. \tag*{(376)} λ N k ∫ S N c j = 1 ∏ k H j ( y j ) d y ≤ C η N log N . (376) We now prove the upper bound in (372) . Suppose first that two of the intervals [ y j , y j + ℓ j ] [y_j,y_j+\ell_j] [ y j , y j + ℓ j ] P ( y ∈ Y N ) = 0 \mathbb{P}(\mathbf{y}\in\mathcal{Y}_N)=0 P ( y ∈ Y N ) = 0
P ( y ∈ Y N ) ≤ H ( y ) , (377) \mathbb{P}(\mathbf{y}\in\mathcal{Y}_N)\le\mathcal{H}(\mathbf{y}), \tag*{(377)} P ( y ∈ Y N ) ≤ H ( y ) , (377) and the intervals ( y j , y j + ℓ j ) (y_j,y_j+\ell_j) ( y j , y j + ℓ j ) I I I
L = N ∑ j ℓ j ≤ C log N . (378) L=N\sum_j\ell_j\le C\sqrt{\log N}. \tag*{(378)} L = N j ∑ ℓ j ≤ C log N . (378) By (341) , we have
H ( y ) ≤ C e C log N ∏ j H j ( y j ) . (379) \mathcal{H}(\mathbf{y})\le Ce^{C\sqrt{\log N}}\prod_j\mathcal{H}_j(y_j). \tag*{(379)} H ( y ) ≤ C e C l o g N j ∏ H j ( y j ) . (379) When moreover y ∈ S N \mathbf{y}\in S_N y ∈ S N 3 η N ≥ N − 1 / 2 3\eta_N\ge N^{-1/2} 3 η N ≥ N − 1/2 (342) gives
H ( y ) = ( 1 + O ( e C log N N − 1 / 4 ) ) ∏ j H j ( y j ) . (380) \mathcal{H}(\mathbf{y})=\left(1+O\left(e^{C\sqrt{\log N}}N^{-1/4}\right)\right)\prod_j\mathcal{H}_j(y_j). \tag*{(380)} H ( y ) = ( 1 + O ( e C l o g N N − 1/4 ) ) j ∏ H j ( y j ) . (380) Inserting these bounds into (367) and using (376) , we find
M N , k ( x ) ≤ ( 1 + O ( e C log N N − 1 / 4 ) ) ∏ j = 1 k ( λ N ∫ y − y + − ℓ j H j ) + C e C log N η N log N . (381) M_{N,k}(x)\le\left(1+O\left(e^{C\sqrt{\log N}}N^{-1/4}\right)\right)\prod_{j=1}^{k}\left(\lambda_N\int_{y_-}^{y_+-\ell_j}\mathcal{H}_j\right)+Ce^{C\sqrt{\log N}}\eta_N\log N. \tag*{(381)} M N , k ( x ) ≤ ( 1 + O ( e C l o g N N − 1/4 ) ) j = 1 ∏ k ( λ N ∫ y − y + − ℓ j H j ) + C e C l o g N η N log N . (381) By (374) , the right side converges to ∏ j Λ I e − x j \prod_j\Lambda_I e^{-x_j} ∏ j Λ I e − x j
To prove the lower bound, suppose that y ∈ S N y \in\mathcal{S}_N y ∈ S N ⋃ j ( y j , y j + ℓ j ) \bigcup_j (y_j,y_j+\ell_j) ⋃ j ( y j , y j + ℓ j ) R N \mathcal{R}_N R N y − y_- y − y + y_+ y + 3 η N 3\eta_N 3 η N I I I y j y_j y j y j + ℓ j y_j+\ell_j y j + ℓ j [ y j , y j + ℓ j ] [y_j,y_j+\ell_j] [ y j , y j + ℓ j ] ( λ i , λ i + 1 ) (\lambda_i,\lambda_{i+1}) ( λ i , λ i + 1 ) λ i , λ i + 1 ∈ I \lambda_i,\lambda_{i+1} \in I λ i , λ i + 1 ∈ I y ∈ Y N y \in\mathcal{Y}_N y ∈ Y N
