Authors’ Note
With the exception of this note, the entire document is machine written. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. However, the resulting document remains lacking in exposition and overall coherence. We are nonetheless releasing this manuscript on Hexagon to disseminate the result as quickly as possible, because we believe it may be of interest to the mathematics and physics communities. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.
P.L. was partially supported by NSF grant DMS-2450004.
Introduction
Background
The Edwards–Anderson (EA) model [58 ] is the basic model of a short-range spin glass. Ising spins σ x ∈ { − 1 , 1 } \sigma_x \in\{-1,1\} σ x ∈ { − 1 , 1 } [127 ] , in which every pair of spins interacts. For the SK model, Parisi’s theory of replica symmetry breaking [118 , 119 ] has to a large extent been made rigorous. The free energy is given by the Parisi formula [76 , 134 ] , its minimizer is unique [7 ] , and Panchenko proved that overlap arrays satisfying the Ghirlanda–Guerra identities are ultrametric [113 , 114 ] . For the EA model in a fixed dimension, by contrast, the existence of a spin-glass phase at positive temperature remains open. At zero temperature, Chatterjee has proved several signatures of glassy behavior, including disorder chaos and low-energy macroscopic excitations [30 ] . There are competing physical descriptions of the low-temperature states: the replica-symmetry-breaking picture [102 ] , the droplet picture [68 ] , and intermediate pictures such as the trivial–nontrivial picture [95 , 111 ] ; see [129 ] for a discussion. The pictures disagree in particular about a uniform external field: replica symmetry breaking predicts a phase transition at a de Almeida–Thouless line [53 , 72 ] , whereas the droplet picture predicts that there is no transition in any positive field.
A natural way to relate the two models is to let the number of neighbors of each spin grow. On the torus ( Z / L Z ) d (\mathbb{Z}/L\mathbb{Z})^d ( Z / L Z ) d 1 / ( 2 d ) 1/(2d) 1/ ( 2 d ) ∑ y ∼ x J x y σ y \sum_{y\sim x} J_{xy}\sigma_y ∑ y ∼ x J x y σ y 2 d 2d 2 d 1 / 2 1/2 1/2 d d d [82 ] , and the low-temperature phase of the EA model on Z d \mathbb{Z}^d Z d 1 / d 1/d 1/ d [72 ] . The EA model on diluted hypercubes of large dimension has also been studied numerically as a mean-field spin glass [67 ] . In particular, the free energy per site of the EA model is expected to converge to that of the SK model as d → ∞ d \to\infty d → ∞
Rigorous results on this limit have so far required high temperature, interactions of diverging range, or graphs that are random or locally tree-like. For β ≤ 1 \beta\le1 β ≤ 1 log 2 + β 2 / 4 \log2 + \beta^2/4 log 2 + β 2 /4 [1 ] . The same limit holds on any regular graph of diverging degree. For β < 1 \beta< 1 β < 1 D D D n n n det ( I − β 2 S + ) − 1 / 2 \operatorname{det}(I-\beta^2S_+)^{-1/2} det ( I − β 2 S + ) − 1/2 S + S_+ S + D D D tr S + ≤ n / D \operatorname{tr} S_+ \le n/\sqrt{D} tr S + ≤ n / D e o ( n ) e^{o(n)} e o ( n ) D → ∞ D \to\infty D → ∞ β = 1 \beta= 1 β = 1 β \beta β β 2 < log 2 \beta^2 < \log2 β 2 < log 2 [81 Corollary 3] . Fröhlich and Zegarlinski [71 ] proved high-temperature mean-field limits of short-range spin glasses as the interaction range tends to infinity. In the Kac limit, the dimension is fixed, the strength of the couplings decays to zero over a distance γ − 1 \gamma^{-1} γ − 1 γ → 0 \gamma\to0 γ → 0 [69 , 70 ] proved that the free energy converges to the SK free energy. One of the two inequalities, proved earlier by Guerra and Toninelli [77 ] , requires the interaction kernel to be positive semidefinite. For sparse random graphs, Dembo, Montanari, and Sen [54 ] showed that the maximum cut of random regular and Erdős–Rényi graphs of large average degree is governed by the SK ground-state energy. Their comparison uses the randomness of the graph. It was extended to other optimization problems on sparse random hypergraphs by Sen [126 ] , and to inhomogeneous random graphs, whose large-degree limit is an inhomogeneous Potts spin glass, by Jagannath, Ko, and Sen [89 ] . El Cheairi and Gamarnik showed that algorithms given by low-degree polynomials with a certain tree structure perform equally well on the SK model and on sparse Erdős–Rényi graphs of large average degree [63 ] . Approximating message passing for the SK model by such polynomials, they obtained nearly maximal cuts on these graphs, assuming that the support of the Parisi measure is an interval [ 0 , q ∗ ] [0,q_*] [ 0 , q ∗ ] β \beta β [62 ] constructed cuts of the same size by local algorithms, under a similar assumption at zero temperature.
None of these methods applies to a deterministic lattice at low temperature. The lattice is not locally tree-like: it has many short cycles. The methods that prove the Parisi formula, namely Guerra’s interpolation [76 ] , the Aizenman–Sims–Starr scheme [2 ] , and Panchenko’s analysis of the overlap distribution [114 ] , use the fact that the covariance of the SK Hamiltonian is a function of the overlap of two configurations. On the lattice, the covariance of H ( σ ) H(\sigma) H ( σ ) H ( τ ) H(\tau) H ( τ ) x y xy x y σ x σ y = τ x τ y \sigma_x\sigma_y = \tau_x\tau_y σ x σ y = τ x τ y − 1 -1 − 1
The goal of this paper is to prove that the free energy of the EA model on the torus ( Z / L Z ) d (\mathbb{Z}/L\mathbb{Z})^d ( Z / L Z ) d d → ∞ d \to\infty d → ∞ L L L { 0 , 1 } d \{0,1\}^d { 0 , 1 } d
Main results
The model. Throughout, G = ( V , E ) G=(V,E) G = ( V , E ) D D D n = ∣ V ∣ n=|V| n = ∣ V ∣ D ≥ 1 D\ge1 D ≥ 1 G D \mathcal{G}_D G D G G G ( g x y ) x y ∈ E (g_{xy})_{xy\in E} ( g x y ) x y ∈ E J x y = g x y / D J_{xy}=g_{xy}/\sqrt{D} J x y = g x y / D β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R
H G ( σ ) = ∑ x y ∈ E J x y σ x σ y , p G ( β , h ) = 1 n E log ∑ σ ∈ { − 1 , 1 } V exp ( β H G ( σ ) + h ∑ x ∈ V σ x ) , e ( G ; h ) = 1 n E max σ ∈ { − 1 , 1 } V ( H G ( σ ) + h ∑ x ∈ V σ x ) . (1) \begin{aligned}
H_G(\sigma)&=\sum_{xy\in E}J_{xy}\sigma_x\sigma_y, \qquad p_G(\beta,h)=\frac{1}{n}\mathbb{E}\log\sum_{\sigma\in\{-1,1\}^V}\exp\left(\beta H_G(\sigma)+h\sum_{x\in V}\sigma_x\right),\\
e(G;h)&=\frac{1}{n}\mathbb{E}\max_{\sigma\in\{-1,1\}^V}\left(H_G(\sigma)+h\sum_{x\in V}\sigma_x\right). \tag*{(1)}
\end{aligned} H G ( σ ) e ( G ; h ) = x y ∈ E ∑ J x y σ x σ y , p G ( β , h ) = n 1 E log σ ∈ { − 1 , 1 } V ∑ exp ( β H G ( σ ) + h x ∈ V ∑ σ x ) , = n 1 E σ ∈ { − 1 , 1 } V max ( H G ( σ ) + h x ∈ V ∑ σ x ) . (1) As in the literature on the SK model, the field enters the pressure through the Gibbs exponent and is not multiplied by β \beta β p G ( β , β h ) / β → e ( G ; h ) p_G(\beta,\beta h)/\beta\to e(G;h) p G ( β , β h ) / β → e ( G ; h ) β → ∞ \beta\to\infty β → ∞ E H G ( σ ) 2 = ∣ E ∣ / D = n / 2 \mathbb{E}H_G(\sigma)^2=|E|/D=n/2 E H G ( σ ) 2 = ∣ E ∣/ D = n /2 σ \sigma σ ∑ y ∼ x E J x y 2 = 1 \sum_{y\sim x}\mathbb{E}J_{xy}^2=1 ∑ y ∼ x E J x y 2 = 1 x x x
The torus T L d = ( Z / L Z ) d \mathbb{T}_L^d=(\mathbb{Z}/L\mathbb{Z})^d T L d = ( Z / L Z ) d ± 1 \pm1 ± 1 2 d 2d 2 d L ≥ 3 L\ge3 L ≥ 3 n = L d n=L^d n = L d H G H_G H G g x y / 2 d g_{xy}/\sqrt{2d} g x y / 2 d p d , L = p T L d p_{d,L}=p_{\mathbb{T}_L^d} p d , L = p T L d
The hypercube Q d = { 0 , 1 } d Q_d=\{0,1\}^d Q d = { 0 , 1 } d d d d n = 2 d n=2^d n = 2 d
No relation between n n n D D D D = 2 d D=2d D = 2 d log n \log n log n T L d \mathbb{T}_L^d T L d D = log 2 n D=\log_2 n D = log 2 n Q d Q_d Q d
The SK model. The SK Hamiltonian on N N N H N S K ( σ ) = N − 1 / 2 ∑ 1 ≤ i < j ≤ N g i j σ i σ j H_N^{\mathrm{SK}}(\sigma)=N^{-1/2}\sum_{1\le i<j\le N}g_{ij}\sigma_i\sigma_j H N SK ( σ ) = N − 1/2 ∑ 1 ≤ i < j ≤ N g ij σ i σ j g i j g_{ij} g ij ( N − 1 ) / 2 (N-1)/2 ( N − 1 ) /2 H G H_G H G ( N − 1 ) / N (N-1)/N ( N − 1 ) / N
P ( β , h ) = lim N → ∞ 1 N E log ∑ σ e β H N S K ( σ ) + h ∑ i σ i , P ∗ ( h ) = lim N → ∞ 1 N E max σ ( H N S K ( σ ) + h ∑ i σ i ) (2) \mathcal{P}(\beta,h)=\lim_{N\to\infty}\frac{1}{N}\mathbb{E}\log\sum_{\sigma}e^{\beta H_N^{\mathrm{SK}}(\sigma)+h\sum_i\sigma_i}, \qquad\mathcal{P}_*(h)=\lim_{N\to\infty}\frac{1}{N}\mathbb{E}\max_{\sigma}\left(H_N^{\mathrm{SK}}(\sigma)+h\sum_i\sigma_i\right) \tag*{(2)} P ( β , h ) = N → ∞ lim N 1 E log σ ∑ e β H N SK ( σ ) + h ∑ i σ i , P ∗ ( h ) = N → ∞ lim N 1 E σ max ( H N SK ( σ ) + h i ∑ σ i ) (2) exist. The first exists by [75 , 134 ] and is given by the Parisi formula [76 , 134 ] , which we recall in § 3.3 . The second exists by a comparison with the first (see (8) ), and P ∗ ( h ) = lim β → ∞ P ( β , β h ) / β P_*(h) = \lim_{\beta\to\infty} \mathcal{P}(\beta,\beta h)/\beta P ∗ ( h ) = lim β → ∞ P ( β , β h ) / β P ∗ ( 0 ) ≈ 0.7632 P_*(0) \approx0.7632 P ∗ ( 0 ) ≈ 0.7632 [51 ] ; see also [54 ] .
Results. Our main theorem states that the pressure of the EA model on the torus converges to the SK pressure, uniformly in the side length.
Theorem 1.1. For every β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R
lim d → ∞ sup L ≥ 3 ∣ p d , L ( β , h ) − P ( β , h ) ∣ = 0. \lim_{d \to\infty}\sup_{L \ge3} \left|p_{d,L}(\beta,h)-\mathcal{P}(\beta,h)\right|=0. d → ∞ lim L ≥ 3 sup ∣ p d , L ( β , h ) − P ( β , h ) ∣ = 0. The same holds on the hypercube.
Theorem 1.2. For every β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R
lim d → ∞ p Q d ( β , h ) = P ( β , h ) . \lim_{d \to\infty}p_{Q_d}(\beta,h)=\mathcal{P}(\beta,h). d → ∞ lim p Q d ( β , h ) = P ( β , h ) . Letting the temperature tend to zero, we obtain the corresponding statement for ground states.
Corollary 1.3. For every h ∈ R h \in\mathbb{R} h ∈ R
lim d → ∞ sup L ≥ 3 ∣ e ( T L d ; h ) − P ∗ ( h ) ∣ = 0 and lim d → ∞ e ( Q d ; h ) = P ∗ ( h ) . \lim_{d \to\infty}\sup_{L \ge3}\left|e(\mathbb{T}_L^d;h)-P_*(h)\right|=0
\qquad\text{and}\qquad
\lim_{d \to\infty}e(Q_d;h)=P_*(h). d → ∞ lim L ≥ 3 sup e ( T L d ; h ) − P ∗ ( h ) = 0 and d → ∞ lim e ( Q d ; h ) = P ∗ ( h ) . We make several remarks on these results.
the derivatives of convex functions that converge pointwise converge at every point where the limit is differentiable. Thus, at every β > 0 \beta> 0 β > 0 P ( ⋅ , h ) \mathcal{P}(\cdot,h) P ( ⋅ , h ) E ⟨ Q ⟩ \mathbb{E}\langle Q\rangle E ⟨ Q ⟩ T L d \mathbb{T}_L^d T L d L L L E ⟨ R 2 ⟩ \mathbb{E}\langle R^2\rangle E ⟨ R 2 ⟩ P ( ⋅ , h ) \mathcal{P}(\cdot,h) P ( ⋅ , h ) β \beta β [43 ] , and Contucci, Mingione, and Starr proved distributional identities after suitable generic Gaussian perturbations whose covariances are powers of the link overlap [48 ] . By Panchenko’s theorem [113 ] , these distributional Ghirlanda–Guerra identities yield ultrametricity of the limiting link-overlap arrays of the generically perturbed models.
Remark 1.7. The proof of the lower bound uses a recent theorem on the support of the Parisi measure in a positive field [99 ] , which we recall in Theorem 2.2 : for h > 0 h > 0 h > 0 [ q − , q + ] [q_-,q_+] [ q − , q + ] q − > 0 q_- > 0 q − > 0 h = 0 h = 0 h = 0 h > 0 h > 0 h > 0 h h h [ 0 , q + ] [0,q_+] [ 0 , q + ] β > 1 \beta> 1 β > 1 β \beta β [141 ] and for every β > 1 \beta> 1 β > 1 [100 ] . At zero temperature, the Stieltjes measure associated with the canonical Parisi order parameter has infinitely many points in its support [10 ] , and its support is [ 0 , 1 ) [0,1) [ 0 , 1 ) [32 ] . We do not use these results. The upper bound uses no information on the support of the Parisi measure.
The upper bound in Theorems 1.1 and 1.2 holds for every regular graph of large degree: Proposition 4.1 states that
lim sup D → ∞ sup G ∈ G D p G ( β , h ) ≤ P ( β , h ) (3) \limsup_{D\to\infty}\sup_{G\in\mathcal{G}_D} p_G(\beta,h) \le\mathcal{P}(\beta,h) \tag*{(3)} D → ∞ lim sup G ∈ G D sup p G ( β , h ) ≤ P ( β , h ) (3) for every β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R D D D n n n D D D K D + 1 K_{D+1} K D + 1
Remark 1.8. The complete bipartite graph K D , D K_{D,D} K D , D D D D D D D β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R
lim D → ∞ p K D , D ( β , h ) = P ( β , h ) . \lim_{D\to\infty} p_{K_{D,D}}(\beta,h) = \mathcal{P}(\beta,h). D → ∞ lim p K D , D ( β , h ) = P ( β , h ) . The lower bound follows from a Gaussian interpolation with the SK model on 2 D 2D 2 D K D , D K_{D,D} K D , D § 9.3 . In the limit D → ∞ D \to\infty D → ∞ [19 Theorem 1.3] , which holds in every field [19 Remark 1.8] . For the bipartite SK model in zero field, a Gaussian interpolation with independent SK models on the two species, which gives a lower bound of this kind, was used by Barra, Genovese, and Guerra [16 Section 3.1] , and for models with exchangeable species this comparison was proved by Issa [86 Theorem 38] . In zero field, the limit of the free energy of balanced multispecies models, which include this one, was identified in [37 , 83 ] ; see § 2.4 .
To our knowledge, for interactions of fixed range, no deterministic family of lattices was previously known on which the free energy of the EA model at low temperature, or its ground-state energy, converges to the SK value. Our results concern the limit in which the dimension tends to infinity, taken after or jointly with the thermodynamic limit; they do not decide between the competing pictures of the EA model in a fixed dimension.
Ideas of the proofs
The upper and lower bounds use different methods. We describe them in turn.
The upper bound. The interpolation identity recalled in § 2.4 gives the upper bound with an error of at most β 2 ϵ / 4 \beta^{2}\epsilon/4 β 2 ϵ /4 λ min ( S ) ≥ − ϵ \lambda_{\min}(S) \ge-\epsilon λ m i n ( S ) ≥ − ϵ S = A G / D S = A_G/D S = A G / D S S S − 1 -1 − 1 [56 , 107 , 109 ] , and in particular the strategy of Mourrat’s upper bounds for models with nonconvex interactions [106 , 108 ] . There, an enriched free energy is shown to be an approximate viscosity supersolution of a Hamilton–Jacobi equation, with Ghirlanda–Guerra identities and synchronization enforced at the points where a test function touches it. Fix a finite Ruelle cascade. We give each vertex x x x q x q_x q x F ( t , q ) F(t,q) F ( t , q ) β t \beta\sqrt{t} β t β 2 t / 4 \beta^{2}t/4 β 2 t /4 q = 0 q = 0 q = 0 t = 1 t = 1 t = 1 β 2 / 4 − p G ( β , h ) \beta^{2}/4 - p_G(\beta,h) β 2 /4 − p G ( β , h ) t = 0 t = 0 t = 0 F F F q q q β 2 ( 4 n ) − 1 ∑ j w j tr ( S C j ) \beta^{2}(4n)^{-1}\sum_j w_j\operatorname{tr}(SC_j) β 2 ( 4 n ) − 1 ∑ j w j tr ( S C j ) C j C_j C j σ 1 ∘ σ 2 \sigma^{1} \circ\sigma^{2} σ 1 ∘ σ 2 j j j S S S
We show instead that it is small. For every η > 0 \eta> 0 η > 0 ∣ tr ( S C j ) ∣ ≤ η tr C j + ( 4 η ) − 1 tr ( S 2 C j ) \lvert\operatorname{tr}(SC_j)\rvert\le\eta\operatorname{tr} C_j + (4\eta)^{-1}\operatorname{tr}(S^{2}C_j) ∣ tr ( S C j )∣ ≤ η tr C j + ( 4 η ) − 1 tr ( S 2 C j ) tr C j ≤ n \operatorname{tr} C_j \le n tr C j ≤ n tr ( S 2 C j ) \operatorname{tr}(S^{2}C_j) tr ( S 2 C j ) ( S ( σ 1 ∘ σ 2 ) ) x (S(\sigma^{1} \circ\sigma^{2}))_x ( S ( σ 1 ∘ σ 2 ) ) x x ∈ V x \in V x ∈ V [42 ] reduces this sum to a set of O ( n / D ) O(n/D) O ( n / D ) [115 ] , due to Mourrat [106 ] , then bounds the conditional variance of each of these row overlaps, given the overlap in the cascade, by a multiple of the mesh of the cascade. The perturbations change the free energy by O ( s 2 / D ) O(s^2/D) O ( s 2 / D ) s s s s → ∞ s \to\infty s → ∞ D → ∞ D \to\infty D → ∞ 1 / n 1/n 1/ n F F F V V V β 2 ( 4 n ) − 1 ∑ j w j p j ⋅ S p j \beta^2(4n)^{-1}\sum_j w_j p_j\cdot Sp_j β 2 ( 4 n ) − 1 ∑ j w j p j ⋅ S p j S S S S S S F F F h = 0 h = 0 h = 0 [107 ] . A comparison of this kind with a one-species Hamilton–Jacobi equation was used by Chen, Issa, and Mourrat [37 Appendix B] for balanced multispecies models in zero field; see also [86 Section 7] . There the initial condition is convex in the paths themselves [37 , 83 ] . This convexity fails in a strong field [106 Section 6.1] , whereas the convexity in the square roots holds in every field.
The lower bound. For m ∈ [ − 1 , 1 ] V m \in[-1,1]^V m ∈ [ − 1 , 1 ] V κ x = 1 − m x 2 \kappa_x = 1 - m_x^2 κ x = 1 − m x 2
Θ G ( m ) = β H G ( m ) + h ∑ x ∈ V m x + ∑ x ∈ V ent ( m x ) + β 2 4 ⟨ κ , S κ ⟩ , \Theta_G(m) = \beta H_G(m) + h \sum_{x \in V} m_x + \sum_{x \in V} \operatorname{ent}(m_x) + \frac{\beta^2}{4}\langle\kappa, S\kappa\rangle, Θ G ( m ) = β H G ( m ) + h x ∈ V ∑ m x + x ∈ V ∑ ent ( m x ) + 4 β 2 ⟨ κ , S κ ⟩ , where S = A G / D S = A_G/D S = A G / D [135 ] of the graph, that is, the expansion to second order in the couplings of the free energy at fixed magnetizations [73 , 121 ] , with each squared coupling replaced by its mean 1 / D 1/D 1/ D { − 1 , 1 } V \{-1,1\}^V { − 1 , 1 } V m m m log Z G ≥ Θ G ( m ) − ( β 2 / 4 ) ⟨ κ , S κ ⟩ \log Z_G \ge\Theta_G(m) - (\beta^2/4)\langle\kappa,S\kappa\rangle log Z G ≥ Θ G ( m ) − ( β 2 /4 ) ⟨ κ , S κ ⟩ m m m Z G Z_G Z G Θ G ( m ) \Theta_G(m) Θ G ( m )
First, we construct m m m [103 ] and Sellke [125 ] , whose nonlinearities are given by the Parisi partial differential equation (PDE). Assuming that the support of the Parisi measure is an interval [ 0 , q ∗ ] [0,q_*] [ 0 , q ∗ ] β \beta β m m m W W W [80 ] shows that n − 1 E Θ G ( m ) n^{-1}\mathbb{E}\,\Theta_G(m) n − 1 E Θ G ( m ) P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) m m m μ + \mu_+ μ + tanh X q + \tanh X_{q_+} tanh X q + X X X q + q_+ q + h > 0 h>0 h > 0 h = 0 h=0 h = 0 h h h
Second, we condition on the products computed by the algorithm. Conditioning on the iterates of message passing goes back to Bolthausen [25 ] , and it was used to compute the free energy of the SK model at high temperature in [26 , 29 ] . Given these products, the vector of couplings is the sum of its conditional mean and a Gaussian vector projected onto a subspace of bounded codimension per vertex. We expand the partition function around m m m σ = m + v \sigma=m+v σ = m + v a ( v ) x y = D − 1 / 2 v x v y a(v)_{xy}=D^{-1/2}v_xv_y a ( v ) x y = D − 1/2 v x v y D ( m ) D(m) D ( m ) β a ( v ∗ ) + Z \beta a(v^*)+Z β a ( v ∗ ) + Z v ∗ = σ ∗ − m v^*=\sigma^*-m v ∗ = σ ∗ − m σ ∗ \sigma^* σ ∗ m m m Z Z Z [132 ] , we show that the loss in the pressure is at most 4 β ( n − 1 E D ( m ) ) 1 / 2 4\beta\bigl(n^{-1}\mathbb{E}\,D(m)\bigr)^{1/2} 4 β ( n − 1 E D ( m ) ) 1/2 E D ( m ) = o ( n ) \mathbb{E}\,D(m)=o(n) E D ( m ) = o ( n ) [26 , 29 ] , the corresponding error is bounded by a conditional second-moment computation at high temperature; the planted model and the transport inequality take its place here. For spherical models, Huang and Sellke [85 ] proved the lower bound in the Parisi formula by constructing an ultrametric tree of pure states, each with approximately the same free energy as the whole model. They build the tree with an optimization algorithm and a truncated second-moment argument.
Third, we prove this relative-entropy estimate (Theorem 7.5). We interpolate between the edge observations and independent scalar Gaussian observations of the coordinates of v ∗ v^* v ∗ [56 , 104 , 105 ] . The Hopf–Lax formula involves the relative entropy ψ μ ( r ) \psi_\mu(r) ψ μ ( r ) r r r μ \mu μ m m m n − 1 D ( m ) n^{-1}D(m) n − 1 D ( m ) sup 0 ≤ s ≤ 1 [ ψ μ ( β 2 s ) − β 2 s 2 / 4 ] \sup_{0\le s\le1}[\psi_\mu(\beta^2s)-\beta^2s^2/4] sup 0 ≤ s ≤ 1 [ ψ μ ( β 2 s ) − β 2 s 2 /4 ] ψ m x \psi_{m_x} ψ m x ψ μ \psi_\mu ψ μ S S S S ⪯ I S \preceq I S ⪯ I S S S μ = μ + \mu=\mu_{+} μ = μ + 0 ≤ s ≤ 1 − q + 0 \le s \le1-q_{+} 0 ≤ s ≤ 1 − q + q + q_{+} q + ψ μ + ( β 2 s ) − β 2 s 2 / 4 \psi_{\mu_{+}}(\beta^{2}s)-\beta^{2}s^{2}/4 ψ μ + ( β 2 s ) − β 2 s 2 /4 [88 ] and by Chen [38 ] . The optimality of the Parisi measure shows that this difference is nonpositive. For s ≥ 1 − q + s \ge1-q_{+} s ≥ 1 − q + Γ μ + ′ ( s ) ≤ β 2 ( 1 − q + − s ) ≤ 0 \Gamma_{\mu_{+}}'(s) \le\beta^{2}(1-q_{+}-s) \le0 Γ μ + ′ ( s ) ≤ β 2 ( 1 − q + − s ) ≤ 0 s ↦ 2 ψ μ ( β 2 s ) − β 2 s 2 / 2 s \mapsto2\psi_{\mu}(\beta^{2}s)-\beta^{2}s^{2}/2 s ↦ 2 ψ μ ( β 2 s ) − β 2 s 2 /2 Γ μ \Gamma_{\mu} Γ μ [41 ] . With q = ∫ m 2 μ ( d m ) < 1 q=\int m^{2}\mu(dm)<1 q = ∫ m 2 μ ( d m ) < 1 [ 0 , 1 − q ] [0,1-q] [ 0 , 1 − q ] ∣ m ∣ |m| ∣ m ∣ [41 Proposition 13] .