P ( y ∈ Y N ) ≥ H ( y ) − P ( R N c ) , y ∈ S N . (382) \mathbb{P}(y \in\mathcal{Y}_N) \ge\mathcal{H}(y)-\mathbb{P}(\mathcal{R}_N^c), \qquad y \in\mathcal{S}_N. \tag*{(382)} P ( y ∈ Y N ) ≥ H ( y ) − P ( R N c ) , y ∈ S N . (382) Using (367) , (342) , Lemma 10.2 with D = k + 1 D=k+1 D = k + 1 χ N ≤ C N log N \chi_N \le CN\sqrt{\log N} χ N ≤ C N log N
M N , k ( x ) ≥ ( 1 − O ( e C log N N − 1 / 4 ) ) χ N k ∫ S N ∏ j = 1 k H j ( y j ) d y − C ( N log N ) k N − k − 1 . (383) M_{N,k}(x) \ge\left(1-O\left(e^{C\sqrt{\log N}}N^{-1/4}\right)\right)\chi_N^k\int_{\mathcal{S}_N}\prod_{j=1}^k\mathcal{H}_j(y_j)\,\mathrm{d}y-C(N\sqrt{\log N})^kN^{-k-1}. \tag*{(383)} M N , k ( x ) ≥ ( 1 − O ( e C l o g N N − 1/4 ) ) χ N k ∫ S N j = 1 ∏ k H j ( y j ) d y − C ( N log N ) k N − k − 1 . (383) Finally, (376) implies that χ N k \chi_N^k χ N k ∏ j H j ( y j ) \prod_j\mathcal{H}_j(y_j) ∏ j H j ( y j ) S N \mathcal{S}_N S N ∏ j [ y − , y + − ℓ j ] \prod_j[y_-,y_+-\ell_j] ∏ j [ y − , y + − ℓ j ] o ( 1 ) o(1) o ( 1 ) (374) , the latter integral, multiplied by χ N k \chi_N^k χ N k ∏ j Λ I e − x j \prod_j\Lambda_I e^{-x_j} ∏ j Λ I e − x j
Proofs of the main results
Proof of Theorem 1.1. Proposition 10.4 shows that the point processes X N \mathcal{X}_N X N Lemma A.1 with Λ = Λ I \Lambda=\Lambda_I Λ = Λ I Theorem 1.1 follows from Lemma A.1 .
Proof of Corollary 1.2. For x ∈ R x \in\mathbb{R} x ∈ R X N ( ( x , ∞ ) ) \mathcal{X}_N((x,\infty)) X N (( x , ∞ )) i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I ) τ i , N > x \tau_{i,N}>x τ i , N > x
{ τ k , N ↓ ≤ x } = { X N ( ( x , ∞ ) ) ≤ k − 1 } , (384) \{\tau_{k,N}^{\downarrow}\le x\}=\{\mathcal{X}_N((x,\infty))\le k-1\}, \tag*{(384)} { τ k , N ↓ ≤ x } = { X N (( x , ∞ )) ≤ k − 1 } , (384) with the convention that τ k , N ↓ = − ∞ \tau_{k,N}^{\downarrow}=-\infty τ k , N ↓ = − ∞ ∣ G N ( I ) ∣ < k |\mathcal{G}_N(I)|<k ∣ G N ( I ) ∣ < k Theorem 1.1 , X N ( ( x , ∞ ) ) \mathcal{X}_N((x,\infty)) X N (( x , ∞ )) Λ I e − x \Lambda_Ie^{-x} Λ I e − x (8) . Taking k = 1 k=1 k = 1 (9) .
We now prove (10) . Let E N = { G N ( I ) ≠ ∅ } \mathcal{E}_N=\{\mathcal{G}_N(I)\ne\varnothing\} E N = { G N ( I ) = ∅ } (8) with k = 1 k=1 k = 1 lim sup N → ∞ P ( E N c ) ≤ exp ( − Λ I e − x ) \limsup_{N\to\infty}\mathbb{P}(\mathcal{E}_N^c)\le\exp(-\Lambda_Ie^{-x}) lim sup N → ∞ P ( E N c ) ≤ exp ( − Λ I e − x ) x ∈ R x\in\mathbb{R} x ∈ R x → − ∞ x\to-\infty x → − ∞ P ( E N ) → 1 \mathbb{P}(\mathcal{E}_N)\to1 P ( E N ) → 1 τ ~ 1 , N \widetilde{\tau}_{1,N} τ 1 , N τ 1 , N ↓ \tau_{1,N}^{\downarrow} τ 1 , N ↓ E N \mathcal{E}_N E N 0 0 0 (9) implies that τ ~ 1 , N \widetilde{\tau}_{1,N} τ 1 , N E N \mathcal{E}_N E N (7) gives
N σ I m 1 , N = b N + a N − 1 ( τ ~ 1 , N − 9 8 log b N ) . (385) N\sigma_I m_{1,N}=b_N+a_N^{-1}\left(\widetilde{\tau}_{1,N}-\frac{9}{8}\log b_N\right). \tag*{(385)} N σ I m 1 , N = b N + a N − 1 ( τ 1 , N − 8 9 log b N ) . (385) From (4) , we have
b N = 8 log N − 4 + O ( ( log N ) − 1 / 2 ) , 1 a N = 4 log N + 1 / 4 = 4 log N + O ( ( log N ) − 3 / 2 ) , (386) \begin{aligned}
b_N&=8\sqrt{\log N}-4+O\left((\log N)^{-1/2}\right),\\
\frac{1}{a_N}&=\frac{4}{\sqrt{\log N+1/4}}=\frac{4}{\sqrt{\log N}}+O\left((\log N)^{-3/2}\right), \tag*{(386)}
\end{aligned} b N a N 1 = 8 log N − 4 + O ( ( log N ) − 1/2 ) , = log N + 1/4 4 = log N 4 + O ( ( log N ) − 3/2 ) , (386) and log b N = ( 1 / 2 ) log log N + log 8 + o ( 1 ) \log b_N = (1/2)\log\log N + \log8 + o(1) log b N = ( 1/2 ) log log N + log 8 + o ( 1 ) E N \mathcal{E}_N E N P ( E N ) → 1 \mathbb{P}(\mathcal{E}_N) \to1 P ( E N ) → 1 (10) .
Proof of the claims in Remark 1.4. Given a real symmetric N × N N \times N N × N M M M μ 1 ≤ ⋯ ≤ μ N \mu_1 \le\cdots\le\mu_N μ 1 ≤ ⋯ ≤ μ N I ′ ⊂ R I' \subset\mathbb{R} I ′ ⊂ R k ≥ 1 k \ge1 k ≥ 1 T k ( M ; I ′ ) T_k(M; I') T k ( M ; I ′ ) k k k μ i + 1 − μ i \mu_{i+1}-\mu_i μ i + 1 − μ i μ i ∈ I ′ \mu_i \in I' μ i ∈ I ′ T k ( M ; I ′ ) = 0 T_k(M; I') = 0 T k ( M ; I ′ ) = 0 k k k
τ ^ k ( M ; I ′ ) = a N ( N σ I T k ( M ; I ′ ) − b N ) + 9 8 log b N , (387) \widehat{\tau}_k(M; I') = a_N\left(N\sigma_I T_k(M; I') - b_N\right) + \frac{9}{8}\log b_N, \tag*{(387)} τ k ( M ; I ′ ) = a N ( N σ I T k ( M ; I ′ ) − b N ) + 8 9 log b N , (387) so that the quantity τ ^ k , N \widehat{\tau}_{k,N} τ k , N τ ^ k ( W N ; I ) \widehat{\tau}_k(W_N; I) τ k ( W N ; I ) k ≥ 1 k \ge1 k ≥ 1 F k ( x ) F_k(x) F k ( x ) (8) .
We first compare W N W_N W N
θ N = ( 1 + 1 / N ) − 1 / 2 (388) \theta_N = (1+1/N)^{-1/2} \tag*{(388)} θ N = ( 1 + 1/ N ) − 1/2 (388) and G N = θ N H N G_N = \theta_N H_N G N = θ N H N ( G N ) i j (G_N)_{ij} ( G N ) ij i ≤ j i \le j i ≤ j ( 1 + 1 i = j ) / ( N + 1 ) (1+\mathbf{1}_{i=j})/(N+1) ( 1 + 1 i = j ) / ( N + 1 ) G N G_N G N 1 1 1 G N G_N G N [33 Definition 1.1] .