The lower bound obtained in this way holds for tori of a fixed side length and for hypercubes. In Section 8, a Gaussian interpolation that adds or removes edges shows that the torus pressure changes by at most β 2 / ( 4 L ) \beta^{2}/(4L) β 2 / ( 4 L ) L L L d d d L L L [44 , 45 ] to study the thermodynamic limit and the surface pressure of the EA model.
Related work
Mean-field comparisons. Gaussian interpolation and comparison arguments are classical [75 , 76 ] . For the SK model, they give the identity
p n S K ( β , h ) − p G ( β , h ) = β 2 4 ∫ 0 1 E ⟨ Q − R 2 ⟩ θ d θ p_n^{\mathrm{SK}}(\beta,h)-p_G(\beta,h)=\frac{\beta^{2}}{4}\int_{0}^{1}\mathbb{E}\left\langle Q-R^{2}\right\rangle_{\theta}\,d\theta p n SK ( β , h ) − p G ( β , h ) = 4 β 2 ∫ 0 1 E ⟨ Q − R 2 ⟩ θ d θ along the interpolation 1 − θ H G + θ H n S K \sqrt{1-\theta}H_G+\sqrt{\theta}H_n^{\mathrm{SK}} 1 − θ H G + θ H n SK Q − R 2 = n − 1 ⟨ u − R 1 , S ( u − R 1 ) ⟩ Q-R^{2}=n^{-1}\langle u-R\mathbf{1},S(u-R\mathbf{1})\rangle Q − R 2 = n − 1 ⟨ u − R 1 , S ( u − R 1 )⟩ λ min ( S ) ≥ − ϵ \lambda_{\min}(S)\ge-\epsilon λ m i n ( S ) ≥ − ϵ Q − R 2 ≥ − ϵ ( 1 − R 2 ) ≥ − ϵ Q-R^{2}\ge-\epsilon(1-R^{2})\ge-\epsilon Q − R 2 ≥ − ϵ ( 1 − R 2 ) ≥ − ϵ p G ( β , h ) ≤ p n S K ( β , h ) + β 2 ϵ / 4 p_G(\beta,h)\le p_n^{\mathrm{SK}}(\beta,h)+\beta^{2}\epsilon/4 p G ( β , h ) ≤ p n SK ( β , h ) + β 2 ϵ /4 G ∈ G D G\in\mathcal{G}_D G ∈ G D S S S [77 ] . If the eigenvalues of S S S 1 \mathbf{1} 1 Q − R 2 Q-R^{2} Q − R 2 S S S 1 1 1 S S S − 1 -1 − 1 Q − R 2 Q-R^{2} Q − R 2
Kac limits and mean-field approximations. High-temperature results for diverging-range mean-field limits were obtained earlier by Fröhlich and Zegarliński [71 ] . In the Kac limit, the lower bound of Franz and Toninelli [69 , 70 ] interpolates between the Kac model and independent SK models on boxes whose side is large compared with the lattice spacing and small compared with the range of the interaction. It uses that each spin interacts with many spins through a kernel that varies slowly on the scale of the boxes. For nearest-neighbor interactions there are no such boxes, and our lower bound uses message passing instead. For Ising and Potts models with an interaction matrix A A A A A A tr ( A 2 ) = o ( n ) \operatorname{tr}(A^{2}) = o(n) tr ( A 2 ) = o ( n ) [18 Theorem 1.1] . In particular, on regular graphs of diverging degree D D D 1 / D 1/D 1/ D [18 Theorem 2.1] . For the EA model, tr ( ( β W ) 2 ) \operatorname{tr}((\beta W)^{2}) tr (( β W ) 2 ) n n n
The Parisi formula. The upper bound in the Parisi formula was proved by Guerra [76 ] and the lower bound by Talagrand [134 ] . Panchenko proved ultrametricity from the Ghirlanda–Guerra identities [74 , 113 , 114 ] . For multispecies models with a positive-definite interaction matrix, Barra, Contucci, Mingione, and Tantari proved the upper bound in the Parisi formula, and Panchenko proved the lower bound by showing that the overlaps of different species are synchronized [17 ] [115 ] . Synchronization was extended to Potts and vector spins in [116 , 117 ] . Connections between the free energy of spin glasses and Hamilton–Jacobi equations were pointed out by Guerra and coauthors; see [15 ] . Mourrat and coauthors developed an approach to the Parisi formula and its extensions through Hamilton–Jacobi equations [56 , 105 , 107 , 108 , 109 ] , and Chen and Xia studied these equations on convex cones [36 ] . Our upper bound combines these tools. The rows of S S S n / D n/D n / D [7 ] and the characterization of the Parisi measure through the first variation of the Parisi functional, obtained by Jagannath and Tobasco [88 ] and independently by Chen [38 ] .
Multispecies models with a nonconvex covariance. For such models, which include the bipartite SK model [16 ] , the Hamilton–Jacobi approach gives an upper bound on the free energy [106 , 108 ] . Chen and Mourrat showed that, if the limit of the free energy of a vector spin glass with nonconvex interactions exists, it is a critical value of a functional of Parisi type [33 ] , and Chen extended this to multispecies models [31 ] . For the bipartite spherical SK model, the limit of the free energy was computed by Auffinger and Chen at high temperature [5 ] and by Baik and Lee in zero field, at every noncritical temperature [11 ] ; the critical value follows by continuity. Subag computed it for pure multispecies spherical models through a TAP representation, assuming that certain free energies converge [130 ] . For Ising spins, in zero field the limit of the free energy was recently identified by Chen, Issa, and Mourrat [37 ] . For balanced models, such as the bipartite SK model with species of equal sizes, this limit is given by a one-species formula. The corresponding lower bound is due to Bates and Sohn [19 ] , and for models with exchangeable species to Issa [86 ] . The matching upper bound was proved in [37 , 83 ] , and the lower bound of Bates and Sohn holds in every field [19 Remark 1.8] . The model on the complete bipartite graph K D , D K_{D,D} K D , D
Message passing and the TAP free energy. AMP was introduced by Donoho, Maleki, and Montanari [57 ] . For the SK model, an iteration with Gaussian state evolution was introduced by Bolthausen [25 ] , and general AMP algorithms were analyzed in [21 , 22 , 91 ] . Hachem extended polynomial AMP to sparse variance profiles [80 ] . Montanari [103 ] showed that AMP with nonlinearities given by the Parisi PDE approximately maximizes the SK Hamiltonian if, for all large β \beta β [ 0 , q ∗ ] [0,q_*] [ 0 , q ∗ ] [125 ] extended the algorithm to nonzero external fields under a corresponding no-overlap-gap assumption. El Alaoui, Montanari, and Sellke [60 ] extended these algorithms to mixed p p p [61 ] . Fan, Li, and Sen [66 ] proved the TAP equations for spin glasses with orthogonally invariant couplings at high temperature, through a conditional second-moment analysis of the free energy restricted to a thin band around the output of AMP. The functional β H ( m ) + h ∑ x m x + ∑ x ent ( m x ) + ( β 2 / 4 ) ∑ x , y S x y ( 1 − m x 2 ) ( 1 − m y 2 ) \beta H(m) + h \sum_x m_x + \sum_x \operatorname{ent}(m_x) + (\beta^2/4) \sum_{x,y} S_{xy}(1-m_x^2)(1-m_y^2) β H ( m ) + h ∑ x m x + ∑ x ent ( m x ) + ( β 2 /4 ) ∑ x , y S x y ( 1 − m x 2 ) ( 1 − m y 2 ) p p p [39 Theorem 1] , and this representation was extended by Chen, Panchenko, and Subag [40 , 41 ] ; see also [9 , 65 ] . For spherical models, a TAP representation of the free energy was proved by Subag [131 ] .
Organization
Section 2 collects notation and the facts about the SK model and the Parisi formula that we use. Sections 3 and 4 prove the upper bound ((3) ) for all regular graphs. Sections 5–7 prove the lower bound for tori of fixed side length and for hypercubes: Section 5 constructs the centers by message passing, Section 6 reduces the lower bound to a sublinear relative-entropy bound for a planted model, and Section 7 proves this statement. Section 8 removes the restriction to a fixed side length and proves Theorems 1.1 and 1.2 and Corollary 1.3.
Preliminaries
This section fixes notation and collects the inputs used in the rest of the paper. Section 2.1 defines the model and records its elementary properties. Section 2.2 lists classical facts about Gaussian vectors. Section 2.3 recalls the Sherrington–Kirkpatrick model, the Parisi formula, and the properties of the Parisi PDE that we use. It ends with Lemma 2.6, which writes P ( β , h ) \mathcal{P}(\beta,h) P ( β , h )
The model
For D ≥ 1 D \ge1 D ≥ 1 G D \mathcal{G}_D G D D D D G = ( V , E ) G=(V,E) G = ( V , E ) n = ∣ V ∣ n=|V| n = ∣ V ∣ x , y , z x,y,z x , y , z e e e x y xy x y G ∈ G D G \in\mathcal{G}_D G ∈ G D S = A G / D S=A_G/D S = A G / D
S 1 = 1 , ∥ S ∥ ≤ 1 , tr S 2 = n D , max x , y ∈ V S x y = 1 D . (4) S\mathbf{1}=\mathbf{1}, \qquad\lVert S\rVert\le1, \qquad\operatorname{tr} S^2=\frac{n}{D}, \qquad\max_{x,y\in V} S_{xy}=\frac{1}{D}. \tag*{(4)} S 1 = 1 , ∥ S ∥ ≤ 1 , tr S 2 = D n , x , y ∈ V max S x y = D 1 . (4) where ∥ ⋅ ∥ \lVert\cdot\rVert ∥ ⋅ ∥ ∑ x ( ∑ y S x y w y ) 2 ≤ ∑ x , y S x y w y 2 = ∑ y w y 2 \sum_x\left(\sum_y S_{xy}w_y\right)^2\le\sum_{x,y}S_{xy}w_y^2=\sum_y w_y^2 ∑ x ( ∑ y S x y w y ) 2 ≤ ∑ x , y S x y w y 2 = ∑ y w y 2
Let ( g e ) e ∈ E (g_e)_{e\in E} ( g e ) e ∈ E J x y = g x y / D J_{xy}=g_{xy}/\sqrt{D} J x y = g x y / D W W W V × V V\times V V × V W x y = J x y W_{xy}=J_{xy} W x y = J x y x y ∈ E xy\in E x y ∈ E W x y = 0 W_{xy}=0 W x y = 0 W x x = 0 W_{xx}=0 W xx = 0 β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R
H G ( w ) = ∑ x y ∈ E J x y w x w y = 1 2 ⟨ w , W w ⟩ ( w ∈ R V ) , p G ( β , h ) = 1 n E log ∑ σ ∈ { − 1 , 1 } V exp ( β H G ( σ ) + h ∑ x ∈ V σ x ) , e ( G ; h ) = 1 n E max σ ∈ { − 1 , 1 } V ( H G ( σ ) + h ∑ x ∈ V σ x ) . (5) \begin{aligned}
H_G(w)&=\sum_{xy\in E}J_{xy}w_xw_y=\frac{1}{2}\langle w,Ww\rangle&&(w\in\mathbb{R}^V),\\
p_G(\beta,h)&=\frac{1}{n}\mathbb{E}\log\sum_{\sigma\in\{-1,1\}^V}\exp\left(\beta H_G(\sigma)+h\sum_{x\in V}\sigma_x\right),\\
e(G;h)&=\frac{1}{n}\mathbb{E}\max_{\sigma\in\{-1,1\}^V}\left(H_G(\sigma)+h\sum_{x\in V}\sigma_x\right). \tag*{(5)}
\end{aligned} H G ( w ) p G ( β , h ) e ( G ; h ) = x y ∈ E ∑ J x y w x w y = 2 1 ⟨ w , W w ⟩ = n 1 E log σ ∈ { − 1 , 1 } V ∑ exp ( β H G ( σ ) + h x ∈ V ∑ σ x ) , = n 1 E σ ∈ { − 1 , 1 } V max ( H G ( σ ) + h x ∈ V ∑ σ x ) . ( w ∈ R V ) , (5) The field is not multiplied by β \beta β σ \sigma σ E H G ( σ ) 2 = ∣ E ∣ / D = n / 2 \mathbb{E}H_G(\sigma)^2=|E|/D=n/2 E H G ( σ ) 2 = ∣ E ∣/ D = n /2
The torus T L d = ( Z / L Z ) d T_L^d = (\mathbb{Z}/L\mathbb{Z})^d T L d = ( Z / L Z ) d L ≥ 3 L \ge3 L ≥ 3 ± 1 \pm1 ± 1 G 2 d \mathcal{G}_{2d} G 2 d n = L d n = L^d n = L d p d , L p_{d,L} p d , L Q d = { 0 , 1 } d Q_d = \{0,1\}^d Q d = { 0 , 1 } d G d \mathcal{G}_d G d n = 2 d n = 2^d n = 2 d
For fixed disorder, ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ σ \sigma σ exp ( β H G ( σ ) + h ∑ x σ x ) \exp(\beta H_G(\sigma) + h\sum_x \sigma_x) exp ( β H G ( σ ) + h ∑ x σ x ) σ 1 , σ 2 , … \sigma^1,\sigma^2,\ldots σ 1 , σ 2 , … u = σ 1 ⊙ σ 2 u = \sigma^1 \odot\sigma^2 u = σ 1 ⊙ σ 2
R = 1 n ∑ x ∈ V u x , Q = 1 ∣ E ∣ ∑ x y ∈ E u x u y = 1 n ⟨ u , S u ⟩ R = \frac{1}{n}\sum_{x\in V}u_x,\qquad Q = \frac{1}{|E|}\sum_{xy\in E}u_xu_y = \frac{1}{n}\langle u,Su\rangle R = n 1 x ∈ V ∑ u x , Q = ∣ E ∣ 1 x y ∈ E ∑ u x u y = n 1 ⟨ u , S u ⟩ for the site overlap and the link overlap. The second expression for Q Q Q ∣ E ∣ = n D / 2 |E| = nD/2 ∣ E ∣ = n D /2
The following notation is used in the lower bound. For m ∈ [ − 1 , 1 ] m\in[-1,1] m ∈ [ − 1 , 1 ]
ent ( m ) = − 1 + m 2 log 1 + m 2 − 1 − m 2 log 1 − m 2 ∈ [ 0 , log 2 ] , \operatorname{ent}(m)=-\frac{1+m}{2}\log\frac{1+m}{2}-\frac{1-m}{2}\log\frac{1-m}{2}\in[0,\log2], ent ( m ) = − 2 1 + m log 2 1 + m − 2 1 − m log 2 1 − m ∈ [ 0 , log 2 ] , with 0 log 0 = 0 0\log0=0 0 log 0 = 0 { − 1 , 1 } \{-1,1\} { − 1 , 1 } m m m m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V π m \pi_m π m { − 1 , 1 } V \{-1,1\}^V { − 1 , 1 } V m m m κ x = 1 − m x 2 \kappa_x=1-m_x^2 κ x = 1 − m x 2 m x = tanh y m_x=\tanh y m x = tanh y
ent ( m x ) = log 2 cosh y − y tanh y . (6) \operatorname{ent}(m_x)=\log2\cosh y-y\tanh y. \tag*{(6)} ent ( m x ) = log 2 cosh y − y tanh y . (6) Lemma 2.1. Let G ∈ G D G\in\mathcal{G}_D G ∈ G D
(a) The function ( β , h ) ↦ p G ( β , h ) (\beta,h)\mapsto p_G(\beta,h) ( β , h ) ↦ p G ( β , h ) R 2 \mathbb{R}^2 R 2 h h h p G ( 0 , h ) = log 2 cosh h p_G(0,h)=\log2\cosh h p G ( 0 , h ) = log 2 cosh h
(b) For β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R
∂ β p G ( β , h ) = β 2 ( 1 − E ⟨ Q ⟩ ) ∈ [ 0 , β 2 ] , ∂ h p G ( β , h ) = 1 n ∑ x ∈ V E ⟨ σ x ⟩ ∈ [ − 1 , 1 ] . \partial_\beta p_G(\beta,h)=\frac{\beta}{2}\left(1-\mathbb{E}\langle Q\rangle\right)\in\left[0,\frac{\beta}{2}\right],\qquad\partial_h p_G(\beta,h)=\frac{1}{n}\sum_{x\in V}\mathbb{E}\langle\sigma_x\rangle\in[-1,1]. ∂ β p G ( β , h ) = 2 β ( 1 − E ⟨ Q ⟩ ) ∈ [ 0 , 2 β ] , ∂ h p G ( β , h ) = n 1 x ∈ V ∑ E ⟨ σ x ⟩ ∈ [ − 1 , 1 ] . (c) For β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R log 2 cosh h ≤ p G ( β , h ) ≤ log 2 cosh h + β 2 / 4 \log2\cosh h\le p_G(\beta,h)\le\log2\cosh h+\beta^2/4 log 2 cosh h ≤ p G ( β , h ) ≤ log 2 cosh h + β 2 /4
(d) For B ≥ 0 B\ge0 B ≥ 0 β , β ′ ∈ [ 0 , B ] \beta,\beta'\in[0,B] β , β ′ ∈ [ 0 , B ] h , h ′ ∈ R h,h'\in\mathbb{R} h , h ′ ∈ R ∣ p G ( β , h ) − p G ( β ′ , h ′ ) ∣ ≤ B ∣ β − β ′ ∣ / 2 + ∣ h − h ′ ∣ |p_G(\beta,h)-p_G(\beta',h')|\le B|\beta-\beta'|/2+|h-h'| ∣ p G ( β , h ) − p G ( β ′ , h ′ ) ∣ ≤ B ∣ β − β ′ ∣/2 + ∣ h − h ′ ∣
Proof. For fixed disorder, the function ( β , h ) ↦ log ∑ σ exp ( β H G ( σ ) + h ∑ x σ x ) (\beta,h)\mapsto\log\sum_\sigma\exp(\beta H_G(\sigma)+h\sum_x\sigma_x) ( β , h ) ↦ log ∑ σ exp ( β H G ( σ ) + h ∑ x σ x ) ⟨ H G ⟩ \langle H_G\rangle ⟨ H G ⟩ ∑ x ⟨ σ x ⟩ \sum_x\langle\sigma_x\rangle ∑ x ⟨ σ x ⟩ max σ ∣ H G ( σ ) ∣ \max_\sigma|H_G(\sigma)| max σ ∣ H G ( σ ) ∣ n n n σ ↦ − σ \sigma\mapsto-\sigma σ ↦ − σ p G p_G p G h h h β = 0 \beta=0 β = 0 ( 2 cosh h ) n (2\cosh h)^n ( 2 cosh h ) n ∂ h p G \partial_h p_G ∂ h p G
For the β \beta β g x y g_{xy} g x y
E [ g x y ⟨ σ x σ y ⟩ ] = E ∂ g x y ⟨ σ x σ y ⟩ = β D E ( 1 − ⟨ σ x σ y ⟩ 2 ) . \mathbb{E}\left[g_{xy}\langle\sigma_x\sigma_y\rangle\right]=\mathbb{E}\,\partial_{g_{xy}}\langle\sigma_x\sigma_y\rangle=\frac{\beta}{\sqrt{D}}\mathbb{E}\left(1-\langle\sigma_x\sigma_y\rangle^2\right). E [ g x y ⟨ σ x σ y ⟩ ] = E ∂ g x y ⟨ σ x σ y ⟩ = D β E ( 1 − ⟨ σ x σ y ⟩ 2 ) . Since ⟨ σ x σ y ⟩ 2 = ⟨ u x u y ⟩ \langle\sigma_x \sigma_y\rangle^2=\langle u_xu_y\rangle ⟨ σ x σ y ⟩ 2 = ⟨ u x u y ⟩ ∣ E ∣ = n D / 2 |E|=nD/2 ∣ E ∣ = n D /2
∂ β p G = 1 n ∑ x y ∈ E 1 D E [ g x y ⟨ σ x σ y ⟩ ] = β n D ∑ x y ∈ E ( 1 − E [ u x u y ] ) = β 2 ( 1 − E [ Q ] ) . \partial_\beta p_G=\frac{1}{n}\sum_{xy\in E}\frac{1}{\sqrt{D}}\mathbb{E}\left[g_{xy}\langle\sigma_x\sigma_y\rangle\right]=\frac{\beta}{nD}\sum_{xy\in E}\left(1-\mathbb{E}\left[u_xu_y\right]\right)=\frac{\beta}{2}\left(1-\mathbb{E}\left[Q\right]\right). ∂ β p G = n 1 x y ∈ E ∑ D 1 E [ g x y ⟨ σ x σ y ⟩ ] = n D β x y ∈ E ∑ ( 1 − E [ u x u y ] ) = 2 β ( 1 − E [ Q ] ) . Since ⟨ Q ⟩ = ∣ E ∣ − 1 ∑ x y ∈ E ⟨ σ x σ y ⟩ 2 ∈ [ 0 , 1 ] \langle Q\rangle=|E|^{-1}\sum_{xy\in E}\langle\sigma_x\sigma_y\rangle^2\in[0,1] ⟨ Q ⟩ = ∣ E ∣ − 1 ∑ x y ∈ E ⟨ σ x σ y ⟩ 2 ∈ [ 0 , 1 ] β \beta β 0 0 0
Classical tools
For a random variable Y Y Y ∥ Y ∥ p = ( E ∣ Y ∣ p ) 1 / p \|Y\|_p=(\mathbb{E}|Y|^p)^{1/p} ∥ Y ∥ p = ( E ∣ Y ∣ p ) 1/ p R k \mathbb{R}^k R k polynomial growth if it is bounded by C ( 1 + ∣ w ∣ p ) C(1+|w|^p) C ( 1 + ∣ w ∣ p ) C , p C,p C , p
(F1) Gaussian integration by parts. Let ( U 1 , … , U k ) (U_1,\ldots,U_k) ( U 1 , … , U k ) Ψ : R k → R \Psi:\mathbb{R}^k\to\mathbb{R} Ψ : R k → R C 1 C^1 C 1 Ψ \Psi Ψ ∇ Ψ \nabla\Psi ∇Ψ j ≤ k j\leq k j ≤ k
E [ U j Ψ ( U 1 , … , U k ) ] = ∑ ℓ = 1 k Cov ( U j , U ℓ ) E ∂ ℓ Ψ ( U 1 , … , U k ) . \mathbb{E}\left[U_j\Psi(U_1,\ldots,U_k)\right]=\sum_{\ell=1}^{k}\operatorname{Cov}(U_j,U_\ell)\mathbb{E}\,\partial_\ell\Psi(U_1,\ldots,U_k). E [ U j Ψ ( U 1 , … , U k ) ] = ℓ = 1 ∑ k Cov ( U j , U ℓ ) E ∂ ℓ Ψ ( U 1 , … , U k ) . For one standard Gaussian variable this is Stein’s identity [128 ] , and the general case follows by writing U ℓ = a ℓ U j + V ℓ U_\ell=a_\ell U_j+V_\ell U ℓ = a ℓ U j + V ℓ a ℓ = Cov ( U ℓ , U j ) / Var U j a_\ell=\operatorname{Cov}(U_\ell,U_j)/\operatorname{Var}U_j a ℓ = Cov ( U ℓ , U j ) / Var U j ( V ℓ ) (V_\ell) ( V ℓ ) U j U_j U j ( V ℓ ) (V_\ell) ( V ℓ ) Var U j = 0 \operatorname{Var}U_j=0 Var U j = 0
(F2) Method of moments. A Gaussian law on R k \mathbb{R}^k R k Z ℓ Z_\ell Z ℓ E ψ ( Z ℓ ) → E ψ ( Z ) \mathbb{E}\psi(Z_\ell)\to\mathbb{E}\psi(Z) E ψ ( Z ℓ ) → E ψ ( Z ) ψ \psi ψ Z Z Z Z ℓ → Z Z_\ell\to Z Z ℓ → Z [24 Theorems 30.1 and 30.2] , and the general case follows by the Cramér–Wold device [24 Theorem 29.4] .