Let Φ ∈ C c ∞ ( R ) \Phi\in C^\infty_c(\mathbb{R}) Φ ∈ C c ∞ ( R ) ν = N / log N \nu= N/\sqrt{\log N} ν = N / log N
S N ( t ) = Φ ( a N σ I log N t − a N b N + 9 8 log b N ) , t ∈ R , (389) S_N(t) = \Phi\left(a_N\sigma_I\sqrt{\log N}\,t-a_Nb_N+\frac{9}{8}\log b_N\right), \qquad t \in\mathbb{R}, \tag*{(389)} S N ( t ) = Φ ( a N σ I log N t − a N b N + 8 9 log b N ) , t ∈ R , (389) so that
Φ ( τ ^ k ( M ; I ) ) = S N ( ν T k ( M ; I ) ) . (390) \Phi(\widehat{\tau}_k(M; I)) = S_N(\nu T_k(M; I)). \tag*{(390)} Φ ( τ k ( M ; I )) = S N ( ν T k ( M ; I )) . (390) The derivatives of S N S_N S N C ( log N ) 5 C(\log N)^5 C ( log N ) 5 a N σ I log N ≤ C log N a_N\sigma_I\sqrt{\log N} \le C\log N a N σ I log N ≤ C log N [33 Corollary 1.6] , whose error term involves derivatives of the test function of order at most five, we find
∣ E Φ ( τ ^ k ( W N ; I ) ) − E Φ ( τ ^ k ( G N ; I ) ) ∣ ≤ C N − c ( log N ) 5 . (391) \left|\mathbb{E}\Phi(\widehat{\tau}_k(W_N; I))-\mathbb{E}\Phi(\widehat{\tau}_k(G_N; I))\right| \le C N^{-c}(\log N)^5. \tag*{(391)} ∣ E Φ ( τ k ( W N ; I )) − E Φ ( τ k ( G N ; I )) ∣ ≤ C N − c ( log N ) 5 . (391) Next, we show that P ( τ ^ k ( G N ; I ) ≤ x ) → F k ( x ) \mathbb{P}(\widehat{\tau}_k(G_N; I) \le x) \to F_k(x) P ( τ k ( G N ; I ) ≤ x ) → F k ( x ) x ∈ R x \in\mathbb{R} x ∈ R I N = θ N − 1 I I_N = \theta_N^{-1}I I N = θ N − 1 I G N G_N G N θ N λ i \theta_N\lambda_i θ N λ i T k ( G N ; I ) = θ N T k ( H N ; I N ) T_k(G_N; I) = \theta_N T_k(H_N; I_N) T k ( G N ; I ) = θ N T k ( H N ; I N ) x ∈ R x \in\mathbb{R} x ∈ R η > 0 \eta> 0 η > 0 (387) and (344) , the events { τ ^ k ( G N ; I ) > x } \{\widehat{\tau}_k(G_N; I) > x\} { τ k ( G N ; I ) > x } { T k ( H N ; I N ) > ℓ N ( x ) / θ N } \{T_k(H_N; I_N) > \ell_N(x)/\theta_N\} { T k ( H N ; I N ) > ℓ N ( x ) / θ N } N N N ℓ N ( x + η ) − ℓ N ( x ) = η / χ N \ell_N(x+\eta)-\ell_N(x)=\eta/\chi_N ℓ N ( x + η ) − ℓ N ( x ) = η / χ N ℓ N ( x ) ( θ N − 1 − 1 ) ≤ C log N / N 2 \ell_N(x)(\theta_N^{-1}-1) \le C\sqrt{\log N}/N^2 ℓ N ( x ) ( θ N − 1 − 1 ) ≤ C log N / N 2 N N N
ℓ N ( x ) ≤ ℓ N ( x ) / θ N ≤ ℓ N ( x + η ) , (392) \ell_N(x) \le\ell_N(x)/\theta_N \le\ell_N(x+\eta), \tag*{(392)} ℓ N ( x ) ≤ ℓ N ( x ) / θ N ≤ ℓ N ( x + η ) , (392) and therefore
P ( τ ^ k ( H N ; I N ) > x + η ) ≤ P ( τ ^ k ( G N ; I ) > x ) ≤ P ( τ ^ k ( H N ; I N ) > x ) . (393) \mathbb{P}(\widehat{\tau}_k(H_N; I_N)>x+\eta) \le\mathbb{P}(\widehat{\tau}_k(G_N; I)>x) \le\mathbb{P}(\widehat{\tau}_k(H_N; I_N)>x). \tag*{(393)} P ( τ k ( H N ; I N ) > x + η ) ≤ P ( τ k ( G N ; I ) > x ) ≤ P ( τ k ( H N ; I N ) > x ) . (393) It remains to compare τ ^ k ( H N ; I N ) \widehat{\tau}_k(H_N;I_N) τ k ( H N ; I N ) τ k , N ↓ \tau^{\downarrow}_{k,N} τ k , N ↓ x ′ ∈ R x' \in\mathbb{R} x ′ ∈ R { τ ^ k ( H N ; I N ) > x ′ } \{\widehat{\tau}_k(H_N;I_N)>x'\} { τ k ( H N ; I N ) > x ′ } { τ k , N ↓ > x ′ } \{\tau^{\downarrow}_{k,N}>x'\} { τ k , N ↓ > x ′ } ( λ i , λ i + 1 ) (\lambda_i,\lambda_{i+1}) ( λ i , λ i + 1 ) ℓ N ( x ′ ) \ell_N(x') ℓ N ( x ′ ) λ i ∈ I N \lambda_i \in I_N λ i ∈ I N i ∈ G N ( I ) i \in\mathcal{G}_N(I) i ∈ G N ( I ) 2 / N 2/N 2/ N y − y_- y − y + y_+ y + ∣ θ − 1 y ± − y ± ∣ ≤ 2 / N |\theta^{-1}y_\pm-y_\pm|\le2/N ∣ θ − 1 y ± − y ± ∣ ≤ 2/ N ℓ N ( x ′ ) \ell_N(x') ℓ N ( x ′ ) 2 / N 2/N 2/ N y ± y_\pm y ±
( y ± − ℓ N ( x ′ ) 2 + 2 N , y ± − 2 N ) , ( y ± + 2 N , y ± + ℓ N ( x ′ ) 2 − 2 N ) . (394) \left(y_\pm-\frac{\ell_N(x')}{2}+\frac{2}{N},y_\pm-\frac{2}{N}\right),\qquad\left(y_\pm+\frac{2}{N},y_\pm+\frac{\ell_N(x')}{2}-\frac{2}{N}\right). \tag*{(394)} ( y ± − 2 ℓ N ( x ′ ) + N 2 , y ± − N 2 ) , ( y ± + N 2 , y ± + 2 ℓ N ( x ′ ) − N 2 ) . (394) These four intervals lie in [ − 2 + δ , 2 − δ ] [-2+\delta,2-\delta] [ − 2 + δ , 2 − δ ] δ > 0 \delta>0 δ > 0 I I I σ ( y ) ≥ σ I − C ℓ N ( x ′ ) \sigma(y)\ge\sigma_I-C\ell_N(x') σ ( y ) ≥ σ I − C ℓ N ( x ′ ) ( y , y + s / ( N σ ( y ) ) ) (y,y+s/(N\sigma(y))) ( y , y + s / ( N σ ( y ))) s ≥ s N ( x ′ ) / 2 − C s\ge s_N(x')/2-C s ≥ s N ( x ′ ) /2 − C
s N ( x ′ ) = N σ I ℓ N ( x ′ ) . (395) s_N(x')=N\sigma_I\ell_N(x'). \tag*{(395)} s N ( x ′ ) = N σ I ℓ N ( x ′ ) . (395) Using Corollary 7.5 , Lemma 7.1 (a), and the monotonicity of q 1 q_1 q 1 C q 1 ( s N ( x ′ ) / 2 − C ) ≤ N − 1 / 4 + o ( 1 ) Cq_1(s_N(x')/2-C)\le N^{-1/4+o(1)} C q 1 ( s N ( x ′ ) /2 − C ) ≤ N − 1/4 + o ( 1 )