(F3) Polynomial density. For every Gaussian measure γ \gamma γ R k \mathbb{R}^k R k 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ L p ( γ ) L^p(\gamma) L p ( γ ) γ \gamma γ γ \gamma γ f ∈ L p ′ ( γ ) f\in L^{p'}(\gamma) f ∈ L p ′ ( γ ) 1 / p + 1 / p ′ = 1 1/p+1/p'=1 1/ p + 1/ p ′ = 1 ∫ f P d γ = 0 \int fP\,\mathrm{d}\gamma=0 ∫ f P d γ = 0 P P P ∫ ∣ f ( w ) ∣ e c ∣ w ∣ γ ( d w ) < ∞ \int|f(w)|e^{c|w|}\gamma(\mathrm{d}w)<\infty ∫ ∣ f ( w ) ∣ e c ∣ w ∣ γ ( d w ) < ∞ c > 0 c>0 c > 0 ∫ f ( w ) e i ⟨ t , w ⟩ γ ( d w ) = 0 \int f(w)e^{i\langle t,w\rangle}\gamma(\mathrm{d}w)=0 ∫ f ( w ) e i ⟨ t , w ⟩ γ ( d w ) = 0 t t t ∫ f d γ = 0 \int f\,\mathrm{d}\gamma=0 ∫ f d γ = 0 f γ f\gamma f γ [24 Section 29] ; if it is zero, both parts vanish. In either case f γ = 0 f\gamma=0 f γ = 0
(F4) Gaussian concentration. Let Ψ : R k → R \Psi:\mathbb{R}^k \to\mathbb{R} Ψ : R k → R L L L g g g R k \mathbb{R}^k R k Var Ψ ( g ) ≤ L 2 \operatorname{Var}\Psi(g) \le L^2 Var Ψ ( g ) ≤ L 2 P ( ∣ Ψ ( g ) − E Ψ ( g ) ∣ ≥ t ) ≤ 2 e − t 2 / ( 2 L 2 ) \mathbb{P}(|\Psi(g)-\mathbb{E}\Psi(g)| \ge t) \le2e^{-t^2/(2L^2)} P ( ∣Ψ ( g ) − E Ψ ( g ) ∣ ≥ t ) ≤ 2 e − t 2 / ( 2 L 2 ) t ≥ 0 t \ge0 t ≥ 0 [28 Chapter 3 and Section 5.4] . In particular, ∥ Ψ ( g ) − E Ψ ( g ) ∥ p ≤ C p L \|\Psi(g)-\mathbb{E}\Psi(g)\|_p \le C_pL ∥Ψ ( g ) − E Ψ ( g ) ∥ p ≤ C p L p ≥ 1 p \ge1 p ≥ 1 C p C_p C p p p p
(F5) Transport inequality. Let γ k \gamma_k γ k R k \mathbb{R}^k R k ρ \rho ρ R k \mathbb{R}^k R k W 2 ( ρ , γ k ) 2 ≤ 2 KL ( ρ ∥ γ k ) W_2(\rho,\gamma_k)^2 \le2\operatorname{KL}(\rho\mathbin\Vert\gamma_k) W 2 ( ρ , γ k ) 2 ≤ 2 KL ( ρ ∥ γ k ) [132 ] .
(F6) Maximum of the Hamiltonian. Let G ∈ G D G \in\mathcal{G}_D G ∈ G D H G H_G H G sup w ∈ [ − 1 , 1 ] V ∣ H G ( w ) ∣ = max σ ∣ H G ( σ ) ∣ \sup_{w\in[-1,1]^V}|H_G(w)|=\max_{\sigma}|H_G(\sigma)| sup w ∈ [ − 1 , 1 ] V ∣ H G ( w ) ∣ = max σ ∣ H G ( σ ) ∣ p ≥ 1 p \ge1 p ≥ 1 C p C_p C p p p p E max σ ∣ H G ( σ ) ∣ p ≤ C p n p \mathbb{E}\max_{\sigma}|H_G(\sigma)|^p \le C_p n^p E max σ ∣ H G ( σ ) ∣ p ≤ C p n p max σ ∣ H G ( σ ) ∣ \max_{\sigma}|H_G(\sigma)| max σ ∣ H G ( σ ) ∣ 2 n + 1 2^{n+1} 2 n + 1 ± H G ( σ ) \pm H_G(\sigma) ± H G ( σ ) n / 2 n/2 n /2 ( 2 ⋅ ( n / 2 ) log 2 n + 1 ) 1 / 2 ≤ 2 n (2\cdot(n/2)\log2^{n+1})^{1/2}\le2n ( 2 ⋅ ( n /2 ) log 2 n + 1 ) 1/2 ≤ 2 n ( n / 2 ) 1 / 2 (n/2)^{1/2} ( n /2 ) 1/2 ( g e ) (g_e) ( g e )
The Sherrington–Kirkpatrick model and the Parisi formula
The constants. Let H N S K ( σ ) = N − 1 / 2 ∑ 1 ≤ i < j ≤ N g i j σ i σ j H_N^{\mathrm{SK}}(\sigma)=N^{-1/2}\sum_{1\le i<j\le N}g_{ij}\sigma_i\sigma_j H N SK ( σ ) = N − 1/2 ∑ 1 ≤ i < j ≤ N g ij σ i σ j { − 1 , 1 } N \{-1,1\}^N { − 1 , 1 } N ( g i j ) (g_{ij}) ( g ij )
p N S K ( β , h ) = 1 N E log ∑ σ e β H N S K ( σ ) + h ∑ i σ i , m N ( h ) = 1 N E max σ ( H N S K ( σ ) + h ∑ i σ i ) . p_N^{\mathrm{SK}}(\beta,h)=\frac{1}{N}\mathbb{E}\log\sum_{\sigma}e^{\beta H_N^{\mathrm{SK}}(\sigma)+h\sum_i\sigma_i},\qquad m_N(h)=\frac{1}{N}\mathbb{E}\max_{\sigma}\left(H_N^{\mathrm{SK}}(\sigma)+h\sum_i\sigma_i\right). p N SK ( β , h ) = N 1 E log σ ∑ e β H N SK ( σ ) + h ∑ i σ i , m N ( h ) = N 1 E σ max ( H N SK ( σ ) + h i ∑ σ i ) . The proof of Lemma 2.1 applies to p N S K p_N^{\mathrm{SK}} p N SK R R R R 2 R^2 R 2 Q Q Q ∂ β p N S K = ( β / 2 ) ( 1 − E ⟨ R 2 ⟩ ) \partial_\beta p_N^{\mathrm{SK}}=(\beta/2)(1-\mathbb{E}\langle R^2\rangle) ∂ β p N SK = ( β /2 ) ( 1 − E ⟨ R 2 ⟩) p N S K p_N^{\mathrm{SK}} p N SK P ( β , h ) = lim N → ∞ p N S K ( β , h ) P(\beta,h)=\lim_{N\to\infty}p_N^{\mathrm{SK}}(\beta,h) P ( β , h ) = lim N → ∞ p N SK ( β , h ) [75 ] , and it is given by the Parisi formula (2.7) below. By Lemma 2.1, P P P h h h
Bounding the partition function at inverse temperature β \beta β β h \beta h β h 2 N 2^N 2 N β > 0 \beta>0 β > 0
β m N ( h ) ≤ 1 N E log ∑ σ e β ( H N S K ( σ ) + h ∑ i σ i ) ≤ β m N ( h ) + log 2. (7) \beta m_N(h)\le\frac{1}{N}\mathbb{E}\log\sum_{\sigma}e^{\beta(H_N^{\mathrm{SK}}(\sigma)+h\sum_i\sigma_i)}\le\beta m_N(h)+\log2. \tag*{(7)} β m N ( h ) ≤ N 1 E log σ ∑ e β ( H N SK ( σ ) + h ∑ i σ i ) ≤ β m N ( h ) + log 2. (7) By the same argument, we have β e ( G ; h ) ≤ p G ( β , β h ) ≤ β e ( G ; h ) + log 2 \beta e(G;h)\le p_G(\beta,\beta h)\le\beta e(G;h)+\log2 β e ( G ; h ) ≤ p G ( β , β h ) ≤ β e ( G ; h ) + log 2 G ∈ G D G\in\mathcal{G}_D G ∈ G D N → ∞ N\to\infty N → ∞ (7) , we obtain lim sup N m N ( h ) ≤ P ( β , β h ) / β ≤ lim inf N m N ( h ) + ( log 2 ) / β \limsup_Nm_N(h)\le P(\beta,\beta h)/\beta\le\liminf_Nm_N(h)+(\log2)/\beta lim sup N m N ( h ) ≤ P ( β , β h ) / β ≤ lim inf N m N ( h ) + ( log 2 ) / β β > 0 \beta>0 β > 0 β → ∞ \beta\to\infty β → ∞ m N ( h ) m_N(h) m N ( h ) P ∗ ( h ) P_*(h) P ∗ ( h ) P ∗ = P ∗ ( 0 ) P_*=P_*(0) P ∗ = P ∗ ( 0 ) P ∗ ( h ) = lim β → ∞ P ( β , β h ) / β P_*(h)=\lim_{\beta\to\infty}P(\beta,\beta h)/\beta P ∗ ( h ) = lim β → ∞ P ( β , β h ) / β β > 0 \beta>0 β > 0
β P ∗ ( h ) ≤ P ( β , β h ) ≤ β P ∗ ( h ) + log 2. (8) \beta P_*(h)\le P(\beta,\beta h)\le\beta P_*(h)+\log2. \tag*{(8)} β P ∗ ( h ) ≤ P ( β , β h ) ≤ β P ∗ ( h ) + log 2. (8) The Parisi formula. For a probability measure μ \mu μ [ 0 , 1 ] [0,1] [ 0 , 1 ] μ ( t ) = μ ( [ 0 , t ] ) \mu(t)=\mu([0,t]) μ ( t ) = μ ([ 0 , t ]) Φ μ \Phi_\mu Φ μ
∂ t Φ + β 2 2 ( ∂ x 2 Φ + μ ( t ) ( ∂ x Φ ) 2 ) = 0 on [ 0 , 1 ) × R , Φ ( 1 , x ) = log 2 cosh x . (9) \partial_t\Phi+\frac{\beta^2}{2}\left(\partial_x^2\Phi+\mu(t)(\partial_x\Phi)^2\right)=0\quad\text{on }[0,1)\times\mathbb{R},\qquad\Phi(1,x)=\log2\cosh x. \tag*{(9)} ∂ t Φ + 2 β 2 ( ∂ x 2 Φ + μ ( t ) ( ∂ x Φ ) 2 ) = 0 on [ 0 , 1 ) × R , Φ ( 1 , x ) = log 2 cosh x . (9) For atomic μ \mu μ [7 (4)–(5)] define Φ μ \Phi_\mu Φ μ (9) classically between consecutive atoms. They extend μ ↦ Φ μ \mu\mapsto\Phi_\mu μ ↦ Φ μ μ \mu μ [76 ] , and this is the function we use. The PDE does not involve the field. The Parisi functional is
P β , h ( μ ) = Φ μ ( 0 , h ) − β 2 2 ∫ 0 1 t μ ( t ) d t . \mathcal{P}_{\beta,h}(\mu)=\Phi_\mu(0,h)-\frac{\beta^2}{2}\int_0^1t\mu(t)\,\mathrm{d}t. P β , h ( μ ) = Φ μ ( 0 , h ) − 2 β 2 ∫ 0 1 t μ ( t ) d t . Parisi predicted the formula (10) below [118 , 119 ] ; Guerra proved the upper bound [76 ] , and Talagrand proved the formula [134 ] . By Talagrand’s theorem and the uniqueness theorem of Auffinger and Chen [7 Corollary 1] , for every β > 0 \beta>0 β > 0 h ∈ R h\in\mathbb{R} h ∈ R
P ( β , h ) = min μ P β , h ( μ ) = P β , h ( ν β , h ) (10) \mathcal{P}(\beta,h)=\min_{\mu}\mathcal{P}_{\beta,h}(\mu)=\mathcal{P}_{\beta,h}(\nu_{\beta,h}) \tag*{(10)} P ( β , h ) = μ min P β , h ( μ ) = P β , h ( ν β , h ) (10) for a unique minimizer ν β , h \nu_{\beta,h} ν β , h Φ μ ( t , ⋅ ) \Phi_\mu(t,\cdot) Φ μ ( t , ⋅ ) P β , h = P β , − h \mathcal{P}_{\beta,h}=\mathcal{P}_{\beta,-h} P β , h = P β , − h ν β , − h = ν β , h \nu_{\beta,-h}=\nu_{\beta,h} ν β , − h = ν β , h q − = min supp ν β , h q_-=\min\operatorname{supp}\nu_{\beta,h} q − = min supp ν β , h q + = max supp ν β , h q_+=\max\operatorname{supp}\nu_{\beta,h} q + = max supp ν β , h
Most of the cited papers use the terminal condition log cosh x \log\cosh x log cosh x log 2 \log2 log 2 N ξ ( R ) N\xi(R) N ξ ( R ) ξ ( s ) = β 2 s 2 / 2 \xi(s)=\beta^2s^2/2 ξ ( s ) = β 2 s 2 /2 [99 Sections 1.1 and 2] the Gibbs weight is exp ( β H N S K ( σ ) + h ∑ i σ i ) \exp(\beta H_N^{\mathrm{SK}}(\sigma)+h\sum_i\sigma_i) exp ( β H N SK ( σ ) + h ∑ i σ i ) β 2 t \beta^2t β 2 t Φ ( t , x ) = log 2 + U ( β 2 t , x ) \Phi(t,x)=\log2+U(\beta^2t,x) Φ ( t , x ) = log 2 + U ( β 2 t , x ) P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) ν β , h \nu_{\beta,h} ν β , h [99 Theorem 1.1] . We need it only for h > 0 h>0 h > 0 h h h
Theorem 2.2. For every β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 0 < q − ≤ q + < 1 0<q_-\le q_+<1 0 < q − ≤ q + < 1 supp ν β , h = [ q − , q + ] \operatorname{supp}\nu_{\beta,h}=[q_-,q_+] supp ν β , h = [ q − , q + ]
The Parisi diffusion. For the rest of this section we fix β > 0 \beta>0 β > 0 h ≥ 0 h\ge0 h ≥ 0 ν = ν β , h \nu=\nu_{\beta,h} ν = ν β , h Φ = Φ ν \Phi=\Phi_\nu Φ = Φ ν
v ( t , x ) = β 2 ν ( t ) ∂ x Φ ( t , x ) , a ( t , x ) = β ∂ x 2 Φ ( t , x ) , v(t,x)=\beta^2\nu(t)\,\partial_x\Phi(t,x),\qquad a(t,x)=\beta\,\partial_x^2\Phi(t,x), v ( t , x ) = β 2 ν ( t ) ∂ x Φ ( t , x ) , a ( t , x ) = β ∂ x 2 Φ ( t , x ) , let B B B X X X
d X t = v ( t , X t ) d t + β d B t , X 0 = h , (11) \mathrm{d}X_t=v(t,X_t)\,\mathrm{d}t+\beta\,\mathrm{d}B_t,\qquad X_0=h, \tag*{(11)} d X t = v ( t , X t ) d t + β d B t , X 0 = h , (11) and let M t = ∂ x Φ ( t , X t ) M_t=\partial_x\Phi(t,X_t) M t = ∂ x Φ ( t , X t ) X X X v v v x x x [88 ] . We call X X X M M M ν ( t ) = 0 \nu(t)=0 ν ( t ) = 0 t < q − t<q_{-} t < q −
X t = h + β B t ( 0 ≤ t ≤ q − ) . (12) X_t=h+\beta B_t \qquad(0\le t\le q_{-}). \tag*{(12)} X t = h + β B t ( 0 ≤ t ≤ q − ) . (12) We use the following facts. Property (P1) and Lemma 2.5 hold for every probability measure μ \mu μ ν \nu ν
(P1) Regularity [7 Proposition 2(i),(ii),(iv)] . For j ≤ 4 j\le4 j ≤ 4 ∂ x j Φ \partial_x^j\Phi ∂ x j Φ [ 0 , 1 ] × R [0,1]\times\mathbb{R} [ 0 , 1 ] × R ∣ ∂ x Φ ∣ ≤ 1 |\partial_x\Phi|\le1 ∣ ∂ x Φ∣ ≤ 1 0 < ∂ x 2 Φ ≤ 1 0<\partial_x^2\Phi\le1 0 < ∂ x 2 Φ ≤ 1 ∣ ∂ x 3 Φ ∣ ≤ 4 |\partial_x^3\Phi|\le4 ∣ ∂ x 3 Φ∣ ≤ 4 μ ℓ → μ \mu_\ell\to\mu μ ℓ → μ ∂ x j Φ μ ℓ → ∂ x j Φ μ \partial_x^j\Phi_{\mu_\ell}\to\partial_x^j\Phi_\mu ∂ x j Φ μ ℓ → ∂ x j Φ μ j ≤ 2 j\le2 j ≤ 2 ∣ v ∣ ≤ β 2 |v|\le\beta^2 ∣ v ∣ ≤ β 2 0 ≤ ∂ x v ≤ β 2 0\le\partial_xv\le\beta^2 0 ≤ ∂ x v ≤ β 2 0 < a ≤ β 0<a\le\beta 0 < a ≤ β ∣ ∂ x a ∣ ≤ 4 β |\partial_xa|\le4\beta ∣ ∂ x a ∣ ≤ 4 β [7 ] these facts are deduced from [6 Propositions 1 and 2] and an estimate of Talagrand. Jagannath and Tobasco [87 Theorem 4 and Lemma 16] gave a PDE proof of the well-posedness and regularity of the Parisi PDE.
(P2) Martingales [7 Lemma 2] . For 0 ≤ s ≤ t ≤ 1 0\le s\le t\le1 0 ≤ s ≤ t ≤ 1
M t − M s = ∫ s t a ( r , X r ) d B r , M_t-M_s=\int_s^t a(r,X_r)\,\mathrm{d}B_r, M t − M s = ∫ s t a ( r , X r ) d B r , ∂ x 2 Φ ( t , X t ) − ∂ x 2 Φ ( s , X s ) = − ∫ s t β 2 ν ( r ) ( ∂ x 2 Φ ( r , X r ) ) 2 d r + ∫ s t β ∂ x 3 Φ ( r , X r ) d B r . \partial_x^2\Phi(t,X_t)-\partial_x^2\Phi(s,X_s)=-\int_s^t\beta^2\nu(r)\left(\partial_x^2\Phi(r,X_r)\right)^2\,\mathrm{d}r+\int_s^t\beta\partial_x^3\Phi(r,X_r)\,\mathrm{d}B_r. ∂ x 2 Φ ( t , X t ) − ∂ x 2 Φ ( s , X s ) = − ∫ s t β 2 ν ( r ) ( ∂ x 2 Φ ( r , X r ) ) 2 d r + ∫ s t β ∂ x 3 Φ ( r , X r ) d B r . (P3) Variational representation at the optimizer [7 Theorem 3] , applied on [ 0 , q + ] [0,q_{+}] [ 0 , q + ] x = h x=h x = h r ↦ M r r\mapsto M_r r ↦ M r
Φ ( 0 , h ) = E Φ ( q + , X q + ) − β 2 2 ∫ 0 q + ν ( r ) E M r 2 d r . \Phi(0,h)=\mathbb{E}\,\Phi(q_{+},X_{q_{+}})-\frac{\beta^2}{2}\int_0^{q_{+}}\nu(r)\mathbb{E}M_r^2\,\mathrm{d}r. Φ ( 0 , h ) = E Φ ( q + , X q + ) − 2 β 2 ∫ 0 q + ν ( r ) E M r 2 d r . In [7 Theorem 3] , Φ μ ( s , x ) \Phi_\mu(s,x) Φ μ ( s , x ) [87 Lemma 18] gave a second proof of this representation by dynamic programming.
(P4) Cole–Hopf above q + q_{+} q + Since ν ( t ) = 1 \nu(t)=1 ν ( t ) = 1 t ≥ q + t\ge q_{+} t ≥ q + Φ ( t , x ) = log 2 cosh x + β 2 ( 1 − t ) / 2 \Phi(t,x)=\log2\cosh x+\beta^2(1-t)/2 Φ ( t , x ) = log 2 cosh x + β 2 ( 1 − t ) /2 t ∈ [ q + , 1 ] t\in[q_{+},1] t ∈ [ q + , 1 ] [ 0 , q + ] [0,q_{+}] [ 0 , q + ] [ q + , 1 ] [q_{+},1] [ q + , 1 ] [7 (4)] ), since E 2 cosh ( x + β 1 − t Z ) = 2 cosh ( x ) e β 2 ( 1 − t ) / 2 \mathbb{E}\,2\cosh(x+\beta\sqrt{1-t}Z)=2\cosh(x)e^{\beta^2(1-t)/2} E 2 cosh ( x + β 1 − t Z ) = 2 cosh ( x ) e β 2 ( 1 − t ) /2 Z ∼ N ( 0 , 1 ) Z\sim N(0,1) Z ∼ N ( 0 , 1 ) ν \nu ν ν \nu ν [ 0 , q + ] [0,q_{+}] [ 0 , q + ]
Optimality of the Parisi measure. Let Σ ( t ) = E a ( t , X t ) 2 \Sigma(t)=\mathbb{E}a(t,X_t)^2 Σ ( t ) = E a ( t , X t ) 2 [ 0 , 1 ] [0,1] [ 0 , 1 ] a a a t ↦ X t t\mapsto X_t t ↦ X t
E M t 2 = M 0 2 + ∫ 0 t Σ ( r ) d r ( t ∈ [ 0 , 1 ] ) . (13) \mathbb{E}M_t^2=M_0^2+\int_0^t\Sigma(r)\,\mathrm{d}r \qquad(t\in[0,1]). \tag*{(13)} E M t 2 = M 0 2 + ∫ 0 t Σ ( r ) d r ( t ∈ [ 0 , 1 ]) . (13) We define
Ψ ( x ) = β 2 2 ∫ x 1 ( E M t 2 − t ) d t ( x ∈ [ 0 , 1 ] ) . (14) \Psi(x)=\frac{\beta^2}{2}\int_x^1\left(\mathbb{E}M_t^2-t\right)\,\mathrm{d}t \qquad(x\in[0,1]). \tag*{(14)} Ψ ( x ) = 2 β 2 ∫ x 1 ( E M t 2 − t ) d t ( x ∈ [ 0 , 1 ]) . (14) By (13) , Ψ \Psi Ψ [ 0 , 1 ] [0,1] [ 0 , 1 ]
Ψ ′ ( x ) = − β 2 2 ( E M x 2 − x ) , Ψ ′ ′ ( x ) = − β 2 2 ( Σ ( x ) − 1 ) . (15) \Psi'(x)=-\frac{\beta^2}{2}\left(\mathbb{E}M_x^2-x\right),\qquad\Psi''(x)=-\frac{\beta^2}{2}\left(\Sigma(x)-1\right). \tag*{(15)} Ψ ′ ( x ) = − 2 β 2 ( E M x 2 − x ) , Ψ ′′ ( x ) = − 2 β 2 ( Σ ( x ) − 1 ) . (15) Jagannath and Tobasco showed that Ψ \Psi Ψ ν \nu ν [88 Lemma 3.2 and Example 3.4] , for every probability measure ρ \rho ρ [ 0 , 1 ] [0,1] [ 0 , 1 ] ε ↦ P β , h ( ( 1 − ε ) ν + ε ρ ) \varepsilon\mapsto\mathcal{P}_{\beta,h}((1-\varepsilon)\nu+\varepsilon\rho) ε ↦ P β , h (( 1 − ε ) ν + ερ ) ε = 0 \varepsilon=0 ε = 0 ∫ Ψ d ρ − ∫ Ψ d ν \int\Psi\,\mathrm{d}\rho-\int\Psi\,\mathrm{d}\nu ∫ Ψ d ρ − ∫ Ψ d ν ν \nu ν P β , h \mathcal{P}_{\beta,h} P β , h ρ \rho ρ [88 Proposition 1.1] ; see also [88 Corollary 3.6] . Chen obtained the first variation and this characterization independently [38 Theorem 2] . For the spherical models, a characterization of this type was obtained earlier by Talagrand; see [88 Remark 1.4] .
Theorem 2.3. Let β > 0 \beta>0 β > 0 h ≥ 0 h\geq0 h ≥ 0 Ψ ( q ) = min x ∈ [ 0 , 1 ] Ψ ( x ) \Psi(q)=\min_{x\in[0,1]}\Psi(x) Ψ ( q ) = min x ∈ [ 0 , 1 ] Ψ ( x ) q ∈ supp ν β , h q\in\operatorname{supp}\nu_{\beta,h} q ∈ supp ν β , h
The statement in [88 Proposition 1.1] is that ν ( { x : Ψ ( x ) = min Ψ } ) = 1 \nu(\{x:\Psi(x)=\min\Psi\})=1 ν ({ x : Ψ ( x ) = min Ψ }) = 1 Ψ \Psi Ψ supp ν \operatorname{supp}\nu supp ν [88 ] , the model is ξ ~ = β 2 ξ 0 \widetilde{\xi}=\beta^2\xi_0 ξ = β 2 ξ 0 ξ 0 ( t ) = t 2 / 2 \xi_0(t)=t^2/2 ξ 0 ( t ) = t 2 /2 (9) with terminal condition log cosh x \log\cosh x log cosh x u μ u_\mu u μ [88 Section 8.1] , where the results of [87 ] on the Parisi PDE are recalled. For atomic μ \mu μ Φ μ − log 2 \Phi_\mu-\log2 Φ μ − log 2 u μ u_\mu u μ [88 Proposition 8.1] . Both μ ↦ u μ \mu\mapsto u_\mu μ ↦ u μ μ ↦ Φ μ \mu\mapsto\Phi_\mu μ ↦ Φ μ [88 Proposition 8.2] and (P1); for the first, weak convergence gives convergence of distribution functions at all but countably many points, which implies convergence in the metric d ( μ , μ ′ ) = ∫ 0 1 ∣ μ ( [ 0 , s ] ) − μ ′ ( [ 0 , s ] ) ∣ d s d(\mu,\mu')=\int_0^1\lvert\mu([0,s])-\mu'([0,s])\rvert\,\mathrm{d}s d ( μ , μ ′ ) = ∫ 0 1 ∣ μ ([ 0 , s ]) − μ ′ ([ 0 , s ])∣ d s μ \mu μ u μ = Φ μ − log 2 u_\mu=\Phi_\mu-\log2 u μ = Φ μ − log 2 μ \mu μ [88 ] is P β , h − log 2 \mathcal{P}_{\beta,h}-\log2 P β , h − log 2 [88 (1.1.2)] is (11) , the field is the same, and the function G ν G_\nu G ν [88 (1.1.1)] is Ψ \Psi Ψ
The conditions E M t 2 = t \mathbb{E}M_t^2=t E M t 2 = t Σ ( t ) ≤ 1 \Sigma(t)\leq1 Σ ( t ) ≤ 1 [88 Proposition 1.1] and [38 Proposition 1] . If ν \nu ν [53 ] ; see [88 Remarks 1.2 and 1.4] and [136 ] . In zero field, they appear earlier in [6 Theorem 5] , where the expectations are written under a Girsanov change of measure. We include the short derivation from Theorem 2.3; the bound q + < 1 q_+<1 q + < 1 [88 Lemma 3.8] , with the same proof.