∣ P ( τ ^ k ( H N ; I N ) > x ′ ) − P ( τ k , N ↓ > x ′ ) ∣ ≤ N − 1 / 4 + o ( 1 ) (396) \left|\mathbb{P}(\widehat{\tau}_k(H_N;I_N)>x')-\mathbb{P}(\tau^{\downarrow}_{k,N}>x')\right|\le N^{-1/4+o(1)} \tag*{(396)} P ( τ k ( H N ; I N ) > x ′ ) − P ( τ k , N ↓ > x ′ ) ≤ N − 1/4 + o ( 1 ) (396) for every fixed x ′ x' x ′ x ′ = x x'=x x ′ = x x ′ = x + η x'=x+\eta x ′ = x + η (393) and using (8) , we find that every limit point of P ( τ ^ k ( G N ; I ) ≤ x ) \mathbb{P}(\widehat{\tau}_k(G_N;I)\le x) P ( τ k ( G N ; I ) ≤ x ) [ F k ( x ) , F k ( x + η ) ] [F_k(x),F_k(x+\eta)] [ F k ( x ) , F k ( x + η )] η ↓ 0 \eta\downarrow0 η ↓ 0 F k F_k F k
P ( τ ^ k ( G N ; I ) ≤ x ) ⟶ F k ( x ) . (397) \mathbb{P}(\widehat{\tau}_k(G_N;I)\le x)\longrightarrow F_k(x). \tag*{(397)} P ( τ k ( G N ; I ) ≤ x ) ⟶ F k ( x ) . (397) Since F k F_k F k R \mathbb{R} R
E Φ ( τ ^ k ( G N ; I ) ) ⟶ ∫ Φ d F k (398) \mathbb{E}\Phi(\widehat{\tau}_k(G_N;I))\longrightarrow\int\Phi\,\mathrm{d}F_k \tag*{(398)} E Φ ( τ k ( G N ; I )) ⟶ ∫ Φ d F k (398) for all Φ ∈ C c ∞ ( R ) \Phi\in C^\infty_c(\mathbb{R}) Φ ∈ C c ∞ ( R ) (391) implies that the same holds for W N W_N W N G N G_N G N E Φ \mathbb{E}\Phi E Φ Φ ∈ C c ∞ ( R ) \Phi\in C^\infty_c(\mathbb{R}) Φ ∈ C c ∞ ( R )
P ( τ ^ k ( W N ; I ) ≤ x ) ⟶ F k ( x ) (399) \mathbb{P}(\widehat{\tau}_k(W_N;I)\le x)\longrightarrow F_k(x) \tag*{(399)} P ( τ k ( W N ; I ) ≤ x ) ⟶ F k ( x ) (399) for all x ∈ R x\in\mathbb{R} x ∈ R
A criterion for Poisson convergence
In this appendix, we prove the criterion used in the proof of Theorem 1.1 . It is a variant of [21 Lemma 1] , where the conclusion is stated for half-lines.
Lemma A.1. Let Λ > 0 \Lambda>0 Λ > 0 N N N X N = ∑ i ∈ I N δ X i , N \mathcal{X}_N=\sum_{i\in\mathcal{I}_N}\delta_{X_{i,N}} X N = ∑ i ∈ I N δ X i , N R \mathbb{R} R I N \mathcal{I}_N I N k ≥ 1 k\geq1 k ≥ 1 x = ( x 1 , … , x k ) ∈ R k \mathbf{x}=(x_1,\ldots,x_k)\in\mathbb{R}^k x = ( x 1 , … , x k ) ∈ R k
lim N → ∞ E ∑ i 1 , … , i k ∈ I N d i s t i n c t ∏ j = 1 k ( X i j , N − x j ) + = ∏ j = 1 k Λ e − x j . (400) \lim_{N\to\infty}\mathbb{E}\sum_{\substack{i_1,\ldots,i_k\in\mathcal{I}_N\\ \mathrm{distinct}}}\prod_{j=1}^{k}(X_{i_j,N}-x_j)_+=\prod_{j=1}^{k}\Lambda e^{-x_j}. \tag*{(400)} N → ∞ lim E i 1 , … , i k ∈ I N distinct ∑ j = 1 ∏ k ( X i j , N − x j ) + = j = 1 ∏ k Λ e − x j . (400) Let A 1 , … , A m ⊂ R A_1,\ldots,A_m\subset\mathbb{R} A 1 , … , A m ⊂ R ( a , b ] (a,b] ( a , b ] a < b a<b a < b ( a , ∞ ) (a,\infty) ( a , ∞ ) ( X N ( A 1 ) , … , X N ( A m ) ) (\mathcal{X}_N(A_1),\ldots,\mathcal{X}_N(A_m)) ( X N ( A 1 ) , … , X N ( A m )) Λ ∫ A l e − x d x \Lambda\int_{A_l}e^{-x}\,dx Λ ∫ A l e − x d x 1 ≤ l ≤ m 1\leq l\leq m 1 ≤ l ≤ m X N \mathcal{X}_N X N PPP ( Λ e − x d x ) \operatorname{PPP}(\Lambda e^{-x}\,dx) PPP ( Λ e − x d x )