Corollary 2.4. Let β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 t ∈ supp ν ∩ ( 0 , 1 ) t\in\operatorname{supp}\nu\cap(0,1) t ∈ supp ν ∩ ( 0 , 1 ) E M t 2 = t \mathbb{E}M_t^2=t E M t 2 = t Σ ( t ) ≤ 1 \Sigma(t)\leq1 Σ ( t ) ≤ 1 q + < 1 q_+<1 q + < 1 E M q + 2 = q + \mathbb{E}M_{q_+}^2=q_+ E M q + 2 = q +
Proof. If t ∈ supp ν ∩ ( 0 , 1 ) t \in\operatorname{supp}\nu\cap(0,1) t ∈ supp ν ∩ ( 0 , 1 ) t t t Ψ \Psi Ψ Theorem 2.3 , so Ψ ′ ( t ) = 0 \Psi'(t)=0 Ψ ′ ( t ) = 0 Ψ ′ ′ ( t ) ≥ 0 \Psi''(t)\ge0 Ψ ′′ ( t ) ≥ 0 q + = 1 q_+=1 q + = 1 Ψ \Psi Ψ 1 1 1 E M 1 2 ≥ 1 \mathbb{E}M_1^2\ge1 E M 1 2 ≥ 1 M 1 = ∂ x Φ ( 1 , X 1 ) = tanh X 1 M_1=\partial_x\Phi(1,X_1)=\tanh X_1 M 1 = ∂ x Φ ( 1 , X 1 ) = tanh X 1 ∣ tanh X 1 ∣ < 1 |\tanh X_1|<1 ∣ tanh X 1 ∣ < 1 q + < 1 q_+<1 q + < 1 q + > 0 q_+>0 q + > 0 E M q + 2 = q + \mathbb{E}M_{q_+}^2=q_+ E M q + 2 = q + q + = 0 q_+=0 q + = 0 Ψ \Psi Ψ 0 0 0 E M 0 2 ≤ 0 \mathbb{E}M_0^2\le0 E M 0 2 ≤ 0 E M q + 2 = 0 = q + \mathbb{E}M_{q_+}^2=0=q_+ E M q + 2 = 0 = q +
We also need the regularity of ∂ x Φ \partial_x\Phi ∂ x Φ ∂ x 2 Φ \partial_x^2\Phi ∂ x 2 Φ
Lemma 2.5. For every probability measure μ \mu μ [ 0 , 1 ] [0,1] [ 0 , 1 ] s , t ∈ [ 0 , 1 ] s,t\in[0,1] s , t ∈ [ 0 , 1 ] x ∈ R x\in\mathbb{R} x ∈ R
∣ ∂ x Φ μ ( t , x ) − ∂ x Φ μ ( s , x ) ∣ ≤ 3 β 2 ∣ t − s ∣ , ∣ ∂ x 2 Φ μ ( t , x ) − ∂ x 2 Φ μ ( s , x ) ∣ ≤ 4 3 β ∣ t − s ∣ 1 / 2 . |\partial_x\Phi_\mu(t,x)-\partial_x\Phi_\mu(s,x)|\le3\beta^2|t-s|,\qquad|\partial_x^2\Phi_\mu(t,x)-\partial_x^2\Phi_\mu(s,x)|\le4\sqrt{3\beta}\,|t-s|^{1/2}. ∣ ∂ x Φ μ ( t , x ) − ∂ x Φ μ ( s , x ) ∣ ≤ 3 β 2 ∣ t − s ∣ , ∣ ∂ x 2 Φ μ ( t , x ) − ∂ x 2 Φ μ ( s , x ) ∣ ≤ 4 3 β ∣ t − s ∣ 1/2 . Proof. We write Φ = Φ μ \Phi=\Phi_\mu Φ = Φ μ μ \mu μ Φ \Phi Φ μ ( t ) \mu(t) μ ( t ) x x x
∂ t ∂ x Φ = − β 2 2 ( ∂ x 3 Φ + 2 μ ( t ) ∂ x Φ ∂ x 2 Φ ) . \partial_t\partial_x\Phi=-\frac{\beta^2}{2}\left(\partial_x^3\Phi+2\mu(t)\partial_x\Phi\partial_x^2\Phi\right). ∂ t ∂ x Φ = − 2 β 2 ( ∂ x 3 Φ + 2 μ ( t ) ∂ x Φ ∂ x 2 Φ ) . which is at most β 2 ( 4 + 2 ) / 2 = 3 β 2 \beta^2(4+2)/2=3\beta^2 β 2 ( 4 + 2 ) /2 = 3 β 2 ∂ x Φ \partial_x\Phi ∂ x Φ [ 0 , 1 ] × R [0,1]\times\mathbb{R} [ 0 , 1 ] × R [ 0 , 1 ] [0,1] [ 0 , 1 ] s , t s,t s , t f = ∂ x Φ ( t , ⋅ ) − ∂ x Φ ( s , ⋅ ) f=\partial_x\Phi(t,\cdot)-\partial_x\Phi(s,\cdot) f = ∂ x Φ ( t , ⋅ ) − ∂ x Φ ( s , ⋅ ) ∥ f ∥ ∞ ≤ 3 β 2 ∣ t − s ∣ \|f\|_\infty\le3\beta^2|t-s| ∥ f ∥ ∞ ≤ 3 β 2 ∣ t − s ∣ ∥ f ′ ′ ∥ ∞ ≤ 8 \|f''\|_\infty\le8 ∥ f ′′ ∥ ∞ ≤ 8 ∥ f ′ ′ ∥ ∞ = 0 \|f''\|_\infty=0 ∥ f ′′ ∥ ∞ = 0 f f f x ± ϵ x\pm\epsilon x ± ϵ 2 ϵ ∣ f ′ ( x ) ∣ ≤ 2 ∥ f ∥ ∞ + ϵ 2 ∥ f ′ ′ ∥ ∞ 2\epsilon|f'(x)|\le2\|f\|_\infty+\epsilon^2\|f''\|_\infty 2 ϵ ∣ f ′ ( x ) ∣ ≤ 2∥ f ∥ ∞ + ϵ 2 ∥ f ′′ ∥ ∞ ϵ = ( 2 ∥ f ∥ ∞ / ∥ f ′ ′ ∥ ∞ ) 1 / 2 \epsilon=(2\|f\|_\infty/\|f''\|_\infty)^{1/2} ϵ = ( 2∥ f ∥ ∞ /∥ f ′′ ∥ ∞ ) 1/2 ∥ f ′ ∥ ∞ 2 ≤ 2 ∥ f ∥ ∞ ∥ f ′ ′ ∥ ∞ ≤ 48 β 2 ∣ t − s ∣ \|f'\|_\infty^2\le2\|f\|_\infty\|f''\|_\infty\le48\beta^2|t-s| ∥ f ′ ∥ ∞ 2 ≤ 2∥ f ∥ ∞ ∥ f ′′ ∥ ∞ ≤ 48 β 2 ∣ t − s ∣ f ′ = ∂ x 2 Φ ( t , ⋅ ) − ∂ x 2 Φ ( s , ⋅ ) f'=\partial_x^2\Phi(t,\cdot)-\partial_x^2\Phi(s,\cdot) f ′ = ∂ x 2 Φ ( t , ⋅ ) − ∂ x 2 Φ ( s , ⋅ ) μ \mu μ μ \mu μ □ \square □
The law of the Parisi center. We write μ + \mu_+ μ + M q + M_{q_+} M q +
E ( β , h ) = ∫ 0 q + E a ( t , X t ) d t . \mathcal{E}(\beta,h)=\int_0^{q_+}\mathbb{E}a(t,X_t)\,\mathrm{d}t. E ( β , h ) = ∫ 0 q + E a ( t , X t ) d t . The next lemma collects the properties of the Parisi martingale used in the lower bound. Part (c) writes P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) h > 0 h>0 h > 0 f : R → [ − 1 , 1 ] f:\mathbb{R}\to[-1,1] f : R → [ − 1 , 1 ] φ f ( t ) = E f ( Y ) f ( Y ′ ) \varphi_f(t)=\mathbb{E}f(Y)f(Y') φ f ( t ) = E f ( Y ) f ( Y ′ ) t ∈ [ 0 , q − ] t\in[0,q_-] t ∈ [ 0 , q − ] ( Y , Y ′ ) (Y,Y') ( Y , Y ′ ) q − q_- q − t t t [135 ] at a magnetization whose coordinates have law μ + \mu_+ μ + E ( β , h ) \mathcal{E}(\beta,h) E ( β , h ) p p p q + q_+ q + [39 Theorem 1] , and a computation close to our proof of (c) appears in the proof of [40 Theorem 7] , without
external field. After the change of variables s = β 2 t s=\beta^{2}t s = β 2 t ϕ f ( t ) > t \phi_f(t)>t ϕ f ( t ) > t [125 Lemma 3.2] . There the root map is defined through the Parisi PDE at zero temperature. If q − = q + q_{-}=q_{+} q − = q + f ( y ) = tanh ( h + β y ) f(y)=\tanh(h+\beta y) f ( y ) = tanh ( h + β y ) [25 Lemma 2.2 and Corollary 2.3] . Our proof uses the same convexity argument as [25 , 125 ] . The proof of (b) follows [103 Lemma 3.7] . Since a ( q + , x ) = β ( 1 − tanh 2 x ) a(q_{+},x)=\beta(1-\tanh^{2}x) a ( q + , x ) = β ( 1 − tanh 2 x ) E a ( q + , X q + ) 2 ≤ 1 \mathbb{E}a(q_{+},X_{q_{+}})^{2}\leq1 E a ( q + , X q + ) 2 ≤ 1 [121 ] for the law of M q + M_{q_{+}} M q + [41 Proposition 14] .
Lemma 2.6. Let β > 0 \beta>0 β > 0 h > 0 h>0 h > 0
(a) We have E M t 2 = t \mathbb{E}M_t^{2}=t E M t 2 = t t ∈ [ q − , q + ] t\in[q_{-},q_{+}] t ∈ [ q − , q + ] q − < q + q_{-}<q_{+} q − < q + E a ( t , X t ) 2 = 1 \mathbb{E}a(t,X_t)^{2}=1 E a ( t , X t ) 2 = 1 t ∈ [ q − , q + ] t\in[q_{-},q_{+}] t ∈ [ q − , q + ] f ( y ) = ∂ x Φ ( q − , h + β y ) f(y)=\partial_x\Phi(q_{-},h+\beta y) f ( y ) = ∂ x Φ ( q − , h + β y ) C 1 C^{1} C 1 ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) f ′ ∈ ( 0 , β ] f'\in(0,\beta] f ′ ∈ ( 0 , β ] f ( B q − ) = M q − f(B_{q_{-}})=M_{q_{-}} f ( B q − ) = M q − ϕ f ( q − ) = q − \phi_f(q_{-})=q_{-} ϕ f ( q − ) = q − ϕ f ( t ) > t \phi_f(t)>t ϕ f ( t ) > t t ∈ [ 0 , q − ) t\in[0,q_{-}) t ∈ [ 0 , q − )
(b) We have M q + = tanh X q + M_{q_{+}}=\tanh X_{q_{+}} M q + = tanh X q + β 2 ( 1 − q + ) 2 ≤ 1 \beta^{2}(1-q_{+})^{2}\leq1 β 2 ( 1 − q + ) 2 ≤ 1
(c) We have
P ( β , h ) = β E ( β , h ) + h E M q + + E ent ( M q + ) + β 2 4 ( 1 − q + ) 2 . (16) \mathcal{P}(\beta,h)=\beta\mathcal{E}(\beta,h)+h\mathbb{E}M_{q_{+}}+\mathbb{E}\operatorname{ent}(M_{q_{+}})+\frac{\beta^{2}}{4}(1-q_{+})^{2}. \tag*{(16)} P ( β , h ) = β E ( β , h ) + h E M q + + E ent ( M q + ) + 4 β 2 ( 1 − q + ) 2 . (16) Proof. (a) By Theorem 2.2 , [ q − , q + ] = supp ν ⊂ ( 0 , 1 ) [q_{-},q_{+}]=\operatorname{supp}\nu\subset(0,1) [ q − , q + ] = supp ν ⊂ ( 0 , 1 ) E M t 2 = t \mathbb{E}M_t^{2}=t E M t 2 = t t ∈ [ q − , q + ] t\in[q_{-},q_{+}] t ∈ [ q − , q + ] q − < q + q_{-}<q_{+} q − < q + (13) gives Σ = 1 \Sigma=1 Σ = 1 ( q − , q + ) (q_{-},q_{+}) ( q − , q + ) [ q − , q + ] [q_{-},q_{+}] [ q − , q + ]
Next, by (P1), f ′ ( y ) = β ∂ x 2 Φ ( q − , h + β y ) ∈ ( 0 , β ] f'(y)=\beta\partial_x^{2}\Phi(q_{-},h+\beta y)\in(0,\beta] f ′ ( y ) = β ∂ x 2 Φ ( q − , h + β y ) ∈ ( 0 , β ] f f f [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] f f f ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) (12) , M t = ∂ x Φ ( t , h + β B t ) M_t=\partial_x\Phi(t,h+\beta B_t) M t = ∂ x Φ ( t , h + β B t ) t ≤ q − t\leq q_{-} t ≤ q − f ( B q − ) = M q − f(B_{q_{-}})=M_{q_{-}} f ( B q − ) = M q − ϕ f ( q − ) = E M q − 2 = q − \phi_f(q_{-})=\mathbb{E}M_{q_{-}}^{2}=q_{-} ϕ f ( q − ) = E M q − 2 = q − t ∈ [ 0 , q − ) t\in[0,q_{-}) t ∈ [ 0 , q − ) M M M M t M_t M t B t B_t B t E [ f ( B q − ) ∣ B t ] = M t \mathbb{E}[f(B_{q_{-}})\mid B_t]=M_t E [ f ( B q − ) ∣ B t ] = M t B ′ B' B ′ B B B A ′ = B t + B q − ′ − B t ′ A'=B_t+B'_{q_{-}}-B'_t A ′ = B t + B q − ′ − B t ′ ( B q − , A ′ ) (B_{q_{-}},A') ( B q − , A ′ ) q − q_{-} q − t t t B t B_t B t
ϕ f ( t ) = E [ E [ f ( B q − ) ∣ B t ] 2 ] = E M t 2 = q − − ∫ t q − Σ ( r ) d r . \phi_f(t)=\mathbb{E}\left[\mathbb{E}[f(B_{q_{-}})\mid B_t]^2\right]=\mathbb{E}M_t^2=q_{-}-\int_t^{q_{-}}\Sigma(r)\,\mathrm{d}r. ϕ f ( t ) = E [ E [ f ( B q − ) ∣ B t ] 2 ] = E M t 2 = q − − ∫ t q − Σ ( r ) d r . It remains to show that Σ ( r ) < 1 \Sigma(r)<1 Σ ( r ) < 1 r < q − r<q_{-} r < q − ν ( r ) = 0 \nu(r)=0 ν ( r ) = 0 r < q − r<q_{-} r < q − ∂ x 2 Φ ( t , X t ) \partial_x^{2}\Phi(t,X_t) ∂ x 2 Φ ( t , X t ) [ 0 , q − ] [0,q_{-}] [ 0 , q − ]
Σ ( t ) = Σ ( 0 ) + β 4 ∫ 0 t E ( ∂ x 3 Φ ( r , X r ) ) 2 d r ( t ≤ q − ) . \Sigma(t)=\Sigma(0)+\beta^{4}\int_0^t\mathbb{E}\left(\partial_x^{3}\Phi(r,X_r)\right)^2\,\mathrm{d}r\qquad(t\leq q_{-}). Σ ( t ) = Σ ( 0 ) + β 4 ∫ 0 t E ( ∂ x 3 Φ ( r , X r ) ) 2 d r ( t ≤ q − ) . The integrand is positive for r ∈ ( 0 , q − ] r\in(0,q_{-}] r ∈ ( 0 , q − ] X r = h + β B r X_r=h+\beta B_r X r = h + β B r ∂ x 3 Φ ( r , ⋅ ) \partial_x^{3}\Phi(r,\cdot) ∂ x 3 Φ ( r , ⋅ ) ∂ x 2 Φ ( r , ⋅ ) \partial_x^{2}\Phi(r,\cdot) ∂ x 2 Φ ( r , ⋅ ) ∂ x Φ ( r , ⋅ ) \partial_x\Phi(r,\cdot) ∂ x Φ ( r , ⋅ ) Σ ( r ) < Σ ( q − ) ≤ 1 \Sigma(r)<\Sigma(q_{-})\leq1 Σ ( r ) < Σ ( q − ) ≤ 1 r < q − r<q_{-} r < q − q − q_{-} q − ϕ f ( t ) > q − − ( q − − t ) = t \phi_f(t)>q_{-}-(q_{-}-t)=t ϕ f ( t ) > q − − ( q − − t ) = t ∂ x Φ ( q + , x ) = tanh x \partial_x\Phi(q_+,x)=\tanh x ∂ x Φ ( q + , x ) = tanh x a ( q + , x ) = β ( 1 − tanh 2 x ) a(q_+,x)=\beta(1-\tanh^2x) a ( q + , x ) = β ( 1 − tanh 2 x ) M q + = tanh X q + M_{q_+}=\tanh X_{q_+} M q + = tanh X q + E tanh 2 X q + = q + \mathbb{E}\tanh^2X_{q_+}=q_+ E tanh 2 X q + = q + β ( 1 − q + ) = E a ( q + , X q + ) \beta(1-q_+)=\mathbb{E}a(q_+,X_{q_+}) β ( 1 − q + ) = E a ( q + , X q + ) q + ∈ ( 0 , 1 ) q_+\in(0,1) q + ∈ ( 0 , 1 ) q + q_+ q + β 2 ( 1 − q + ) 2 ≤ E a ( q + , X q + ) 2 ≤ 1 \beta^2(1-q_+)^2\leq\mathbb{E}a(q_+,X_{q_+})^2\leq1 β 2 ( 1 − q + ) 2 ≤ E a ( q + , X q + ) 2 ≤ 1
(c) We compute each term of P β , h ( ν ) \mathcal{P}_{\beta,h}(\nu) P β , h ( ν ) P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) (10) .
Since ν ( r ) = 0 \nu(r)=0 ν ( r ) = 0 r < q − r<q_- r < q − E M r 2 = r \mathbb{E}M_r^2=r E M r 2 = r [ q − , q + ] [q_-,q_+] [ q − , q + ]
Φ ( 0 , h ) = E log 2 cosh X q + + β 2 2 ( 1 − q + ) − β 2 2 ∫ 0 q + t ν ( t ) d t \Phi(0,h)=\mathbb{E}\log2\cosh X_{q_+}+\frac{\beta^2}{2}(1-q_+)-\frac{\beta^2}{2}\int_0^{q_+}t\nu(t)\,\mathrm{d}t Φ ( 0 , h ) = E log 2 cosh X q + + 2 β 2 ( 1 − q + ) − 2 β 2 ∫ 0 q + t ν ( t ) d t Since ν ( t ) = 1 \nu(t)=1 ν ( t ) = 1 [ q + , 1 ] [q_+,1] [ q + , 1 ]
β 2 2 ∫ 0 1 t ν ( t ) d t = β 2 2 ∫ 0 q + t ν ( t ) d t + β 2 4 ( 1 − q + 2 ) . \frac{\beta^2}{2}\int_0^1t\nu(t)\,\mathrm{d}t=\frac{\beta^2}{2}\int_0^{q_+}t\nu(t)\,\mathrm{d}t+\frac{\beta^2}{4}(1-q_+^2). 2 β 2 ∫ 0 1 t ν ( t ) d t = 2 β 2 ∫ 0 q + t ν ( t ) d t + 4 β 2 ( 1 − q + 2 ) . By (11) , X q + = h + ∫ 0 q + β 2 ν ( t ) M t d t + β B q + X_{q_+}=h+\int_0^{q_+}\beta^2\nu(t)M_t\,\mathrm{d}t+\beta B_{q_+} X q + = h + ∫ 0 q + β 2 ν ( t ) M t d t + β B q + M M M E [ M t M q + ] = E M t 2 \mathbb{E}[M_tM_{q_+}]=\mathbb{E}M_t^2 E [ M t M q + ] = E M t 2 t t t ν ( t ) ≠ 0 \nu(t)\ne0 ν ( t ) = 0 E [ B q + M q + ] = ∫ 0 q + E a ( t , X t ) d t = E ( β , h ) \mathbb{E}[B_{q_+}M_{q_+}]=\int_0^{q_+}\mathbb{E}a(t,X_t)\,\mathrm{d}t=\mathcal{E}(\beta,h) E [ B q + M q + ] = ∫ 0 q + E a ( t , X t ) d t = E ( β , h )
E [ X q + M q + ] = h E M q + + β 2 ∫ 0 q + t ν ( t ) d t + β E ( β , h ) . \mathbb{E}[X_{q_+}M_{q_+}]=h\mathbb{E}M_{q_+}+\beta^2\int_0^{q_+}t\nu(t)\,\mathrm{d}t+\beta\mathcal{E}(\beta,h). E [ X q + M q + ] = h E M q + + β 2 ∫ 0 q + t ν ( t ) d t + β E ( β , h ) . When we substitute these expressions into the Parisi functional, the integrals ∫ 0 q + t ν ( t ) d t \int_0^{q_+}t\nu(t)\,\mathrm{d}t ∫ 0 q + t ν ( t ) d t
P ( β , h ) = E ent ( M q + ) + h E M q + + β E ( β , h ) + β 2 2 ( 1 − q + ) − β 2 4 ( 1 − q + 2 ) . \mathcal{P}(\beta,h)=\mathbb{E}\operatorname{ent}(M_{q_+})+h\mathbb{E}M_{q_+}+\beta\mathcal{E}(\beta,h)+\frac{\beta^2}{2}(1-q_+)-\frac{\beta^2}{4}(1-q_+^2). P ( β , h ) = E ent ( M q + ) + h E M q + + β E ( β , h ) + 2 β 2 ( 1 − q + ) − 4 β 2 ( 1 − q + 2 ) . This is (2.13), because 2 ( 1 − q + ) − ( 1 − q + 2 ) = ( 1 − q + ) 2 2(1-q_+)-(1-q_+^2)=(1-q_+)^2 2 ( 1 − q + ) − ( 1 − q + 2 ) = ( 1 − q + ) 2
By Lemma 2.6(a,b), μ + \mu_+ μ + tanh X q + \tanh X_{q_+} tanh X q + μ + ( ( − 1 , 1 ) ) = 1 \mu_+((-1,1))=1 μ + (( − 1 , 1 )) = 1 ∫ m 2 μ + ( d m ) = q + \int m^2\mu_+(\mathrm{d}m)=q_+ ∫ m 2 μ + ( d m ) = q +
The goal of this section is to develop the calculus of finite cascades that is used in the proof of Proposition 4.1 . § 4.1 fixes the notation. § 4.2 records derivative identities for cascade recursions with auxiliary randomness; they are applied to the scalar transform in § 4.3 and to an enriched free energy in § 5 . § 4.4 proves that the curvature and the complements of the transported squared magnetizations of an Ising cascade are completely monotone functions of the square of the field. § 4.5 deduces that the scalar transform is convex in the square roots of its variances, for every value of the field. § 4.6 studies a Hopf–Lax formula built from the scalar transform and compares its value at the origin with the Parisi formula.
Notation
Throughout this section and the next, we fix an integer K ≥ 1 K \ge1 K ≥ 1
0 < ζ 0 < ζ 1 < ⋯ < ζ K − 1 < ζ K = 1 , (17) 0 < \zeta_0 < \zeta_1 < \cdots< \zeta_{K-1} < \zeta_K = 1, \tag*{(17)} 0 < ζ 0 < ζ 1 < ⋯ < ζ K − 1 < ζ K = 1 , (17) and we set ζ − 1 = 0 \zeta_{-1} = 0 ζ − 1 = 0
w j = ζ j − ζ j − 1 ( 0 ≤ j ≤ K ) , w ˉ = max 0 ≤ j ≤ K w j . (18) w_j = \zeta_j - \zeta_{j-1}\quad(0 \le j \le K),\qquad\bar{w} = \max_{0 \le j \le K} w_j. \tag*{(18)} w j = ζ j − ζ j − 1 ( 0 ≤ j ≤ K ) , w ˉ = 0 ≤ j ≤ K max w j . (18) Then w j > 0 w_j > 0 w j > 0 ∑ j w j = 1 \sum_j w_j = 1 ∑ j w j = 1 x , y ∈ R K + 1 x,y \in\mathbb{R}^{K+1} x , y ∈ R K + 1 ⟨ x , y ⟩ w = ∑ j = 0 K w j x j y j \langle x,y\rangle_w = \sum_{j=0}^{K} w_j x_j y_j ⟨ x , y ⟩ w = ∑ j = 0 K w j x j y j ∥ x ∥ 1 , w = ∑ j w j ∣ x j ∣ \|x\|_{1,w} = \sum_j w_j |x_j| ∥ x ∥ 1 , w = ∑ j w j ∣ x j ∣ ∥ x ∥ 2 , w = ⟨ x , x ⟩ w 1 / 2 \|x\|_{2,w} = \langle x,x\rangle_w^{1/2} ∥ x ∥ 2 , w = ⟨ x , x ⟩ w 1/2 ∑ j w j = 1 \sum_j w_j = 1 ∑ j w j = 1 ∥ x ∥ 1 , w ≤ ∥ x ∥ 2 , w \|x\|_{1,w} \le\|x\|_{2,w} ∥ x ∥ 1 , w ≤ ∥ x ∥ 2 , w
C K = { q ∈ R K + 1 : 0 ≤ q 0 ≤ q 1 ≤ ⋯ ≤ q K } , C_K = \{q \in\mathbb{R}^{K+1}: 0 \le q_0 \le q_1 \le\cdots\le q_K\}, C K = { q ∈ R K + 1 : 0 ≤ q 0 ≤ q 1 ≤ ⋯ ≤ q K } , and its dual cone is C K ∗ = { z ∈ R K + 1 : ⟨ z , q ⟩ w ≥ 0 for all q ∈ C K } C_K^* = \{z \in\mathbb{R}^{K+1}: \langle z,q\rangle_w \ge0\text{ for all }q \in C_K\} C K ∗ = { z ∈ R K + 1 : ⟨ z , q ⟩ w ≥ 0 for all q ∈ C K } q ∈ C K q \in C_K q ∈ C K q = ∑ ℓ = 0 K ( q ℓ − q ℓ − 1 ) 1 [ ℓ , K ] q = \sum_{\ell=0}^{K}(q_\ell-q_{\ell-1})\mathbf{1}_{[\ell,K]} q = ∑ ℓ = 0 K ( q ℓ − q ℓ − 1 ) 1 [ ℓ , K ] q − 1 = 0 q_{-1}=0 q − 1 = 0 1 [ ℓ , K ] \mathbf{1}_{[\ell,K]} 1 [ ℓ , K ] ℓ \ell ℓ
C K ∗ = { z ∈ R K + 1 : ∑ j = ℓ K w j z j ≥ 0 for 0 ≤ ℓ ≤ K } . (19) C_K^* = \left\{z \in\mathbb{R}^{K+1}: \sum_{j=\ell}^{K} w_j z_j \ge0\text{ for }0 \le\ell\le K\right\}. \tag*{(19)} C K ∗ = ⎩ ⎨ ⎧ z ∈ R K + 1 : j = ℓ ∑ K w j z j ≥ 0 for 0 ≤ ℓ ≤ K ⎭ ⎬ ⎫ . (19) We write z ⪰ ∗ z ′ z \succeq_* z' z ⪰ ∗ z ′ z − z ′ ∈ C K ∗ z-z' \in C_K^* z − z ′ ∈ C K ∗ u : C K → R u:C_K \to\mathbb{R} u : C K → R differentiable on C K C_K C K D u : C K → R K + 1 Du:C_K \to\mathbb{R}^{K+1} D u : C K → R K + 1 o ( ∣ q ′ − q ∣ ) o(|q^\prime-q|) o ( ∣ q ′ − q ∣ ) q ′ → q q^\prime\to q q ′ → q C K C_K C K D u Du D u u u u ( D u ) j = w j − 1 ∂ u / ∂ q j (Du)_j = w_j^{-1}\partial u/\partial q_j ( D u ) j = w j − 1 ∂ u / ∂ q j u u u C K C_K C K q , q ′ ∈ C K q,q^\prime\in C_K q , q ′ ∈ C K q q q q ′ q^\prime q ′ C K C_K C K
u ( q ′ ) − u ( q ) = ∫ 0 1 ⟨ D u ( q + θ ( q ′ − q ) ) , q ′ − q ⟩ w d θ . (20) u(q^\prime)-u(q)=\int_0^1 \langle Du(q+\theta(q^\prime-q)),q^\prime-q\rangle_w\,d\theta. \tag*{(20)} u ( q ′ ) − u ( q ) = ∫ 0 1 ⟨ D u ( q + θ ( q ′ − q )) , q ′ − q ⟩ w d θ . (20) We write Z , Z 0 , Z 1 , … Z,Z_0,Z_1,\ldots Z , Z 0 , Z 1 , … b ≥ 0 b\ge0 b ≥ 0 g g g γ b ∗ g ( x ) = E g ( x + b Z ) \gamma_b*g(x)=\mathbb{E}g(x+\sqrt{b}Z) γ b ∗ g ( x ) = E g ( x + b Z ) g g g b ↦ γ b ∗ g ( x ) b\mapsto\gamma_b*g(x) b ↦ γ b ∗ g ( x ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) b = 0 b=0 b = 0
∂ b ( γ b ∗ g ) = 1 2 γ b ∗ g ′ ′ = 1 2 ∂ x 2 ( γ b ∗ g ) . (21) \partial_b(\gamma_b*g)=\frac{1}{2}\gamma_b*g^{\prime\prime}=\frac{1}{2}\partial_x^2(\gamma_b*g). \tag*{(21)} ∂ b ( γ b ∗ g ) = 2 1 γ b ∗ g ′′ = 2 1 ∂ x 2 ( γ b ∗ g ) . (21) Cascade recursions
We consider recursions in which, at each level, a Gaussian increment of the field is integrated together with an auxiliary random element. Let ( Ω ℓ , A ℓ , P ℓ ) (\Omega_\ell,\mathcal{A}_\ell,\mathbb{P}_\ell) ( Ω ℓ , A ℓ , P ℓ ) 1 ≤ ℓ ≤ K 1\le\ell\le K 1 ≤ ℓ ≤ K ω = ( ω 1 , … , ω K ) \omega=(\omega_1,\ldots,\omega_K) ω = ( ω 1 , … , ω K ) ω ℓ ∼ P ℓ \omega_\ell\sim\mathbb{P}_\ell ω ℓ ∼ P ℓ Z 1 , … , Z K Z_1,\ldots,Z_K Z 1 , … , Z K ω \omega ω ω ≤ k = ( ω 1 , … , ω k ) \omega_{\le k}=(\omega_1,\ldots,\omega_k) ω ≤ k = ( ω 1 , … , ω k ) L : R × Ω 1 × ⋯ × Ω K → R L:\mathbb{R}\times\Omega_1\times\cdots\times\Omega_K\to\mathbb{R} L : R × Ω 1 × ⋯ × Ω K → R L ( ⋅ , ω ) L(\cdot,\omega) L ( ⋅ , ω ) ω \omega ω
∣ ∂ x L ∣ ≤ 1 , ∂ x 2 L = 1 − ( ∂ x L ) 2 , E exp ( c ∣ L ( 0 , ω ) ∣ ) < ∞ for every c > 0. (22) |\partial_xL|\le1,\qquad\partial_x^2L=1-(\partial_xL)^2,\qquad\mathbb{E}\exp(c|L(0,\omega)|)<\infty\ \text{for every }c>0. \tag*{(22)} ∣ ∂ x L ∣ ≤ 1 , ∂ x 2 L = 1 − ( ∂ x L ) 2 , E exp ( c ∣ L ( 0 , ω ) ∣ ) < ∞ for every c > 0. (22) The two main examples are L ( x ) = log 2 cosh x L(x)=\log2\cosh x L ( x ) = log 2 cosh x L ( x , ω ) = log ∑ σ exp ( x σ x 0 + A ( σ , ω ) ) L(x,\omega)=\log\sum_\sigma\exp(x\sigma_{x_0}+A(\sigma,\omega)) L ( x , ω ) = log ∑ σ exp ( x σ x 0 + A ( σ , ω )) σ \sigma σ x 0 x_0 x 0 ∂ x L = ⟨ σ x 0 ⟩ \partial_xL=\langle\sigma_{x_0}\rangle ∂ x L = ⟨ σ x 0 ⟩ ∂ x 2 L = 1 − ⟨ σ x 0 ⟩ 2 \partial_x^2L=1-\langle\sigma_{x_0}\rangle^2 ∂ x 2 L = 1 − ⟨ σ x 0 ⟩ 2 L ( ⋅ , ω ) L(\cdot,\omega) L ( ⋅ , ω ) r ≥ 1 r\ge1 r ≥ 1 ∂ x L \partial_xL ∂ x L r r r
Let Δ 1 , … , Δ K ≥ 0 \Delta_1,\ldots,\Delta_K\ge0 Δ 1 , … , Δ K ≥ 0 L K = L L_K=L L K = L 1 ≤ ℓ ≤ K 1\le\ell\le K 1 ≤ ℓ ≤ K
L ℓ − 1 ( x , ω ≤ ℓ − 1 ) = 1 ζ ℓ − 1 log E [ exp ( ζ ℓ − 1 L ℓ ( x + Δ ℓ Z ℓ , ω ≤ ℓ − 1 , ω ℓ ) ) ] . (23) L_{\ell-1}(x,\omega_{\le\ell-1})=\frac{1}{\zeta_{\ell-1}}\log\mathbb{E}\left[\exp\left(\zeta_{\ell-1}L_\ell\left(x+\sqrt{\Delta_\ell}Z_\ell,\omega_{\le\ell-1},\omega_\ell\right)\right)\right]. \tag*{(23)} L ℓ − 1 ( x , ω ≤ ℓ − 1 ) = ζ ℓ − 1 1 log E [ exp ( ζ ℓ − 1 L ℓ ( x + Δ ℓ Z ℓ , ω ≤ ℓ − 1 , ω ℓ ) ) ] . (23) where the expectation is over ( Z ℓ , ω ℓ ) (Z_\ell,\omega_\ell) ( Z ℓ , ω ℓ ) L 0 L_0 L 0 x x x 0 ≤ k < K 0\le k<K 0 ≤ k < K ( X k , ω ≤ k ) (X_k,\omega_{\le k}) ( X k , ω ≤ k ) ( X k + 1 , ω k + 1 ) (X_{k+1},\omega_{k+1}) ( X k + 1 , ω k + 1 ) ( X k + Δ k + 1 Z k + 1 , ω k + 1 ) (X_k+\sqrt{\Delta_{k+1}}Z_{k+1},\omega_{k+1}) ( X k + Δ k + 1 Z k + 1 , ω k + 1 )
exp ( ζ k ( L k + 1 ( X k + 1 , ω ≤ k + 1 ) − L k ( X k , ω ≤ k ) ) ) . (24) \exp\left(\zeta_k\left(L_{k+1}(X_{k+1},\omega_{\le k+1})-L_k(X_k,\omega_{\le k})\right)\right). \tag*{(24)} exp ( ζ k ( L k + 1 ( X k + 1 , ω ≤ k + 1 ) − L k ( X k , ω ≤ k ) ) ) . (24) which has unit mean by (23) . This defines a Markov chain ( X k , ω ≤ k ) k ≤ K (X_k,\omega_{\leq k})_{k\leq K} ( X k , ω ≤ k ) k ≤ K X 0 X_0 X 0 E ^ \widehat{\mathbb{E}} E
M j = ∂ x L j ( X j , ω ≤ j ) , Λ k , j ( x , ω ≤ k ) = E ^ [ M j 2 ∣ X k = x , ω ≤ k ] ( 0 ≤ k ≤ j ≤ K ) . M_j=\partial_xL_j(X_j,\omega_{\leq j}),\qquad\Lambda_{k,j}(x,\omega_{\leq k})=\widehat{\mathbb{E}}\left[M_j^2\mid X_k=x,\omega_{\leq k}\right]\qquad(0\leq k\leq j\leq K). M j = ∂ x L j ( X j , ω ≤ j ) , Λ k , j ( x , ω ≤ k ) = E [ M j 2 ∣ X k = x , ω ≤ k ] ( 0 ≤ k ≤ j ≤ K ) . In particular Λ k , k = ( ∂ x L k ) 2 \Lambda_{k,k}=(\partial_xL_k)^2 Λ k , k = ( ∂ x L k ) 2
Lemma 3.1. The following hold for every 0 ≤ k ≤ K 0\leq k\leq K 0 ≤ k ≤ K
(a) The function L k ( ⋅ , ω ≤ k ) L_k(\cdot,\omega_{\leq k}) L k ( ⋅ , ω ≤ k ) ∣ ∂ x L k ∣ ≤ 1 |\partial_xL_k|\leq1 ∣ ∂ x L k ∣ ≤ 1 r ≥ 1 r\geq1 r ≥ 1 r r r K K K ∂ x L k ( X k , ω ≤ k ) = E ^ [ M k + 1 ∣ X k , ω ≤ k ] \partial_xL_k(X_k,\omega_{\leq k})=\widehat{\mathbb{E}}[M_{k+1}\mid X_k,\omega_{\leq k}] ∂ x L k ( X k , ω ≤ k ) = E [ M k + 1 ∣ X k , ω ≤ k ] k < K k<K k < K 0 ≤ Λ k , k ≤ Λ k , k + 1 ≤ ⋯ ≤ Λ k , K ≤ 1 0\leq\Lambda_{k,k}\leq\Lambda_{k,k+1}\leq\cdots\leq\Lambda_{k,K}\leq1 0 ≤ Λ k , k ≤ Λ k , k + 1 ≤ ⋯ ≤ Λ k , K ≤ 1
(b) We have
∂ x 2 L k = 1 − ζ k Λ k , k − ∑ j = k + 1 K w j Λ k , j . (25) \partial_x^2L_k=1-\zeta_k\Lambda_{k,k}-\sum_{j=k+1}^{K}w_j\Lambda_{k,j}. \tag*{(25)} ∂ x 2 L k = 1 − ζ k Λ k , k − j = k + 1 ∑ K w j Λ k , j . (25) (c) For k ≤ ℓ ≤ K k\leq\ell\leq K k ≤ ℓ ≤ K L k L_k L k Δ ℓ ∈ [ 0 , ∞ ) \Delta_\ell\in[0,\infty) Δ ℓ ∈ [ 0 , ∞ ) Δ ℓ = 0 \Delta_\ell=0 Δ ℓ = 0
∂ Δ ℓ L k = 1 2 ( 1 − ∑ j = ℓ K w j Λ k , j ) . (26) \partial_{\Delta_\ell}L_k=\frac{1}{2}\left(1-\sum_{j=\ell}^{K}w_j\Lambda_{k,j}\right). \tag*{(26)} ∂ Δ ℓ L k = 2 1 1 − j = ℓ ∑ K w j Λ k , j . (26) Proof. For ℓ ≤ K \ell\leq K ℓ ≤ K L ℓ L_\ell L ℓ E exp ( c ∣ L ℓ ( 0 , ω ≤ ℓ ) ∣ ) < ∞ \mathbb{E}\exp(c|L_\ell(0,\omega_{\leq\ell})|)<\infty E exp ( c ∣ L ℓ ( 0 , ω ≤ ℓ ) ∣ ) < ∞ c > 0 c>0 c > 0 ∣ L ℓ ( x , ⋅ ) − L ℓ ( 0 , ⋅ ) ∣ ≤ ∣ x ∣ |L_\ell(x,\cdot)-L_\ell(0,\cdot)|\leq|x| ∣ L ℓ ( x , ⋅ ) − L ℓ ( 0 , ⋅ ) ∣ ≤ ∣ x ∣ (23) is justified by dominated convergence, and
∂ x L ℓ − 1 = E ^ ℓ − 1 [ ∂ x L ℓ ] , ∂ x 2 L ℓ − 1 = E ^ ℓ − 1 [ ∂ x 2 L ℓ ] + ζ ℓ − 1 ( E ^ ℓ − 1 [ ( ∂ x L ℓ ) 2 ] − ( ∂ x L ℓ − 1 ) 2 ) . (27) \partial_xL_{\ell-1}=\widehat{\mathbb{E}}_{\ell-1}[\partial_xL_\ell],\qquad
\partial_x^2L_{\ell-1}=\widehat{\mathbb{E}}_{\ell-1}[\partial_x^2L_\ell]+\zeta_{\ell-1}\left(\widehat{\mathbb{E}}_{\ell-1}[(\partial_xL_\ell)^2]-(\partial_xL_{\ell-1})^2\right). \tag*{(27)} ∂ x L ℓ − 1 = E ℓ − 1 [ ∂ x L ℓ ] , ∂ x 2 L ℓ − 1 = E ℓ − 1 [ ∂ x 2 L ℓ ] + ζ ℓ − 1 ( E ℓ − 1 [( ∂ x L ℓ ) 2 ] − ( ∂ x L ℓ − 1 ) 2 ) . (27) where E ^ ℓ − 1 \widehat{\mathbb{E}}_{\ell-1} E ℓ − 1 ℓ − 1 \ell-1 ℓ − 1 ℓ \ell ℓ (24) . Higher derivatives of L ℓ − 1 L_{\ell-1} L ℓ − 1 L ℓ L_\ell L ℓ K K K Y = ∣ L ℓ ( 0 , ⋅ ) ∣ + Δ ℓ ∣ Z ℓ ∣ Y=|L_\ell(0,\cdot)|+\sqrt{\Delta_\ell}|Z_\ell| Y = ∣ L ℓ ( 0 , ⋅ ) ∣ + Δ ℓ ∣ Z ℓ ∣ (23) and Jensen’s inequality,
∣ L ℓ − 1 ( 0 , ⋅ ) ∣ ≤ ζ ℓ − 1 − 1 log E ℓ e ζ ℓ − 1 Y |L_{\ell-1}(0,\cdot)|\leq\zeta_{\ell-1}^{-1}\log\mathbb{E}_\ell e^{\zeta_{\ell-1}Y} ∣ L ℓ − 1 ( 0 , ⋅ ) ∣ ≤ ζ ℓ − 1 − 1 log E ℓ e ζ ℓ − 1 Y E ℓ \mathbb{E}_\ell E ℓ ( Z ℓ , ω ℓ ) (Z_\ell,\omega_\ell) ( Z ℓ , ω ℓ ) c ≥ ζ ℓ − 1 c\geq\zeta_{\ell-1} c ≥ ζ ℓ − 1 E exp ( c ∣ L ℓ − 1 ( 0 , ⋅ ) ∣ ) ≤ E e c Y < ∞ \mathbb{E}\exp(c|L_{\ell-1}(0,\cdot)|)\leq\mathbb{E}e^{cY}<\infty E exp ( c ∣ L ℓ − 1 ( 0 , ⋅ ) ∣ ) ≤ E e c Y < ∞ M k = E ^ [ M j ∣ X k , ω ≤ k ] M_k=\widehat{\mathbb{E}}[M_j\mid X_k,\omega_{\leq k}] M k = E [ M j ∣ X k , ω ≤ k ] k ≤ j k\leq j k ≤ j j ↦ Λ k , j j\mapsto\Lambda_{k,j} j ↦ Λ k , j ∣ M j ∣ ≤ 1 |M_j|\leq1 ∣ M j ∣ ≤ 1 Λ k , K ≤ 1 \Lambda_{k,K}\leq1 Λ k , K ≤ 1
We prove (b) by downward induction. For k = K k=K k = K ζ K = 1 \zeta_K=1 ζ K = 1 k + 1 k+1 k + 1 E ^ k [ Λ k + 1 , j ] = Λ k , j \widehat{\mathbb{E}}_k[\Lambda_{k+1,j}]=\Lambda_{k,j} E k [ Λ k + 1 , j ] = Λ k , j j ≥ k + 1 j\geq k+1 j ≥ k + 1
∂ x 2 L k = 1 − ζ k + 1 Λ k , k + 1 − ∑ j ≥ k + 2 w j Λ k , j + ζ k Λ k , k + 1 − ζ k Λ k , k . \partial_x^2L_k=1-\zeta_{k+1}\Lambda_{k,k+1}-\sum_{j\geq k+2}w_j\Lambda_{k,j}+\zeta_k\Lambda_{k,k+1}-\zeta_k\Lambda_{k,k}. ∂ x 2 L k = 1 − ζ k + 1 Λ k , k + 1 − j ≥ k + 2 ∑ w j Λ k , j + ζ k Λ k , k + 1 − ζ k Λ k , k . Since ζ k + 1 − ζ k = w k + 1 \zeta_{k+1}-\zeta_k=w_{k+1} ζ k + 1 − ζ k = w k + 1 (25) at level k k k
For (c), let first ℓ = k + 1 \ell=k+1 ℓ = k + 1 (21) , applied for fixed ( ω ≤ k + 1 ) (\omega_{\leq k+1}) ( ω ≤ k + 1 ) ω k + 1 \omega_{k+1} ω k + 1
∂ Δ k + 1 L k = 1 2 ζ k ∂ x 2 exp ( ζ k L k ) exp ( ζ k L k ) = 1 2 ( ∂ x 2 L k + ζ k ( ∂ x L k ) 2 ) . \partial_{\Delta_{k+1}}L_k=\frac{1}{2\zeta_k}\frac{\partial_x^2\exp(\zeta_kL_k)}{\exp(\zeta_kL_k)}=\frac{1}{2}\left(\partial_x^2L_k+\zeta_k(\partial_xL_k)^2\right). ∂ Δ k + 1 L k = 2 ζ k 1 exp ( ζ k L k ) ∂ x 2 exp ( ζ k L k ) = 2 1 ( ∂ x 2 L k + ζ k ( ∂ x L k ) 2 ) . and (25) turns the right side into ( 1 − ∑ j ≥ k + 1 w j Λ k , j ) / 2 (1-\sum_{j\geq k+1}w_j\Lambda_{k,j})/2 ( 1 − ∑ j ≥ k + 1 w j Λ k , j ) /2 ℓ > k + 1 \ell>k+1 ℓ > k + 1 ∂ Δ ℓ L k = E ^ k [ ∂ Δ ℓ L k + 1 ] \partial_{\Delta_\ell}L_k=\widehat{\mathbb{E}}_k[\partial_{\Delta_\ell}L_{k+1}] ∂ Δ ℓ L k = E k [ ∂ Δ ℓ L k + 1 ] Δ ℓ \Delta_\ell Δ ℓ (23) at level k k k L k + 1 L_{k+1} L k + 1 ℓ − k \ell-k ℓ − k E ^ k [ Λ k + 1 , j ] = Λ k , j \widehat{\mathbb{E}}_k[\Lambda_{k+1,j}]=\Lambda_{k,j} E k [ Λ k + 1 , j ] = Λ k , j (26) .