Proof. For x ∈ R k \mathbf{x}\in\mathbb{R}^k x ∈ R k
I N ( x ) = E ∑ i 1 , … , i k ∈ I N d i s t i n c t ∏ j = 1 k 1 { X i j , N > x j } , f ( x ) = Λ e − x . (401) I_N(\mathbf{x})=\mathbb{E}\sum_{\substack{i_1,\ldots,i_k\in\mathcal{I}_N\\ \mathrm{distinct}}}\prod_{j=1}^{k}\mathbf{1}\{X_{i_j,N}>x_j\},\qquad f(x)=\Lambda e^{-x}. \tag*{(401)} I N ( x ) = E i 1 , … , i k ∈ I N distinct ∑ j = 1 ∏ k 1 { X i j , N > x j } , f ( x ) = Λ e − x . (401) We first show that I N ( x ) → ∏ j f ( x j ) I_N(\mathbf{x})\to\prod_j f(x_j) I N ( x ) → ∏ j f ( x j ) h > 0 h>0 h > 0 x ∈ R x\in\mathbb{R} x ∈ R
u x , h ( t ) = h − 1 ( ( t − x ) + − ( t − x − h ) + ) , (402) u_{x,h}(t)=h^{-1}\bigl((t-x)_+-(t-x-h)_+\bigr), \tag*{(402)} u x , h ( t ) = h − 1 ( ( t − x ) + − ( t − x − h ) + ) , (402) so that
1 { t > x + h } ≤ u x , h ( t ) ≤ 1 { t > x } . (403) \mathbf{1}\{t>x+h\}\leq u_{x,h}(t)\leq\mathbf{1}\{t>x\}. \tag*{(403)} 1 { t > x + h } ≤ u x , h ( t ) ≤ 1 { t > x } . (403) Expanding the product ∏ j u x j , h ( X i j , N ) \prod_j u_{x_j,h}(X_{i_j,N}) ∏ j u x j , h ( X i j , N ) 2 k 2^k 2 k
lim N → ∞ E ∑ i 1 , … , i k ∈ I N d i s t i n c t ∏ j = 1 k u x j , h ( X i j , N ) = ∏ j = 1 k f ( x j ) − f ( x j + h ) h . (404) \lim_{N\to\infty}\mathbb{E}\sum_{\substack{i_1,\ldots,i_k\in\mathcal{I}_N\\ \mathrm{distinct}}}\prod_{j=1}^{k}u_{x_j,h}(X_{i_j,N})=\prod_{j=1}^{k}\frac{f(x_j)-f(x_j+h)}{h}. \tag*{(404)} N → ∞ lim E i 1 , … , i k ∈ I N distinct ∑ j = 1 ∏ k u x j , h ( X i j , N ) = j = 1 ∏ k h f ( x j ) − f ( x j + h ) . (404) Combining this with (A.4), we find
lim sup N → ∞ I N ( x ) ≤ ∏ j = 1 k f ( x j − h ) − f ( x j ) h , lim inf N → ∞ I N ( x ) ≥ ∏ j = 1 k f ( x j ) − f ( x j + h ) h . (405) \limsup_{N\to\infty}I_N(\mathbf{x})\leq\prod_{j=1}^{k}\frac{f(x_j-h)-f(x_j)}{h},\qquad
\liminf_{N\to\infty}I_N(\mathbf{x})\geq\prod_{j=1}^{k}\frac{f(x_j)-f(x_j+h)}{h}. \tag*{(405)} N → ∞ lim sup I N ( x ) ≤ j = 1 ∏ k h f ( x j − h ) − f ( x j ) , N → ∞ lim inf I N ( x ) ≥ j = 1 ∏ k h f ( x j ) − f ( x j + h ) . (405) where in the first bound we used (A.4) with x j − h x_j-h x j − h x j x_j x j h ↓ 0 h \downarrow0 h ↓ 0