The scalar transform
We now specialize Lemma 3.1 to L ( x ) = log 2 cosh x L(x)=\log2\cosh x L ( x ) = log 2 cosh x v ∈ C K v\in C_K v ∈ C K Δ ℓ = v ℓ − v ℓ − 1 \Delta_\ell=v_\ell-v_{\ell-1} Δ ℓ = v ℓ − v ℓ − 1 1 ≤ ℓ ≤ K 1\leq\ell\leq K 1 ≤ ℓ ≤ K ϕ k = L k \phi_k=L_k ϕ k = L k X 0 = h + v 0 Z 0 X_0=h+\sqrt{v_0}Z_0 X 0 = h + v 0 Z 0 compensated scalar transform is
Ψ h ( v ) = v K 2 − E ϕ 0 ( h + v 0 Z 0 ) , ψ h ( q ) = Ψ h ( 2 q ) ( v , q ∈ C K ) . (28) \Psi_h(v)=\frac{v_K}{2}-\mathbb{E}\,\phi_0\left(h+\sqrt{v_0}Z_0\right),\qquad\psi_h(q)=\Psi_h(2q)\qquad(v,q\in C_K). \tag*{(28)} Ψ h ( v ) = 2 v K − E ϕ 0 ( h + v 0 Z 0 ) , ψ h ( q ) = Ψ h ( 2 q ) ( v , q ∈ C K ) . (28) A level of ψ h \psi_h ψ h q j q_j q j 2 q j 2q_j 2 q j h = 0 h=0 h = 0 log 2 \log2 log 2 ψ h \psi_h ψ h [107 ] ; see also [56 Lemma 6.4] . We also write
T j ( h ) = T j ( h ; v ) = E Λ 0 , j ( h + v 0 Z 0 ) = E ^ [ ϕ j ′ ( X j ) 2 ] ( 0 ≤ j ≤ K ) . (29) T_j(h)=T_j(h;v)=\mathbb{E}\,\Lambda_{0,j}\left(h+\sqrt{v_0}Z_0\right)=\widehat{\mathbb{E}}\left[\phi_j'(X_j)^2\right]\qquad(0\leq j\leq K). \tag*{(29)} T j ( h ) = T j ( h ; v ) = E Λ 0 , j ( h + v 0 Z 0 ) = E [ ϕ j ′ ( X j ) 2 ] ( 0 ≤ j ≤ K ) . (29) Lemma 3.2. For every h ∈ R h\in\mathbb{R} h ∈ R Ψ h \Psi_h Ψ h C K C_K C K
∂ v j Ψ h ( v ) = w j 2 T j ( h ; v ) ( 0 ≤ j ≤ K ) , 0 ≤ T 0 ≤ T 1 ≤ ⋯ ≤ T K ≤ 1. (30) \partial_{v_j}\Psi_h(v)=\frac{w_j}{2}T_j(h;v)\qquad(0\leq j\leq K),\qquad0\leq T_0\leq T_1\leq\cdots\leq T_K\leq1. \tag*{(30)} ∂ v j Ψ h ( v ) = 2 w j T j ( h ; v ) ( 0 ≤ j ≤ K ) , 0 ≤ T 0 ≤ T 1 ≤ ⋯ ≤ T K ≤ 1. (30) The function ψ h \psi_h ψ h C K C_K C K D ψ h ( q ) = ( T j ( h ; 2 q ) ) j ≤ K ∈ C K ∩ [ 0 , 1 ] K + 1 D\psi_h(q)=(T_j(h;2q))_{j\leq K}\in C_K\cap[0,1]^{K+1} D ψ h ( q ) = ( T j ( h ; 2 q ) ) j ≤ K ∈ C K ∩ [ 0 , 1 ] K + 1 ∣ ψ h ( q ) − ψ h ( q ′ ) ∣ ≤ ∥ q − q ′ ∥ 1 , w |\psi_h(q)-\psi_h(q')|\leq\|q-q'\|_{1,w} ∣ ψ h ( q ) − ψ h ( q ′ ) ∣ ≤ ∥ q − q ′ ∥ 1 , w q , q ′ ∈ C K q,q'\in C_K q , q ′ ∈ C K ψ h ( q ) ≤ ψ h ( q ′ ) \psi_h(q)\leq\psi_h(q') ψ h ( q ) ≤ ψ h ( q ′ ) q ≤ q ′ q\leq q' q ≤ q ′
Proof. Iterating (21) through the finite recursion gives all mixed derivatives of ϕ 0 \phi_0 ϕ 0 x x x Δ ℓ ∈ [ 0 , ∞ ) \Delta_\ell\in[0,\infty) Δ ℓ ∈ [ 0 , ∞ ) γ v 0 ∗ ϕ 0 \gamma_{\sqrt{v_0}}*\phi_0 γ v 0 ∗ ϕ 0 Ψ h \Psi_h Ψ h ( v 0 , Δ 1 , … , Δ K ) (v_0,\Delta_1,\ldots,\Delta_K) ( v 0 , Δ 1 , … , Δ K ) [ 0 , ∞ ) K + 1 [0,\infty)^{K+1} [ 0 , ∞ ) K + 1 C K C_K C K 1 ≤ j < K 1\leq j<K 1 ≤ j < K v j v_j v j Δ j \Delta_j Δ j Δ j + 1 \Delta_{j+1} Δ j + 1
∂ v j Ψ h = − E [ ( ∂ Δ j − ∂ Δ j + 1 ) ϕ 0 ( X 0 ) ] = − 1 2 E [ ∑ i ≥ j + 1 w i Λ 0 , i ( X 0 ) − ∑ i ≥ j w i Λ 0 , i ( X 0 ) ] = w j 2 T j . \partial_{v_j}\Psi_h=-\mathbb{E}\left[(\partial_{\Delta_j}-\partial_{\Delta_{j+1}})\phi_0(X_0)\right]=-\frac{1}{2}\mathbb{E}\left[\sum_{i\geq j+1}w_i\Lambda_{0,i}(X_0)-\sum_{i\geq j}w_i\Lambda_{0,i}(X_0)\right]=\frac{w_j}{2}T_j. ∂ v j Ψ h = − E [ ( ∂ Δ j − ∂ Δ j + 1 ) ϕ 0 ( X 0 ) ] = − 2 1 E [ i ≥ j + 1 ∑ w i Λ 0 , i ( X 0 ) − i ≥ j ∑ w i Λ 0 , i ( X 0 ) ] = 2 w j T j . For j = K j=K j = K Δ K \Delta_K Δ K ν K / 2 \nu_K/2 ν K /2
∂ ν K Ψ h = 1 2 − 1 2 E [ 1 − w K Λ 0 , K ( X 0 ) ] = w K 2 T K . \partial_{\nu_K}\Psi_h=\frac{1}{2}-\frac{1}{2}\mathbb{E}\left[1-w_K\Lambda_{0,K}(X_0)\right]=\frac{w_K}{2}T_K. ∂ ν K Ψ h = 2 1 − 2 1 E [ 1 − w K Λ 0 , K ( X 0 ) ] = 2 w K T K . The endpoint ν 0 \nu_0 ν 0 X 0 X_0 X 0 Δ 1 \Delta_1 Δ 1
∂ ν 0 Ψ h = − 1 2 E [ ϕ 0 ′ ′ ( X 0 ) ] + E [ ∂ Δ 1 ϕ 0 ( X 0 ) ] = − 1 2 E [ 1 − ζ 0 Λ 0 , 0 − ∑ j ≥ 1 w j Λ 0 , j ] + 1 2 E [ 1 − ∑ j ≥ 1 w j Λ 0 , j ] , \begin{aligned}
\partial_{\nu_0}\Psi_h&=-\frac{1}{2}\mathbb{E}\left[\phi_0^{\prime\prime}(X_0)\right]+\mathbb{E}\left[\partial_{\Delta_1}\phi_0(X_0)\right]\\
&=-\frac{1}{2}\mathbb{E}\left[1-\zeta_0\Lambda_{0,0}-\sum_{j\geq1}w_j\Lambda_{0,j}\right]+\frac{1}{2}\mathbb{E}\left[1-\sum_{j\geq1}w_j\Lambda_{0,j}\right],
\end{aligned} ∂ ν 0 Ψ h = − 2 1 E [ ϕ 0 ′′ ( X 0 ) ] + E [ ∂ Δ 1 ϕ 0 ( X 0 ) ] = − 2 1 E [ 1 − ζ 0 Λ 0 , 0 − j ≥ 1 ∑ w j Λ 0 , j ] + 2 1 E [ 1 − j ≥ 1 ∑ w j Λ 0 , j ] , which equals w 0 T 0 / 2 w_0T_0/2 w 0 T 0 /2 ζ 0 = w 0 \zeta_0=w_0 ζ 0 = w 0 T j T_j T j ψ h \psi_h ψ h
Up to a change of variables, the formula (3.14) is [112 Lemma 1] , and for h = 0 h=0 h = 0 [37 Proposition 2.5] . The nonnegativity and the ordering of the derivatives are a special case of [108 Proposition 3.6] .
Complete monotonicity
A function f : [ 0 , ∞ ) → R f\colon[0,\infty)\to\mathbb{R} f : [ 0 , ∞ ) → R completely monotone if it is smooth on [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) 0 0 0 ( − 1 ) k f ( k ) ≥ 0 (-1)^k f^{(k)}\geq0 ( − 1 ) k f ( k ) ≥ 0 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) k ≥ 0 k\geq0 k ≥ 0 g : [ 0 , ∞ ) → [ 0 , ∞ ) g\colon[0,\infty)\to[0,\infty) g : [ 0 , ∞ ) → [ 0 , ∞ ) Bernstein function if it is smooth and g ′ g' g ′ F F F R \mathbb{R} R F ~ ( s ) = F ( s ) \widetilde{F}(s)=F(\sqrt{s}) F ( s ) = F ( s ) s ≥ 0 s\geq0 s ≥ 0 F F F x 2 x^2 x 2 F ~ \widetilde{F} F F F F
∂ s F ~ ( s ) = ( D F ) ( s ) , ( D F ) ( x ) = 1 2 ∫ 0 1 F ′ ′ ( x θ ) d θ , (31) \partial_s\widetilde{F}(s)=(\mathcal{D}F)(\sqrt{s}),\qquad(\mathcal{D}F)(x)=\frac{1}{2}\int_0^1 F^{\prime\prime}(x\theta)\,d\theta, \tag*{(31)} ∂ s F ( s ) = ( D F ) ( s ) , ( D F ) ( x ) = 2 1 ∫ 0 1 F ′′ ( x θ ) d θ , (31) because F ′ ( 0 ) = 0 F'(0)=0 F ′ ( 0 ) = 0 D \mathcal{D} D F ~ \widetilde{F} F [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ∂ s k F ~ ( s ) = ( D k F ) ( s ) \partial_s^k\widetilde{F}(s)=(\mathcal{D}^kF)(\sqrt{s}) ∂ s k F ( s ) = ( D k F ) ( s ) ∣ ∂ s k F ~ ( s ) ∣ ≤ 2 − k sup ∣ y ∣ ≤ s ∣ F ( 2 k ) ( y ) ∣ \lvert\partial_s^k\widetilde{F}(s)\rvert\leq2^{-k}\sup_{\lvert y\rvert\leq\sqrt{s}}\lvert F^{(2k)}(y)\rvert ∣ ∂ s k F ( s )∣ ≤ 2 − k sup ∣ y ∣ ≤ s ∣ F ( 2 k ) ( y )∣
We use the following facts.
Lemma 3.3. (i) Products and nonnegative linear combinations of completely monotone functions are completely monotone. If f f f s ↦ f ( a s ) s\mapsto f(as) s ↦ f ( a s ) a ≥ 0 a\geq0 a ≥ 0 s ↦ ∫ 0 1 f ( s θ 2 ) d θ s\mapsto\int_0^1 f(s\theta^2)\,d\theta s ↦ ∫ 0 1 f ( s θ 2 ) d θ
(ii) If g g g e − g e^{-g} e − g g > 0 g>0 g > 0 1 / g 1/g 1/ g
(iii) A function f f f f ( s ) = ∫ [ 0 , ∞ ) e − λ s μ ( d λ ) f(s)=\int_{[0,\infty)}e^{-\lambda s}\,\mu(d\lambda) f ( s ) = ∫ [ 0 , ∞ ) e − λ s μ ( d λ ) μ \mu μ ∫ λ k μ ( d λ ) < ∞ \int\lambda^k\mu(d\lambda)<\infty ∫ λ k μ ( d λ ) < ∞ k k k
(iv) If G G G x 2 x^2 x 2 b ≥ 0 b \ge0 b ≥ 0 γ b ∗ G \gamma_b * G γ b ∗ G x 2 x^2 x 2 b ↦ γ b ∗ G ( 0 ) b \mapsto\gamma_b * G(0) b ↦ γ b ∗ G ( 0 ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
Proof. Part (i) follows from the Leibniz rule and differentiation under the integral sign. For (ii), let H = e − g H=e^{-g} H = e − g H ′ = − g ′ H H'=-g'H H ′ = − g ′ H
( − 1 ) k + 1 H ( k + 1 ) = ∑ i = 0 k ( k i ) [ ( − 1 ) i g ( i ) ] [ ( − 1 ) k − i H ( k − i ) ] . (-1)^{k+1}H^{(k+1)}=\sum_{i=0}^{k}\binom{k}{i}\left[(-1)^i g^{(i)}\right]\left[(-1)^{k-i}H^{(k-i)}\right]. ( − 1 ) k + 1 H ( k + 1 ) = i = 0 ∑ k ( i k ) [ ( − 1 ) i g ( i ) ] [ ( − 1 ) k − i H ( k − i ) ] . which is nonnegative by induction on k k k g > 0 g>0 g > 0 H = 1 / g H=1/g H = 1/ g H ′ = − g ′ H 2 H'=-g'H^2 H ′ = − g ′ H 2 H H H H 2 H^2 H 2 ( − 1 ) k H ( k ) ≥ 0 (-1)^kH^{(k)}\ge0 ( − 1 ) k H ( k ) ≥ 0 [124 Theorem 1.4] ; the measure is finite because f ( 0 ) < ∞ f(0)<\infty f ( 0 ) < ∞ f f f G ~ ( s ) = ∫ e − λ s μ ( d λ ) \widetilde{G}(s)=\int e^{-\lambda s}\mu(d\lambda) G ( s ) = ∫ e − λ s μ ( d λ ) λ ≥ 0 \lambda\ge0 λ ≥ 0 b ≥ 0 b\ge0 b ≥ 0
E exp ( − λ ( x + b Z ) 2 ) = ( 1 + 2 b λ ) − 1 / 2 exp ( − λ x 2 1 + 2 b λ ) . \mathbb{E}\exp\left(-\lambda\left(x+\sqrt{b}Z\right)^2\right)=(1+2b\lambda)^{-1/2}\exp\left(-\frac{\lambda x^2}{1+2b\lambda}\right). E exp ( − λ ( x + b Z ) 2 ) = ( 1 + 2 bλ ) − 1/2 exp ( − 1 + 2 bλ λ x 2 ) . Integrating against μ \mu μ γ b ∗ G \gamma_b*G γ b ∗ G x ↦ e − λ ′ x 2 x\mapsto e^{-\lambda' x^2} x ↦ e − λ ′ x 2 λ ′ = λ / ( 1 + 2 b λ ) ≥ 0 \lambda'=\lambda/(1+2b\lambda)\ge0 λ ′ = λ / ( 1 + 2 bλ ) ≥ 0 λ ′ ≤ λ \lambda'\le\lambda λ ′ ≤ λ μ \mu μ x = 0 x=0 x = 0 ∫ ( 1 + 2 b λ ) − 1 / 2 μ ( d λ ) \int(1+2b\lambda)^{-1/2}\mu(d\lambda) ∫ ( 1 + 2 bλ ) − 1/2 μ ( d λ ) b ↦ ( 1 + 2 b λ ) − 1 / 2 b\mapsto(1+2b\lambda)^{-1/2} b ↦ ( 1 + 2 bλ ) − 1/2
The derivatives that we must control satisfy linear parabolic equations in the variable s = x 2 s=x^2 s = x 2 s = 0 s=0 s = 0
Lemma 3.4. Let d > 0 d>0 d > 0 z : [ 0 , d ] × [ 0 , ∞ ) → R z:[0,d]\times[0,\infty)\to\mathbb{R} z : [ 0 , d ] × [ 0 , ∞ ) → R ∂ b z \partial_bz ∂ b z ∂ s z \partial_sz ∂ s z ∂ s 2 z \partial_s^2z ∂ s 2 z ( 0 , d ] × [ 0 , ∞ ) (0,d]\times[0,\infty) ( 0 , d ] × [ 0 , ∞ ) s = 0 s=0 s = 0 ∣ z ( b , s ) ∣ ≤ C z e s |z(b,s)|\le C_z e^{\sqrt{s}} ∣ z ( b , s ) ∣ ≤ C z e s z ( 0 , ⋅ ) ≥ 0 z(0,\cdot)\ge0 z ( 0 , ⋅ ) ≥ 0
∂ b z ≥ 2 s ∂ s 2 z + β ( b , s ) ∂ s z + V ( b , s ) z on ( 0 , d ] × [ 0 , ∞ ) , \partial_bz\ge2s\partial_s^2z+\beta(b,s)\partial_sz+V(b,s)z\qquad\text{on }(0,d]\times[0,\infty), ∂ b z ≥ 2 s ∂ s 2 z + β ( b , s ) ∂ s z + V ( b , s ) z on ( 0 , d ] × [ 0 , ∞ ) , where 0 ≤ β ( b , s ) ≤ C β ( 1 + s ) 0\le\beta(b,s)\le C_\beta(1+\sqrt{s}) 0 ≤ β ( b , s ) ≤ C β ( 1 + s ) ∣ V ( b , s ) ∣ ≤ C V |V(b,s)|\le C_V ∣ V ( b , s ) ∣ ≤ C V z ≥ 0 z\ge0 z ≥ 0
Proof. Let Φ ( s ) = cosh ( 2 s ) = ∑ k ≥ 0 4 k s k / ( 2 k ) ! \Phi(s)=\cosh(2\sqrt{s})=\sum_{k\ge0}4^ks^k/(2k)! Φ ( s ) = cosh ( 2 s ) = ∑ k ≥ 0 4 k s k / ( 2 k )! [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) Φ ′ ( s ) = sinh ( 2 s ) / s ≥ 0 \Phi'(s)=\sinh(2\sqrt{s})/\sqrt{s}\ge0 Φ ′ ( s ) = sinh ( 2 s ) / s ≥ 0 2 s Φ ′ ′ ( s ) = 2 cosh ( 2 s ) − sinh ( 2 s ) / s 2s\Phi''(s)=2\cosh(2\sqrt{s})-\sinh(2\sqrt{s})/\sqrt{s} 2 s Φ ′′ ( s ) = 2 cosh ( 2 s ) − sinh ( 2 s ) / s sinh y ≤ y cosh y \sinh y\le y\cosh y sinh y ≤ y cosh y sinh y ≤ cosh y \sinh y\le\cosh y sinh y ≤ cosh y y ≥ 0 y\ge0 y ≥ 0
2 s Φ ′ ′ + β Φ ′ ≤ 2 Φ + C β ( sinh ( 2 s ) s + sinh ( 2 s ) ) ≤ ( 2 + 3 C β ) Φ . 2s\Phi''+\beta\Phi'\le2\Phi+C_\beta\left(\frac{\sinh(2\sqrt{s})}{\sqrt{s}}+\sinh(2\sqrt{s})\right)\le(2+3C_\beta)\Phi. 2 s Φ ′′ + β Φ ′ ≤ 2Φ + C β ( s sinh ( 2 s ) + sinh ( 2 s ) ) ≤ ( 2 + 3 C β ) Φ. Set C Φ = 2 + 3 C β C_\Phi=2+3C_\beta C Φ = 2 + 3 C β Λ > C V \Lambda>C_V Λ > C V ϵ > 0 \epsilon>0 ϵ > 0 z ϵ ( b , s ) = e − Λ b z ( b , s ) + ϵ e C Φ b Φ ( s ) z_\epsilon(b,s)=e^{-\Lambda b}z(b,s)+\epsilon e^{C_\Phi b}\Phi(s) z ϵ ( b , s ) = e − Λ b z ( b , s ) + ϵ e C Φ b Φ ( s ) Φ ( s ) ≥ e 2 s / 2 \Phi(s)\ge e^{2\sqrt{s}}/2 Φ ( s ) ≥ e 2 s /2 z ϵ z_\epsilon z ϵ + ∞ +\infty + ∞ s → ∞ s\to\infty s → ∞ b ∈ [ 0 , d ] b\in[0,d] b ∈ [ 0 , d ] [ 0 , d ] × [ 0 , ∞ ) [0,d]\times[0,\infty) [ 0 , d ] × [ 0 , ∞ ) ( b 1 , s 1 ) (b_1,s_1) ( b 1 , s 1 ) b 1 > 0 b_1>0 b 1 > 0 z ε ( 0 , ⋅ ) > 0 z_\varepsilon(0,\cdot)>0 z ε ( 0 , ⋅ ) > 0 ( b 1 , s 1 ) (b_1,s_1) ( b 1 , s 1 ) ∂ b z ε ≤ 0 \partial_b z_\varepsilon\le0 ∂ b z ε ≤ 0 2 s ∂ s 2 z ε + β ∂ s z ε ≥ 0 2s\partial_s^2z_\varepsilon+\beta\partial_s z_\varepsilon\ge0 2 s ∂ s 2 z ε + β ∂ s z ε ≥ 0 s 1 > 0 s_1>0 s 1 > 0 ∂ s z ε = 0 ≤ ∂ s 2 z ε \partial_s z_\varepsilon=0\le\partial_s^2z_\varepsilon ∂ s z ε = 0 ≤ ∂ s 2 z ε s 1 = 0 s_1=0 s 1 = 0 ∂ s z ε ≥ 0 \partial_s z_\varepsilon\ge0 ∂ s z ε ≥ 0
∂ b z ε ≥ ( 2 s ∂ s 2 z ε + β ∂ s z ε ) + ε e C Φ b ( C Φ Φ − 2 s Φ ′ ′ − β Φ ′ ) + ( V − Λ ) e − Λ b z . \partial_b z_\varepsilon\ge\left(2s\partial_s^2z_\varepsilon+\beta\partial_s z_\varepsilon\right)+\varepsilon e^{C\Phi b}\left(C_\Phi\Phi-2s\Phi^{\prime\prime}-\beta\Phi^\prime\right)+(V-\Lambda)e^{-\Lambda b}z. ∂ b z ε ≥ ( 2 s ∂ s 2 z ε + β ∂ s z ε ) + ε e C Φ b ( C Φ Φ − 2 s Φ ′′ − β Φ ′ ) + ( V − Λ ) e − Λ b z . At ( b 1 , s 1 ) (b_1,s_1) ( b 1 , s 1 ) e − Λ b z = z ε − ε e C Φ b Φ < 0 e^{-\Lambda b}z=z_\varepsilon-\varepsilon e^{C\Phi b}\Phi<0 e − Λ b z = z ε − ε e C Φ b Φ < 0 ∂ b z ε > 0 \partial_bz_\varepsilon>0 ∂ b z ε > 0 z ε ≥ 0 z_\varepsilon\ge0 z ε ≥ 0 ε > 0 \varepsilon>0 ε > 0 z ≥ 0 z\ge0 z ≥ 0
We now show that one step of the recursion (3.7), without auxiliary randomness, preserves the relevant complete monotonicity properties.
Lemma 3.5. Let 0 < m ≤ m ′ ≤ 1 0<m\le m^\prime\le1 0 < m ≤ m ′ ≤ 1 φ i n : R → R \varphi_{\mathrm{in}}:\mathbb{R}\to\mathbb{R} φ in : R → R ∣ φ i n ′ ∣ ≤ 1 |\varphi_{\mathrm{in}}^\prime|\le1 ∣ φ in ′ ∣ ≤ 1 r ≥ 1 r\ge1 r ≥ 1 Λ i n ι \Lambda_{\mathrm{in}}^\iota Λ in ι ι ∈ I \iota\in\mathcal{I} ι ∈ I [ 0 , 1 ] [0,1] [ 0 , 1 ] c ∈ [ 0 , m ′ ] c\in[0,m^\prime] c ∈ [ 0 , m ′ ] ι ∈ I \iota\in\mathcal{I} ι ∈ I
φ i n ′ ′ , e c φ i n φ i n ′ ′ , e c φ i n ( 1 − ( φ i n ′ ) 2 ) , e c φ i n ( 1 − Λ i n ι ) (32) \varphi_{\mathrm{in}}^{\prime\prime},\qquad e^{c\varphi_{\mathrm{in}}}\varphi_{\mathrm{in}}^{\prime\prime},\qquad e^{c\varphi_{\mathrm{in}}}\left(1-\left(\varphi_{\mathrm{in}}^\prime\right)^2\right),\qquad e^{c\varphi_{\mathrm{in}}}\left(1-\Lambda_{\mathrm{in}}^\iota\right) \tag*{(32)} φ in ′′ , e c φ in φ in ′′ , e c φ in ( 1 − ( φ in ′ ) 2 ) , e c φ in ( 1 − Λ in ι ) (32) are completely monotone in x 2 x^2 x 2 b ≥ 0 b\ge0 b ≥ 0
φ ( b , ⋅ ) = 1 m log ( γ b ∗ e m φ i n ) , Λ ι ( b , ⋅ ) = γ b ∗ ( e m φ i n Λ i n ι ) γ b ∗ e m φ i n . \varphi(b,\cdot)=\frac{1}{m}\log\left(\gamma_b*e^{m\varphi_{\mathrm{in}}}\right),\qquad\Lambda^\iota(b,\cdot)=\frac{\gamma_b*\left(e^{m\varphi_{\mathrm{in}}}\Lambda_{\mathrm{in}}^\iota\right)}{\gamma_b*e^{m\varphi_{\mathrm{in}}}}. φ ( b , ⋅ ) = m 1 log ( γ b ∗ e m φ in ) , Λ ι ( b , ⋅ ) = γ b ∗ e m φ in γ b ∗ ( e m φ in Λ in ι ) . Then, for every b ≥ 0 b\ge0 b ≥ 0 φ ( b , ⋅ ) \varphi(b,\cdot) φ ( b , ⋅ ) Λ ι ( b , ⋅ ) \Lambda^\iota(b,\cdot) Λ ι ( b , ⋅ ) m ′ m^\prime m ′ m m m
Proof. Fix d > 0 d>0 d > 0 b ∈ [ 0 , d ] b\in[0,d] b ∈ [ 0 , d ] φ ( b , ⋅ ) \varphi(b,\cdot) φ ( b , ⋅ ) ζ ℓ − 1 \zeta_{\ell-1} ζ ℓ − 1 m m m ∣ ∂ x φ ∣ ≤ 1 |\partial_x\varphi|\le1 ∣ ∂ x φ ∣ ≤ 1 ∂ x 2 φ ≥ 0 \partial_x^2\varphi\ge0 ∂ x 2 φ ≥ 0 r ≥ 1 r\ge1 r ≥ 1 b ∈ [ 0 , d ] b\in[0,d] b ∈ [ 0 , d ] Λ ι ( b , ⋅ ) \Lambda^\iota(b,\cdot) Λ ι ( b , ⋅ ) [ 0 , 1 ] [0,1] [ 0 , 1 ] Λ i n ι \Lambda_{\mathrm{in}}^\iota Λ in ι u = ∂ x φ u=\partial_x\varphi u = ∂ x φ ψ = ∂ x 2 φ \psi=\partial_x^2\varphi ψ = ∂ x 2 φ b > 0 b>0 b > 0
∂ b φ = 1 2 ( ∂ x 2 φ + m ( ∂ x φ ) 2 ) , ∂ b u = 1 2 ∂ x 2 u + m u ∂ x u , ∂ b ψ = 1 2 ∂ x 2 ψ + m u ∂ x ψ + m ψ 2 , (33) \partial_b\varphi=\frac{1}{2}\left(\partial_x^2\varphi+m(\partial_x\varphi)^2\right),\qquad\partial_bu=\frac{1}{2}\partial_x^2u+mu\partial_xu,\qquad\partial_b\psi=\frac{1}{2}\partial_x^2\psi+mu\partial_x\psi+m\psi^2, \tag*{(33)} ∂ b φ = 2 1 ( ∂ x 2 φ + m ( ∂ x φ ) 2 ) , ∂ b u = 2 1 ∂ x 2 u + m u ∂ x u , ∂ b ψ = 2 1 ∂ x 2 ψ + m u ∂ x ψ + m ψ 2 , (33) and 2 ∂ b Λ ι = ∂ x 2 Λ ι + 2 m u ∂ x Λ ι 2\partial_b\Lambda^\iota=\partial_x^2\Lambda^\iota+2mu\partial_x\Lambda^\iota 2 ∂ b Λ ι = ∂ x 2 Λ ι + 2 m u ∂ x Λ ι c ∈ [ 0 , m ] c\in[0,m] c ∈ [ 0 , m ] H c = e c φ ψ H_c=e^{c\varphi}\psi H c = e c φ ψ G c = e c φ ( 1 − u 2 ) G_c=e^{c\varphi}(1-u^2) G c = e c φ ( 1 − u 2 ) Θ c ι = e c φ ( 1 − Λ ι ) \Theta_c^\iota=e^{c\varphi}(1-\Lambda^\iota) Θ c ι = e c φ ( 1 − Λ ι )
∂ b H c = 1 2 ∂ x 2 H c + ( m − c ) u ∂ x H c + ( m ψ − 1 2 c ( m − c ) u 2 ) H c , \partial_bH_c=\frac{1}{2}\partial_x^2H_c+(m-c)u\,\partial_xH_c+\left(m\psi-\frac{1}{2}c(m-c)u^2\right)H_c, ∂ b H c = 2 1 ∂ x 2 H c + ( m − c ) u ∂ x H c + ( m ψ − 2 1 c ( m − c ) u 2 ) H c , ∂ b G c = 1 2 ∂ x 2 G c + ( m − c ) u ∂ x G c − 1 2 c ( m − c ) u 2 G c + H c ψ , \partial_bG_c=\frac{1}{2}\partial_x^2G_c+(m-c)u\,\partial_xG_c-\frac{1}{2}c(m-c)u^2G_c+H_c\psi, ∂ b G c = 2 1 ∂ x 2 G c + ( m − c ) u ∂ x G c − 2 1 c ( m − c ) u 2 G c + H c ψ , ∂ b Θ c ι = 1 2 ∂ x 2 Θ c ι + ( m − c ) u ∂ x Θ c ι − 1 2 c ( m − c ) u 2 Θ c ι . (34) \partial_b\Theta_c^\iota=\frac{1}{2}\partial_x^2\Theta_c^\iota+(m-c)u\,\partial_x\Theta_c^\iota-\frac{1}{2}c(m-c)u^2\Theta_c^\iota. \tag*{(34)} ∂ b Θ c ι = 2 1 ∂ x 2 Θ c ι + ( m − c ) u ∂ x Θ c ι − 2 1 c ( m − c ) u 2 Θ c ι . (34) We pass to the variable s = x 2 s=x^2 s = x 2 f ( b , s ) = ψ ( b , s ) f(b,s)=\psi(b,\sqrt{s}) f ( b , s ) = ψ ( b , s ) b ∗ ( b , s ) = s u ( b , s ) b_*(b,s)=\sqrt{s}\,u(b,\sqrt{s}) b ∗ ( b , s ) = s u ( b , s ) F ( b , ⋅ ) F(b,\cdot) F ( b , ⋅ ) ∂ x 2 F = 2 ∂ s F ~ + 4 s ∂ s 2 F ~ \partial_x^2F=2\partial_s\widetilde{F}+4s\partial_s^2\widetilde{F} ∂ x 2 F = 2 ∂ s F + 4 s ∂ s 2 F γ u ∂ x F = 2 γ b ∗ ∂ s F ~ \gamma u\partial_xF=2\gamma b_*\partial_s\widetilde{F} γ u ∂ x F = 2 γ b ∗ ∂ s F F ~ ( b , s ) = F ( b , s ) \widetilde{F}(b,s)=F(b,\sqrt{s}) F ( b , s ) = F ( b , s ) u ( b , 0 ) = 0 u(b,0)=0 u ( b , 0 ) = 0 u ( b , x ) = x ∫ 0 1 ψ ( b , x θ ) d θ u(b,x)=x\int_0^1\psi(b,x\theta)\,\mathrm{d}\theta u ( b , x ) = x ∫ 0 1 ψ ( b , x θ ) d θ
∂ s b ∗ = 1 2 [ f ( s ) + ∫ 0 1 f ( s θ 2 ) d θ ] , ∂ s ( u ( s ) 2 ) = f ( s ) ∫ 0 1 f ( s θ 2 ) d θ . (35) \partial_s b_*=\frac{1}{2}\left[f(s)+\int_0^1 f(s\theta^2)\,\mathrm{d}\theta\right],\qquad\partial_s\left(u(\sqrt{s})^2\right)=f(s)\int_0^1 f(s\theta^2)\,\mathrm{d}\theta. \tag*{(35)} ∂ s b ∗ = 2 1 [ f ( s ) + ∫ 0 1 f ( s θ 2 ) d θ ] , ∂ s ( u ( s ) 2 ) = f ( s ) ∫ 0 1 f ( s θ 2 ) d θ . (35) Since ϕ ( b , ⋅ ) \phi(b,\cdot) ϕ ( b , ⋅ ) u ( b , x ) u(b,x) u ( b , x ) x x x 0 ≤ b ∗ ≤ s 0\leq b_*\leq\sqrt{s} 0 ≤ b ∗ ≤ s ∣ u ∣ ≤ 1 |u|\leq1 ∣ u ∣ ≤ 1 ψ \psi ψ (35) that ∂ s b ∗ \partial_s b_* ∂ s b ∗
All functions below are smooth in ( b , s ) ∈ ( 0 , d ] × [ 0 , ∞ ) (b,s)\in(0,d]\times[0,\infty) ( b , s ) ∈ ( 0 , d ] × [ 0 , ∞ ) [ 0 , d ] × [ 0 , ∞ ) [0,d]\times[0,\infty) [ 0 , d ] × [ 0 , ∞ ) s s s (31) and the smoothness of the heat semigroup. Since ϕ ( b , x ) ≤ ϕ ( b , 0 ) + ∣ x ∣ \phi(b,x)\leq\phi(b,0)+|x| ϕ ( b , x ) ≤ ϕ ( b , 0 ) + ∣ x ∣ ϕ ( b , 0 ) \phi(b,0) ϕ ( b , 0 ) [ 0 , d ] [0,d] [ 0 , d ] x x x H c H_c H c G c G_c G c Θ c ι \Theta_c^\iota Θ c ι C e c ∣ x ∣ ≤ C e ∣ x ∣ Ce^{c|x|}\leq Ce^{|x|} C e c ∣ x ∣ ≤ C e ∣ x ∣ (31) every s s s C e s Ce^{\sqrt{s}} C e s f f f b ∗ b_* b ∗ u 2 u^2 u 2
Step 1: f ( b , ⋅ ) f(b,\cdot) f ( b , ⋅ ) Set ϖ k = ( − 1 ) k ∂ s k f \varpi_k=(-1)^k\partial_s^k f ϖ k = ( − 1 ) k ∂ s k f j ≥ 1 j\geq1 j ≥ 1 B j = ( − 1 ) j − 1 ∂ s j b ∗ B_j=(-1)^{j-1}\partial_s^j b_* B j = ( − 1 ) j − 1 ∂ s j b ∗ (35) ,
B j = 1 2 [ ϖ j − 1 ( s ) + ∫ 0 1 θ 2 j − 2 ϖ j − 1 ( s θ 2 ) d θ ] . B_j=\frac{1}{2}\left[\varpi_{j-1}(s)+\int_0^1\theta^{2j-2}\varpi_{j-1}(s\theta^2)\,\mathrm{d}\theta\right]. B j = 2 1 [ ϖ j − 1 ( s ) + ∫ 0 1 θ 2 j − 2 ϖ j − 1 ( s θ 2 ) d θ ] . In the variable s s s (33) reads ∂ b f = 2 s ∂ s 2 f + ( 1 + 2 m b ∗ ) ∂ s f + m f 2 \partial_bf=2s\partial_s^2f+(1+2mb_*)\partial_sf+mf^2 ∂ b f = 2 s ∂ s 2 f + ( 1 + 2 m b ∗ ) ∂ s f + m f 2 k ≥ 1 k\geq1 k ≥ 1 s s s
∂ b ϖ k = 2 s ∂ s 2 ϖ k + ( 2 k + 1 + 2 m b ∗ ) ∂ s ϖ k + ( 2 m k ∂ s b ∗ + 2 m f ) ϖ k + m ∑ i = 1 k − 1 ( k i ) ϖ i ϖ k − i + 2 m ∑ i = 2 k ( k i ) B i ϖ k − i + 1 . (36) \begin{aligned}
\partial_b\varpi_k={}&2s\partial_s^2\varpi_k+(2k+1+2mb_*)\partial_s\varpi_k+(2mk\partial_sb_*+2mf)\varpi_k\\
&+m\sum_{i=1}^{k-1}\binom{k}{i}\varpi_i\varpi_{k-i}
+2m\sum_{i=2}^{k}\binom{k}{i}B_i\varpi_{k-i+1}. \tag*{(36)}
\end{aligned} ∂ b ϖ k = 2 s ∂ s 2 ϖ k + ( 2 k + 1 + 2 m b ∗ ) ∂ s ϖ k + ( 2 mk ∂ s b ∗ + 2 m f ) ϖ k + m i = 1 ∑ k − 1 ( i k ) ϖ i ϖ k − i + 2 m i = 2 ∑ k ( i k ) B i ϖ k − i + 1 . (36) We have ϖ 0 = f ≥ 0 \varpi_0=f\geq0 ϖ 0 = f ≥ 0 ϕ ( b , ⋅ ) \phi(b,\cdot) ϕ ( b , ⋅ ) ϖ 0 , … , ϖ k − 1 ≥ 0 \varpi_0,\ldots,\varpi_{k-1}\geq0 ϖ 0 , … , ϖ k − 1 ≥ 0 [ 0 , d ] × [ 0 , ∞ ) [0,d]\times[0,\infty) [ 0 , d ] × [ 0 , ∞ ) B 2 , … , B k ≥ 0 B_2,\ldots,B_k\geq0 B 2 , … , B k ≥ 0 (36) are nonnegative, because they involve only ϖ 1 , … , ϖ k − 1 \varpi_1,\ldots,\varpi_{k-1} ϖ 1 , … , ϖ k − 1 B 2 , … , B k B_2,\ldots,B_k B 2 , … , B k ϖ k ( 0 , ⋅ ) ≥ 0 \varpi_k(0,\cdot)\geq0 ϖ k ( 0 , ⋅ ) ≥ 0 (32) . Lemma 3.4, applied with β = 2 k + 1 + 2 m b ∗ \beta=2k+1+2mb_* β = 2 k + 1 + 2 m b ∗ V = 2 m k ∂ s b ∗ + 2 m f V=2mk\partial_sb_*+2mf V = 2 mk ∂ s b ∗ + 2 m f ϖ k ≥ 0 \varpi_k\geq0 ϖ k ≥ 0 f ( b , ⋅ ) f(b,\cdot) f ( b , ⋅ ) (35) and Lemma 3.3(i), ( − 1 ) j − 1 ∂ s j b ∗ ≥ 0 (-1)^{j-1}\partial_s^jb_*\geq0 ( − 1 ) j − 1 ∂ s j b ∗ ≥ 0 ( − 1 ) j − 1 ∂ s j ( u 2 ) ≥ 0 (-1)^{j-1}\partial_s^j(u^2)\geq0 ( − 1 ) j − 1 ∂ s j ( u 2 ) ≥ 0 j ≥ 1 j\geq1 j ≥ 1
Step 2: a general induction. Let F F F H c H_c H c G c G_c G c Θ c ι \Theta_c^\iota Θ c ι c ∈ [ 0 , m ] c\in[0,m] c ∈ [ 0 , m ] γ = m − c ≥ 0 \gamma=m-c\geq0 γ = m − c ≥ 0 (34) , F ~ \widetilde{F} F
∂ b F ~ = 2 s ∂ s 2 F ~ + ( 1 + 2 γ b ∗ ) ∂ s F ~ + P F ~ + J , \partial_b\widetilde{F}=2s\partial_s^2\widetilde{F}+(1+2\gamma b_*)\partial_s\widetilde{F}+P\widetilde{F}+J, ∂ b F = 2 s ∂ s 2 F + ( 1 + 2 γ b ∗ ) ∂ s F + P F + J , where P = m f − c γ u 2 / 2 P = mf - c\gamma u^2/2 P = m f − c γ u 2 /2 J = 0 J = 0 J = 0 H c H_c H c P = − c γ u 2 / 2 P = -c\gamma u^2/2 P = − c γ u 2 /2 J = H ~ c f J = \widetilde{H}_c f J = H c f G c G_c G c P = − c γ u 2 / 2 P = -c\gamma u^2/2 P = − c γ u 2 /2 J = 0 J = 0 J = 0 Θ c l \Theta_c^l Θ c l ( − 1 ) j P ( j ) ≥ 0 (-1)^jP^{(j)} \ge0 ( − 1 ) j P ( j ) ≥ 0 j ≥ 1 j \ge1 j ≥ 1 s s s z k = ( − 1 ) k ∂ s k F ~ z_k = (-1)^k\partial_s^k\widetilde{F} z k = ( − 1 ) k ∂ s k F k k k
∂ b z k = 2 s ∂ s 2 z k + ( 2 k + 1 + 2 γ b ∗ ) ∂ s z k + ( 2 γ k ∂ s b ∗ + P ) z k + 2 γ ∑ i = 2 k ( k i ) B i z k − i + 1 + ∑ i = 1 k ( k i ) ( − 1 ) i P ( i ) z k − i + ( − 1 ) k J ( k ) . \partial_b z_k = 2s\partial_s^2 z_k + (2k+1+2\gamma b_*)\partial_s z_k + (2\gamma k\partial_s b_*+P)z_k
+ 2\gamma\sum_{i=2}^{k}\binom{k}{i}B_i z_{k-i+1}
+ \sum_{i=1}^{k}\binom{k}{i}(-1)^iP^{(i)}z_{k-i}
+ (-1)^kJ^{(k)}. ∂ b z k = 2 s ∂ s 2 z k + ( 2 k + 1 + 2 γ b ∗ ) ∂ s z k + ( 2 γ k ∂ s b ∗ + P ) z k + 2 γ i = 2 ∑ k ( i k ) B i z k − i + 1 + i = 1 ∑ k ( i k ) ( − 1 ) i P ( i ) z k − i + ( − 1 ) k J ( k ) . If J J J z 0 , … , z k − 1 ≥ 0 z_0,\ldots,z_{k-1}\ge0 z 0 , … , z k − 1 ≥ 0 z k ( 0 , ⋅ ) ≥ 0 z_k(0,\cdot)\ge0 z k ( 0 , ⋅ ) ≥ 0 Lemma 3.4 gives z k ≥ 0 z_k\ge0 z k ≥ 0 z 0 = F ~ ≥ 0 z_0=\widetilde{F}\ge0 z 0 = F ≥ 0 ψ ≥ 0 \psi\ge0 ψ ≥ 0 ∣ u ∣ ≤ 1 |u|\le1 ∣ u ∣ ≤ 1 Λ l ≤ 1 \Lambda^l\le1 Λ l ≤ 1
We apply Step 2 first to H c H_c H c J = 0 J=0 J = 0 H c ( b , ⋅ ) H_c(b,\cdot) H c ( b , ⋅ ) x 2 x^2 x 2 J = H ~ c f J=\widetilde{H}_c f J = H c f G c G_c G c Θ c l \Theta_c^l Θ c l
Lemma 3.6. Let v ∈ C K v\in C_K v ∈ C K ϕ k \phi_k ϕ k Λ k , j \Lambda_{k,j} Λ k , j § 4.3 . For every 0 ≤ k ≤ j ≤ K 0\le k\le j\le K 0 ≤ k ≤ j ≤ K c ∈ [ 0 , ζ k ] c\in[0,\zeta_k] c ∈ [ 0 , ζ k ]
ϕ k ′ ′ , e c ϕ k ϕ k ′ ′ , e c ϕ k ( 1 − Λ k , j ) \phi_k'',\qquad e^{c\phi_k}\phi_k'',\qquad e^{c\phi_k}(1-\Lambda_{k,j}) ϕ k ′′ , e c ϕ k ϕ k ′′ , e c ϕ k ( 1 − Λ k , j ) are completely monotone in x 2 x^2 x 2 ϕ k ′ \phi_k' ϕ k ′ Λ k , j \Lambda_{k,j} Λ k , j ∣ x ∣ |x| ∣ x ∣ j j j ν j \nu_j ν j [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) 1 − T j ( 0 ) 1-T_j(0) 1 − T j ( 0 )
1 − T j ( h ) = ∫ [ 0 , ∞ ) e − λ h 2 ν j ( d λ ) ( h ∈ R ) . (37) 1-T_j(h)=\int_{[0,\infty)}e^{-\lambda h^2}\nu_j(d\lambda)\qquad(h\in\mathbb{R}). \tag*{(37)} 1 − T j ( h ) = ∫ [ 0 , ∞ ) e − λ h 2 ν j ( d λ ) ( h ∈ R ) . (37) Proof. We prove the complete-monotonicity claims by downward induction, carrying also that ϕ k \phi_k ϕ k Λ k , j \Lambda_{k,j} Λ k , j 0 ≤ Λ k , j ≤ 1 0\le\Lambda_{k,j}\le1 0 ≤ Λ k , j ≤ 1 k = K k=K k = K ϕ K = log 2 cosh x \phi_K=\log2\cosh x ϕ K = log 2 cosh x ϕ K ′ ′ = 1 − ( ϕ K ′ ) 2 = sech 2 x \phi_K''=1-(\phi_K')^2=\operatorname{sech}^2x ϕ K ′′ = 1 − ( ϕ K ′ ) 2 = sech 2 x Λ K , K = tanh 2 x \Lambda_{K,K}=\tanh^2x Λ K , K = tanh 2 x ζ K = 1 \zeta_K=1 ζ K = 1 sech 2 x \operatorname{sech}^2x sech 2 x 2 c sech 2 − c x 2^c\operatorname{sech}^{2-c}x 2 c sech 2 − c x 2 − c ≥ 1 2-c\ge1 2 − c ≥ 1 cosh x = ∏ i ≥ 1 ( 1 + 4 x 2 / ( ( 2 i − 1 ) 2 π 2 ) ) \cosh x=\prod_{i\ge1}(1+4x^2/((2i-1)^2\pi^2)) cosh x = ∏ i ≥ 1 ( 1 + 4 x 2 / (( 2 i − 1 ) 2 π 2 )) [55 (4.36.2)] ,
∂ s log cosh s = ∑ i ≥ 1 4 ( 2 i − 1 ) 2 π 2 + 4 s . \partial_s\log\cosh\sqrt{s}=\sum_{i\ge1}\frac{4}{(2i-1)^2\pi^2+4s}. ∂ s log cosh s = i ≥ 1 ∑ ( 2 i − 1 ) 2 π 2 + 4 s 4 . where the series of derivatives of every order converges uniformly on [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) log cosh s \log\cosh\sqrt{s} log cosh s sech p s = e − p log cosh s \operatorname{sech}^p\sqrt{s}=e^{-p\log\cosh\sqrt{s}} sech p s = e − p l o g c o s h s p > 0 p>0 p > 0 Lemma 3.3 (ii).
Suppose that the strengthened claim holds at level k + 1 ≤ K k+1\le K k + 1 ≤ K m = ζ k m=\zeta_k m = ζ k m ′ = ζ k + 1 m'=\zeta_{k+1} m ′ = ζ k + 1 ϕ i n = ϕ k + 1 \phi_{\mathrm{in}}=\phi_{k+1} ϕ in = ϕ k + 1 Λ i n j = Λ k + 1 , j \Lambda_{\mathrm{in}}^j=\Lambda_{k+1,j} Λ in j = Λ k + 1 , j k + 1 ≤ j ≤ K k+1\le j\le K k + 1 ≤ j ≤ K b = Δ k + 1 b=\Delta_{k+1} b = Δ k + 1 Lemma 3.1 (a): the function ϕ k + 1 \phi_{k+1} ϕ k + 1 log 2 cosh \log2\cosh log 2 cosh ϕ k + 1 ′ ′ \phi_{k+1}'' ϕ k + 1 ′′ x 2 x^2 x 2 ( ϕ k + 1 ′ ) 2 = Λ k + 1 , k + 1 (\phi_{k+1}')^2=\Lambda_{k+1,k+1} ( ϕ k + 1 ′ ) 2 = Λ k + 1 , k + 1 ϕ ( Δ k + 1 , ⋅ ) = ϕ k \phi(\Delta_{k+1},\cdot)=\phi_k ϕ ( Δ k + 1 , ⋅ ) = ϕ k Λ j ( Δ k + 1 , ⋅ ) = Λ k , j \Lambda^j(\Delta_{k+1},\cdot)=\Lambda_{k,j} Λ j ( Δ k + 1 , ⋅ ) = Λ k , j j ≥ k + 1 j\ge k+1 j ≥ k + 1 Λ k , k = ( ϕ k ′ ) 2 \Lambda_{k,k}=(\phi_k')^2 Λ k , k = ( ϕ k ′ ) 2 ( ϕ k ′ ) 2 (\phi_k')^2 ( ϕ k ′ ) 2 k k k
Since ϕ k \phi_k ϕ k ϕ k ′ \phi_k' ϕ k ′ ϕ k ′ ′ ≥ 0 \phi_k''\ge0 ϕ k ′′ ≥ 0 1 − Λ k , j 1-\Lambda_{k,j} 1 − Λ k , j x 2 x^2 x 2 ∣ x ∣ |x| ∣ x ∣ 1 − Λ 0 , j 1-\Lambda_{0,j} 1 − Λ 0 , j x 2 x^2 x 2 1 − T j = γ v 0 ∗ ( 1 − Λ 0 , j ) 1-T_j=\gamma_{v_0}*(1-\Lambda_{0,j}) 1 − T j = γ v 0 ∗ ( 1 − Λ 0 , j ) h 2 h^2 h 2
The monotonicity of ϕ k ′ \phi_k' ϕ k ′ Λ k , j \Lambda_{k,j} Λ k , j ∣ x ∣ |x| ∣ x ∣ [112 Lemma 2] . The complete monotonicity is the additional information used in Lemma 3.7 and Proposition 3.8.
Square-root convexity
The next estimate bounds the curvature of T j T_j T j h = 0 h=0 h = 0 v j v_j v j T j ′ ′ T_j'' T j ′′ h h h
Lemma 3.7. For every v ∈ C K v\in C_K v ∈ C K 0 ≤ j ≤ K 0\le j\le K 0 ≤ j ≤ K v j T j ′ ′ ( 0 ) ≤ 2 T j ( 0 ) v_jT_j''(0)\le2T_j(0) v j T j ′′ ( 0 ) ≤ 2 T j ( 0 )
Proof. Fix j j j Λ k , j \Lambda_{k,j} Λ k , j k k k j j j 0 0 0 0 ≤ k < j 0\le k<j 0 ≤ k < j k + 1 k+1 k + 1 k k k b ∈ [ 0 , Δ k + 1 ] b\in[0,\Delta_{k+1}] b ∈ [ 0 , Δ k + 1 ] m = ζ k m=\zeta_k m = ζ k ϕ i n = ϕ k + 1 \phi_{\mathrm{in}}=\phi_{k+1} ϕ in = ϕ k + 1 Λ i n = Λ k + 1 , j \Lambda_{\mathrm{in}}=\Lambda_{k+1,j} Λ in = Λ k + 1 , j D = e ζ k ϕ k + 1 D=e^{\zeta_k\phi_{k+1}} D = e ζ k ϕ k + 1 B ( b ) = γ b ∗ D ( 0 ) B(b)=\gamma_b*D(0) B ( b ) = γ b ∗ D ( 0 )
g ( b ) = Λ ( b , 0 ) , 1 − g ( b ) = N ( b ) B ( b ) , N ( b ) = γ b ∗ ( D ( 1 − Λ k + 1 , j ) ) ( 0 ) . g(b)=\Lambda(b,0),\qquad1-g(b)=\frac{N(b)}{B(b)},\qquad N(b)=\gamma_b*\bigl(D(1-\Lambda_{k+1,j})\bigr)(0). g ( b ) = Λ ( b , 0 ) , 1 − g ( b ) = B ( b ) N ( b ) , N ( b ) = γ b ∗ ( D ( 1 − Λ k + 1 , j ) ) ( 0 ) . By Lemma 3.6 at level k + 1 k+1 k + 1 c = ζ k c=\zeta_k c = ζ k D ( 1 − Λ k + 1 , i ) D(1-\Lambda_{k+1,i}) D ( 1 − Λ k + 1 , i ) x 2 x^2 x 2 i ≥ k + 1 i\ge k+1 i ≥ k + 1 N N N N i ( b ) = γ b ∗ ( D ( 1 − Λ k + 1 , i ) ) ( 0 ) N_i(b)=\gamma_b*(D(1-\Lambda_{k+1,i}))(0) N i ( b ) = γ b ∗ ( D ( 1 − Λ k + 1 , i )) ( 0 ) b b b k + 1 k+1 k + 1 ζ k + ( ζ k + 1 − ζ k ) + ∑ i > k + 1 w i = 1 \zeta_k+(\zeta_{k+1}-\zeta_k)+\sum_{i>k+1}w_i=1 ζ k + ( ζ k + 1 − ζ k ) + ∑ i > k + 1 w i = 1
D ′ ′ = ζ k D ( ϕ k + 1 ′ ′ + ζ k ( ϕ k + 1 ′ ) 2 ) = ζ k 2 D + ζ k ( ζ k + 1 − ζ k ) D ( 1 − Λ k + 1 , k + 1 ) + ζ k ∑ i > k + 1 w i D ( 1 − Λ k + 1 , i ) . \begin{aligned}
D''&=\zeta_kD\bigl(\phi_{k+1}''+\zeta_k(\phi_{k+1}')^2\bigr)\\
&=\zeta_k^2D+\zeta_k(\zeta_{k+1}-\zeta_k)D(1-\Lambda_{k+1,k+1})+\zeta_k\sum_{i>k+1}w_iD(1-\Lambda_{k+1,i}).
\end{aligned} D ′′ = ζ k D ( ϕ k + 1 ′′ + ζ k ( ϕ k + 1 ′ ) 2 ) = ζ k 2 D + ζ k ( ζ k + 1 − ζ k ) D ( 1 − Λ k + 1 , k + 1 ) + ζ k i > k + 1 ∑ w i D ( 1 − Λ k + 1 , i ) . By (3.5),
B ′ = 1 2 γ b ∗ D ′ ′ ( 0 ) = ζ k 2 2 B + ζ k 2 ∑ i ≥ k + 1 c i N i , c i ≥ 0. B'=\frac{1}{2}\gamma_b*D''(0)=\frac{\zeta_k^2}{2}B+\frac{\zeta_k}{2}\sum_{i\ge k+1}c_iN_i,\qquad c_i\ge0. B ′ = 2 1 γ b ∗ D ′′ ( 0 ) = 2 ζ k 2 B + 2 ζ k i ≥ k + 1 ∑ c i N i , c i ≥ 0. The function B ~ = e − ζ k 2 b / 2 B \widetilde B=e^{-\zeta_k^2b/2}B B = e − ζ k 2 b /2 B 1 / B ~ 1/\widetilde B 1/ B 1 − g = N e − ζ k 2 b / 2 / B ~ 1-g=N e^{-\zeta_k^2b/2}/\widetilde B 1 − g = N e − ζ k 2 b /2 / B Lemma 3.3 (i). It follows that g g g [ 0 , Δ k + 1 ] [0,\Delta_{k+1}] [ 0 , Δ k + 1 ] g ( b ) = γ b ∗ Λ 0 , j ( 0 ) g(b)=\gamma_b * \Lambda_{0,j}(0) g ( b ) = γ b ∗ Λ 0 , j ( 0 ) b ∈ [ 0 , v 0 ] b\in[0,v_0] b ∈ [ 0 , v 0 ] Lemma 3.3 (iv).