I N ( x ) → ∏ j ( − f ′ ( x j ) ) = ∏ j f ( x j ) . (406) \mathcal{I}_N(\mathbf{x}) \to\prod_j(-f'(x_j))=\prod_j f(x_j). \tag*{(406)} I N ( x ) → j ∏ ( − f ′ ( x j )) = j ∏ f ( x j ) . (406) Next, let k 1 , … , k m ≥ 0 k_1,\ldots,k_m \ge0 k 1 , … , k m ≥ 0
( n ) k = n ( n − 1 ) ⋯ ( n − k + 1 ) . (407) (n)_k=n(n-1)\cdots(n-k+1). \tag*{(407)} ( n ) k = n ( n − 1 ) ⋯ ( n − k + 1 ) . (407) Since the intervals A l A_l A l E ∏ l = 1 m ( X N ( A l ) ) k l \mathbb{E}\prod_{l=1}^m(\mathcal{X}_N(A_l))_{k_l} E ∏ l = 1 m ( X N ( A l ) ) k l ( i l , p ) 1 ≤ l ≤ m , 1 ≤ p ≤ k l (i_{l,p})_{1\le l\le m,1\le p\le k_l} ( i l , p ) 1 ≤ l ≤ m , 1 ≤ p ≤ k l X i l , p , N ∈ A l \mathcal{X}_{i_{l,p},N}\in A_l X i l , p , N ∈ A l l l l p p p
1 { t ∈ ( a , b ] } = 1 { t > a } − 1 { t > b } (408) \mathbf{1}\{t\in(a,b]\}=\mathbf{1}\{t>a\}-\mathbf{1}\{t>b\} \tag*{(408)} 1 { t ∈ ( a , b ]} = 1 { t > a } − 1 { t > b } (408) and
1 { t ∈ ( a , ∞ ) } = 1 { t > a } , (409) \mathbf{1}\{t\in(a,\infty)\}=\mathbf{1}\{t>a\}, \tag*{(409)} 1 { t ∈ ( a , ∞ )} = 1 { t > a } , (409) and we expand the product of these indicators. This expresses E ∏ l = 1 m ( X N ( A l ) ) k l \mathbb{E}\prod_{l=1}^m(\mathcal{X}_N(A_l))_{k_l} E ∏ l = 1 m ( X N ( A l ) ) k l I N ( x ) \mathcal{I}_N(\mathbf{x}) I N ( x ) x \mathbf{x} x A l A_l A l
lim N → ∞ E ∏ l = 1 m ( X N ( A l ) ) k l = ∏ l = 1 m ( Λ ∫ A l e − x d x ) k l , (410) \lim_{N\to\infty}\mathbb{E}\prod_{l=1}^m(\mathcal{X}_N(A_l))_{k_l}
=\prod_{l=1}^m\left(\Lambda\int_{A_l}e^{-x}\,\mathrm{d}x\right)^{k_l}, \tag*{(410)} N → ∞ lim E l = 1 ∏ m ( X N ( A l ) ) k l = l = 1 ∏ m ( Λ ∫ A l e − x d x ) k l , (410) which is the corresponding factorial moment of the claimed Poisson vector. The method of moments then gives the convergence in distribution of ( X N ( A 1 ) , … , X N ( A m ) ) (\mathcal{X}_N(A_1),\ldots,\mathcal{X}_N(A_m)) ( X N ( A 1 ) , … , X N ( A m ))
Finally, let X \mathcal{X} X R \mathbb{R} R Λ e − x d x \Lambda e^{-x}\,\mathrm{d}x Λ e − x d x X N ( B ) \mathcal{X}_N(B) X N ( B ) X ( B ) \mathcal{X}(B) X ( B ) B B B ( a , b ] (a,b] ( a , b ] [30 ] , as stated in [4 Theorem 2.2] for point processes on R 2 \mathbb{R}^2 R 2 X N \mathcal{X}_N X N X \mathcal{X} X [4 Proposition 2.1] for the same deduction. This completes the proof.
References 