Since ∂ x ϕ ( b , 0 ) = 0 \partial_x\phi(b,0)=0 ∂ x ϕ ( b , 0 ) = 0 2 g ′ ( b ) = ∂ x 2 Λ ( b , 0 ) 2g'(b)=\partial_x^2\Lambda(b,0) 2 g ′ ( b ) = ∂ x 2 Λ ( b , 0 ) Λ \Lambda Λ b 0 g ′ ( 0 ) ≤ g ( 0 ) b_0g'(0)\le g(0) b 0 g ′ ( 0 ) ≤ g ( 0 ) b 0 ≥ 0 b_0\ge0 b 0 ≥ 0 b g ′ ( b ) ≤ g ( b ) − g ( 0 ) bg'(b)\le g(b)-g(0) b g ′ ( b ) ≤ g ( b ) − g ( 0 ) g ′ ( b ) ≤ g ′ ( 0 ) g'(b)\le g'(0) g ′ ( b ) ≤ g ′ ( 0 )
( b 0 + b ) g ′ ( b ) ≤ g ( b ) − g ( 0 ) + b 0 g ′ ( 0 ) ≤ g ( b ) . (b_0+b)g'(b)\le g(b)-g(0)+b_0g'(0)\le g(b). ( b 0 + b ) g ′ ( b ) ≤ g ( b ) − g ( 0 ) + b 0 g ′ ( 0 ) ≤ g ( b ) . At level j j j Λ j , j = ( ϕ j ′ ) 2 \Lambda_{j,j}=(\phi'_j)^2 Λ j , j = ( ϕ j ′ ) 2 0 0 0 b 0 = 0 b_0=0 b 0 = 0 g g g g ′ g' g ′ Λ k , j ( 0 ) \Lambda_{k,j}(0) Λ k , j ( 0 ) Λ k , j ′ ′ ( 0 ) / 2 \Lambda''_{k,j}(0)/2 Λ k , j ′′ ( 0 ) /2 Δ j , … , Δ 1 \Delta_j,\ldots,\Delta_1 Δ j , … , Δ 1 v 0 v_0 v 0 v j v_j v j v j T j ′ ′ ( 0 ) ≤ 2 T j ( 0 ) v_jT''_j(0)\le2T_j(0) v j T j ′′ ( 0 ) ≤ 2 T j ( 0 )
Proposition 3.8. For every h ∈ R h\in\mathbb{R} h ∈ R s ↦ Ψ h ( s 0 2 , … , s K 2 ) s\mapsto\Psi_h(s_0^2,\ldots,s_K^2) s ↦ Ψ h ( s 0 2 , … , s K 2 ) { s ∈ R K + 1 : 0 ≤ s 0 ≤ s 1 ≤ ⋯ ≤ s K } \{s\in\mathbb{R}^{K+1}:0\le s_0\le s_1\le\cdots\le s_K\} { s ∈ R K + 1 : 0 ≤ s 0 ≤ s 1 ≤ ⋯ ≤ s K } s ↦ ψ h ( s 0 2 , … , s K 2 ) s\mapsto\psi_h(s_0^2,\ldots,s_K^2) s ↦ ψ h ( s 0 2 , … , s K 2 )
Proof. The second statement follows from the first, since ψ h ( s 2 ) = Ψ h ( ( 2 s ) 2 ) \psi_h(s^2)=\Psi_h((\sqrt{2}s)^2) ψ h ( s 2 ) = Ψ h (( 2 s ) 2 ) O = { s : 0 < s 0 < s 1 < ⋯ < s K } O=\{s:0<s_0<s_1<\cdots<s_K\} O = { s : 0 < s 0 < s 1 < ⋯ < s K } s ↦ Ψ h ( s 2 ) s\mapsto\Psi_h(s^2) s ↦ Ψ h ( s 2 ) O O O s ∈ O s\in O s ∈ O v = s 2 v=s^2 v = s 2 0 < v 0 < ⋯ < v K 0<v_0<\cdots<v_K 0 < v 0 < ⋯ < v K J i j = ∂ v j T i ( h ; v ) J_{ij}=\partial_{v_j}T_i(h;v) J ij = ∂ v j T i ( h ; v )
We first show that J i j ≤ 0 J_{ij}\le0 J ij ≤ 0 i < j i<j i < j ( X 0 , … , X i ) (X_0,\ldots,X_i) ( X 0 , … , X i ) ∏ ℓ < i exp ( ζ ℓ ( ϕ ℓ + 1 ( X ℓ + 1 ) − ϕ ℓ ( X ℓ ) ) ) \prod_{\ell<i}\exp(\zeta_\ell(\phi_{\ell+1}(X_{\ell+1})-\phi_\ell(X_\ell))) ∏ ℓ < i exp ( ζ ℓ ( ϕ ℓ + 1 ( X ℓ + 1 ) − ϕ ℓ ( X ℓ ))) v 0 , Δ 1 , … , Δ i v_0,\Delta_1,\ldots,\Delta_i v 0 , Δ 1 , … , Δ i v j v_j v j ℓ < j \ell<j ℓ < j ∂ v j ϕ ℓ = c j − w j Λ ℓ , j / 2 \partial_{v_j}\phi_\ell=c_j-w_j\Lambda_{\ell,j}/2 ∂ v j ϕ ℓ = c j − w j Λ ℓ , j /2 Lemma 3.1 (c), where c j = 1 / 2 c_j=1/2 c j = 1/2 j = K j=K j = K c j = 0 c_j=0 c j = 0 T i = E ^ [ ( ϕ i ′ ( X i ) ) 2 ] T_i=\widehat{\mathbb{E}}[(\phi'_i(X_i))^2] T i = E [( ϕ i ′ ( X i ) ) 2 ]
J i j = − w j E ^ [ ϕ i ′ ( X i ) ∂ x Λ i , j ( X i ) ] − w j 2 ∑ ℓ < i ζ ℓ E ^ [ ( ϕ i ′ ( X i ) ) 2 ( Λ ℓ + 1 , j ( X ℓ + 1 ) − Λ ℓ , j ( X ℓ ) ) ] . J_{ij}=-w_j\widehat{\mathbb{E}}\left[\phi'_i(X_i)\partial_x\Lambda_{i,j}(X_i)\right]-\frac{w_j}{2}\sum_{\ell<i}\zeta_\ell\widehat{\mathbb{E}}\left[(\phi'_i(X_i))^2\left(\Lambda_{\ell+1,j}(X_{\ell+1})-\Lambda_{\ell,j}(X_\ell)\right)\right]. J ij = − w j E [ ϕ i ′ ( X i ) ∂ x Λ i , j ( X i ) ] − 2 w j ℓ < i ∑ ζ ℓ E [ ( ϕ i ′ ( X i ) ) 2 ( Λ ℓ + 1 , j ( X ℓ + 1 ) − Λ ℓ , j ( X ℓ ) ) ] . where the constants c j c_j c j E ^ [ ( ϕ i ′ ( X i ) ) 2 ∣ X ℓ + 1 ] = Λ ℓ + 1 , i ( X ℓ + 1 ) \widehat{\mathbb{E}}[(\phi'_i(X_i))^2\mid X_{\ell+1}]=\Lambda_{\ell+1,i}(X_{\ell+1}) E [( ϕ i ′ ( X i ) ) 2 ∣ X ℓ + 1 ] = Λ ℓ + 1 , i ( X ℓ + 1 ) Λ ℓ , j ( X ℓ ) = E ^ [ Λ ℓ + 1 , j ( X ℓ + 1 ) ∣ X ℓ ] \Lambda_{\ell,j}(X_\ell)=\widehat{\mathbb{E}}[\Lambda_{\ell+1,j}(X_{\ell+1})\mid X_\ell] Λ ℓ , j ( X ℓ ) = E [ Λ ℓ + 1 , j ( X ℓ + 1 ) ∣ X ℓ ] Λ ℓ + 1 , i ( X ℓ + 1 ) \Lambda_{\ell+1,i}(X_{\ell+1}) Λ ℓ + 1 , i ( X ℓ + 1 ) Λ ℓ + 1 , j ( X ℓ + 1 ) \Lambda_{\ell+1,j}(X_{\ell+1}) Λ ℓ + 1 , j ( X ℓ + 1 ) X ℓ X_\ell X ℓ ∣ X ℓ + 1 ∣ \lvert X_{\ell+1}\rvert ∣ X ℓ + 1 ∣ F 1 , F 2 F_1,F_2 F 1 , F 2 Y ′ Y' Y ′ Y Y Y 2 Cov ( F 1 ( Y ) , F 2 ( Y ) ) = E [ ( F 1 ( Y ) − F 1 ( Y ′ ) ) ( F 2 ( Y ) − F 2 ( Y ′ ) ) ] ≥ 0 2\operatorname{Cov}(F_1(Y),F_2(Y))=\mathbb{E}[(F_1(Y)-F_1(Y'))(F_2(Y)-F_2(Y'))]\ge0 2 Cov ( F 1 ( Y ) , F 2 ( Y )) = E [( F 1 ( Y ) − F 1 ( Y ′ )) ( F 2 ( Y ) − F 2 ( Y ′ ))] ≥ 0 ϕ i ′ ( x ) \phi'_i(x) ϕ i ′ ( x ) ∂ x Λ i , j ( x ) \partial_x\Lambda_{i,j}(x) ∂ x Λ i , j ( x ) x x x J i j ≤ 0 J_{ij}\le0 J ij ≤ 0 Ψ h \Psi_h Ψ h v v v (30) implies that w i J i j = w j J j i w_iJ_{ij}=w_jJ_{ji} w i J ij = w j J j i J i j ≤ 0 J_{ij}\le0 J ij ≤ 0 i ≠ j i\ne j i = j Δ 1 , … , Δ K \Delta_1,\ldots,\Delta_K Δ 1 , … , Δ K v 0 v_0 v 0 (21) , we find
∑ j = 0 K J i j = d d ϵ ∣ ϵ = 0 E Λ 0 , i ( h + v 0 + ϵ Z ) = 1 2 T i ′ ′ ( h ) . (38) \sum_{j=0}^{K} J_{ij}=\left.\frac{\mathrm{d}}{\mathrm{d}\epsilon}\right|_{\epsilon=0}\mathbb{E}\,\Lambda_{0,i}\left(h+\sqrt{v_0+\epsilon}Z\right)=\frac{1}{2}T_i''(h). \tag*{(38)} j = 0 ∑ K J ij = d ϵ d ϵ = 0 E Λ 0 , i ( h + v 0 + ϵ Z ) = 2 1 T i ′′ ( h ) . (38) By Lemmas 3.6 and 3.7 , with y = λ h 2 y=\lambda h^2 y = λ h 2 ν i \nu_i ν i (37) ,
T i ( h ) + v i T i ′ ′ ( h ) = 1 − ∫ e − λ h 2 ν i ( d λ ) − v i ∫ λ ( 4 y − 2 ) e − y ν i ( d λ ) ≥ T i ( 0 ) − 4 e − 3 / 2 v i ∫ λ ν i ( d λ ) ≥ ( 1 − 4 e − 3 / 2 ) T i ( 0 ) ≥ 0. \begin{aligned}
T_i(h)+v_iT_i''(h)=1-\int e^{-\lambda h^2}\nu_i(\mathrm{d}\lambda)-v_i\int\lambda(4y-2)e^{-y}\nu_i(\mathrm{d}\lambda) \\
&\ge T_i(0)-4e^{-3/2}v_i\int\lambda\nu_i(\mathrm{d}\lambda)\ge\left(1-4e^{-3/2}\right)T_i(0)\ge0.
\end{aligned} T i ( h ) + v i T i ′′ ( h ) = 1 − ∫ e − λ h 2 ν i ( d λ ) − v i ∫ λ ( 4 y − 2 ) e − y ν i ( d λ ) ≥ T i ( 0 ) − 4 e − 3/2 v i ∫ λ ν i ( d λ ) ≥ ( 1 − 4 e − 3/2 ) T i ( 0 ) ≥ 0. Here we used that ν i \nu_i ν i 1 − T i ( 0 ) 1-T_i(0) 1 − T i ( 0 ) sup y ≥ 0 ( 4 y − 2 ) e − y = 4 e − 3 / 2 < 1 \sup_{y\ge0}(4y-2)e^{-y}=4e^{-3/2}<1 sup y ≥ 0 ( 4 y − 2 ) e − y = 4 e − 3/2 < 1 T i ′ ′ ( 0 ) = 2 ∫ λ ν i ( d λ ) T_i''(0)=2\int\lambda\nu_i(\mathrm{d}\lambda) T i ′′ ( 0 ) = 2 ∫ λ ν i ( d λ ) Lemma 3.7 reads v i ∫ λ ν i ( d λ ) ≤ T i ( 0 ) v_i\int\lambda\nu_i(\mathrm{d}\lambda)\le T_i(0) v i ∫ λ ν i ( d λ ) ≤ T i ( 0 )
Let M be the Hessian of s ↦ Ψ h ( s 2 ) s\mapsto\Psi_h(s^2) s ↦ Ψ h ( s 2 ) s s s (30) , ∂ s i Ψ h ( s 2 ) = w i s i T i \partial_{s_i}\Psi_h(s^2)=w_is_iT_i ∂ s i Ψ h ( s 2 ) = w i s i T i
M i j = δ i j w i T i + 2 w i s i s j J i j . M_{ij}=\delta_{ij}w_iT_i+2w_is_is_jJ_{ij}. M ij = δ ij w i T i + 2 w i s i s j J ij . The off-diagonal entries are nonpositive. Let ϱ i = 1 / s i > 0 \varrho_i=1/s_i>0 ϱ i = 1/ s i > 0 (38) ,
( M ϱ ) i = w i T i s i + 2 w i s i ∑ j J i j = w i s i ( T i + v i T i ′ ′ ( h ) ) ≥ 0. (M\varrho)_i=\frac{w_iT_i}{s_i}+2w_is_i\sum_jJ_{ij}=\frac{w_i}{s_i}\left(T_i+v_iT_i''(h)\right)\ge0. ( M ϱ ) i = s i w i T i + 2 w i s i j ∑ J ij = s i w i ( T i + v i T i ′′ ( h ) ) ≥ 0. For y ∈ R K + 1 y\in\mathbb{R}^{K+1} y ∈ R K + 1 ξ i = ϱ i y i \xi_i=\varrho_iy_i ξ i = ϱ i y i
ξ T M ξ = ∑ i < j ( − M i j ) ϱ i ϱ j ( y i − y j ) 2 + ∑ i ϱ i ( M ϱ ) i y i 2 ≥ 0 , \xi^{\mathsf{T}}M\xi=\sum_{i<j}(-M_{ij})\varrho_i\varrho_j(y_i-y_j)^2+\sum_i\varrho_i(M\varrho)_iy_i^2\ge0, ξ T M ξ = i < j ∑ ( − M ij ) ϱ i ϱ j ( y i − y j ) 2 + i ∑ ϱ i ( M ϱ ) i y i 2 ≥ 0 , which shows that M is positive semidefinite.
For h = 0 h=0 h = 0 [37 Proposition 2.2] and Ho [83 Theorem 1.8] proved that ψ 0 \psi_0 ψ 0 C K C_K C K ψ 0 \psi_0 ψ 0 h = 0 h=0 h = 0 ψ 0 \psi_0 ψ 0 [8 Theorem 1] ; see [83 Section 3] . Convexity on C K C_K C K ∣ h ∣ |h| ∣ h ∣ K = 0 K=0 K = 0 ψ h ( s ) = s − E log 2 cosh ( h + 2 s Z ) \psi_h(s)=s-\mathbb{E}\log2\cosh(h+\sqrt{2s}Z) ψ h ( s ) = s − E log 2 cosh ( h + 2 s Z ) [106 Section 6.1] , with the uniform measure on { − 1 , 1 } \{-1,1\} { − 1 , 1 } tanh h \tanh h tanh h ψ h ′ ′ ( 0 ) = 2 ( 1 − tanh 2 h ) ( 1 − 3 tanh 2 h ) \psi_h''(0)=2(1-\tanh^2h)(1-3\tanh^2h) ψ h ′′ ( 0 ) = 2 ( 1 − tanh 2 h ) ( 1 − 3 tanh 2 h ) tanh 2 h > 1 / 3 \tanh^2h>1/3 tanh 2 h > 1/3 [37 Section 1] . For spherical spins, the analogous initial condition is convex in the paths in every field [34 Lemma 2.9] . The proof in [37 Section 2] has the same structure as the proof of Proposition 3.8: the Hessian of ψ 0 \psi_0 ψ 0 [112 Theorem 2] , and nonnegative row sums. For h ≠ 0 h\ne0 h = 0 Ψ h \Psi_h Ψ h T i ′ ′ ( h ) T_i''(h) T i ′′ ( h ) (38) , and they may be negative. In the proof of Proposition 3.8, the change of variables v = s 2 v=s^2 v = s 2 Lemma 3.7 compensate for this.
The Hopf–Lax formula and the Parisi functional
Fix β > 0 \beta> 0 β > 0 h ∈ R h \in\mathbb{R} h ∈ R t > 0 t > 0 t > 0 y ∈ C K y \in C_K y ∈ C K
U ( t , y ) = sup y ′ ∈ C K { ψ h ( y ′ ) − ∥ y ′ − y ∥ 2 , w 2 β 2 t } , U ( 0 , y ) = ψ h ( y ) . (39) U(t,y)=\sup_{y'\in C_K}\left\{\psi_h(y')-\frac{\lVert y'-y\rVert_{2,w}^{2}}{\beta^{2}t}\right\},\qquad U(0,y)=\psi_h(y). \tag*{(39)} U ( t , y ) = y ′ ∈ C K sup { ψ h ( y ′ ) − β 2 t ∥ y ′ − y ∥ 2 , w 2 } , U ( 0 , y ) = ψ h ( y ) . (39) This is the Hopf–Lax formula for the equation ∂ t U − ( β 2 / 4 ) ∥ D U ∥ 2 , w 2 = 0 \partial_t U-(\beta^2/4)\lVert DU\rVert_{2,w}^{2}=0 ∂ t U − ( β 2 /4 ) ∥ D U ∥ 2 , w 2 = 0 ψ h \psi_h ψ h [56 Theorem 3.8] for the corresponding statement on R K + 1 \mathbb{R}^{K+1} R K + 1 [36 Proposition 6.2] for cones. We use only the elementary properties in the next lemma.
Lemma 3.9. The following hold.
(a) For t > 0 t>0 t > 0 y ∈ C K y\in C_K y ∈ C K y ′ y' y ′
P = 2 ( y ′ − y ) β 2 t = D ψ h ( y ′ ) ∈ C K ∩ [ 0 , 1 ] K + 1 . (40) P=\frac{2(y'-y)}{\beta^{2}t}=D\psi_h(y')\in C_K\cap[0,1]^{K+1}. \tag*{(40)} P = β 2 t 2 ( y ′ − y ) = D ψ h ( y ′ ) ∈ C K ∩ [ 0 , 1 ] K + 1 . (40) (b) For t ≥ 0 t\geq0 t ≥ 0 y , y ~ ∈ C K y,\widetilde y\in C_K y , y ∈ C K ∣ U ( t , y ) − U ( t , y ~ ) ∣ ≤ ∥ y − y ~ ∥ 1 , w \lvert U(t,y)-U(t,\widetilde y)\rvert\leq\lVert y-\widetilde y\rVert_{1,w} ∣ U ( t , y ) − U ( t , y )∣ ≤ ∥ y − y ∥ 1 , w
(c) For 0 ≤ t ≤ t ′ 0\leq t\leq t' 0 ≤ t ≤ t ′ y ∈ C K y\in C_K y ∈ C K 0 ≤ U ( t ′ , y ) − U ( t , y ) ≤ β 2 ( t ′ − t ) / 2 0\leq U(t',y)-U(t,y)\leq\beta^2(t'-t)/2 0 ≤ U ( t ′ , y ) − U ( t , y ) ≤ β 2 ( t ′ − t ) /2
In particular, U U U [ 0 , ∞ ) × C K [0,\infty)\times C_K [ 0 , ∞ ) × C K
Proof. (a) By Lemma 3.2, ψ h ( y ′ ) ≤ ψ h ( y ) + ∥ y ′ − y ∥ 2 , w \psi_h(y')\leq\psi_h(y)+\lVert y'-y\rVert_{2,w} ψ h ( y ′ ) ≤ ψ h ( y ) + ∥ y ′ − y ∥ 2 , w C K C_K C K − ∞ -\infty − ∞ ∣ y ′ ∣ → ∞ \lvert y'\rvert\to\infty ∣ y ′ ∣ → ∞ y ′ y' y ′ P = D ψ h ( y ′ ) P=D\psi_h(y') P = D ψ h ( y ′ ) y ′ ′ ∈ C K y''\in C_K y ′′ ∈ C K y ′ y' y ′ y ′ ′ y'' y ′′ C K C_K C K y ′ y' y ′ y ′ ′ − y ′ y''-y' y ′′ − y ′
⟨ P − 2 ( y ′ − y ) β 2 t , y ′ ′ − y ′ ⟩ w ≤ 0. \left\langle P-\frac{2(y'-y)}{\beta^{2}t},y''-y'\right\rangle_w\leq0. ⟨ P − β 2 t 2 ( y ′ − y ) , y ′′ − y ′ ⟩ w ≤ 0. Since P ∈ C K P\in C_K P ∈ C K y ′ ′ = y + β 2 t P / 2 ∈ C K y''=y+\beta^{2}tP/2\in C_K y ′′ = y + β 2 tP /2 ∈ C K ( β 2 t / 2 ) ∥ P − 2 ( y ′ − y ) / ( β 2 t ) ∥ 2 , w 2 (\beta^{2}t/2)\lVert P-2(y'-y)/(\beta^{2}t)\rVert_{2,w}^{2} ( β 2 t /2 ) ∥ P − 2 ( y ′ − y ) / ( β 2 t ) ∥ 2 , w 2
(b) For t = 0 t=0 t = 0 t > 0 t>0 t > 0 y ′ y' y ′ y y y y ′ = y + β 2 t P / 2 y'=y+\beta^{2}tP/2 y ′ = y + β 2 tP /2 P ∈ C K P\in C_K P ∈ C K y ~ + β 2 t P / 2 \widetilde y+\beta^{2}tP/2 y + β 2 tP /2 C K C_K C K
U ( t , y ~ ) ≥ ψ h ( y ~ + β 2 t P 2 ) − β 2 t 4 ∥ P ∥ 2 , w 2 ≥ ψ h ( y ′ ) − ∥ y − y ~ ∥ 1 , w − β 2 t 4 ∥ P ∥ 2 , w 2 = U ( t , y ) − ∥ y − y ~ ∥ 1 , w . \begin{aligned}
U(t,\widetilde y)\geq\psi_h\left(\widetilde y+\frac{\beta^{2}tP}{2}\right)-\frac{\beta^{2}t}{4}\lVert P\rVert_{2,w}^{2} \\
&\geq\psi_h(y')-\lVert y-\widetilde y\rVert_{1,w}-\frac{\beta^{2}t}{4}\lVert P\rVert_{2,w}^{2}=U(t,y)-\lVert y-\widetilde y\rVert_{1,w}.
\end{aligned} U ( t , y ) ≥ ψ h ( y + 2 β 2 tP ) − 4 β 2 t ∥ P ∥ 2 , w 2 ≥ ψ h ( y ′ ) − ∥ y − y ∥ 1 , w − 4 β 2 t ∥ P ∥ 2 , w 2 = U ( t , y ) − ∥ y − y ∥ 1 , w . Exchanging y y y y ~ \widetilde y y t > 0 t>0 t > 0 y ′ y' y ′ t t t U ( t ′ , y ) ≥ ψ h ( y ′ ) − ∥ y ′ − y ∥ 2 , w 2 / ( β 2 t ′ ) ≥ U ( t , y ) U(t',y)\geq\psi_h(y')-\|y'-y\|_{2,w}^2/(\beta^2t')\geq U(t,y) U ( t ′ , y ) ≥ ψ h ( y ′ ) − ∥ y ′ − y ∥ 2 , w 2 / ( β 2 t ′ ) ≥ U ( t , y ) t = 0 t=0 t = 0 y ′ = y y'=y y ′ = y y ′ ′ = y + β 2 t ′ P ′ ′ / 2 y''=y+\beta^2t'P''/2 y ′′ = y + β 2 t ′ P ′′ /2 t ′ t' t ′ P ′ ′ ∈ C K ∩ [ 0 , 1 ] K + 1 P''\in C_K\cap[0,1]^{K+1} P ′′ ∈ C K ∩ [ 0 , 1 ] K + 1 y + β 2 t ′ P ′ ′ / 2 y+\beta^2t'P''/2 y + β 2 t ′ P ′′ /2 C K C_K C K
U ( t ′ , y ) − U ( t , y ) ≤ ψ h ( y + β 2 t ′ 2 P ′ ′ ) − ψ h ( y + β 2 t 2 P ′ ′ ) − β 2 ( t ′ − t ) 4 ∥ P ′ ′ ∥ 2 , w 2 ≤ β 2 ( t ′ − t ) 2 ∥ P ′ ′ ∥ 1 , w , U(t',y)-U(t,y)\leq\psi_h\left(y+\frac{\beta^2t'}{2}P''\right)-\psi_h\left(y+\frac{\beta^2t}{2}P''\right)-\frac{\beta^2(t'-t)}{4}\|P''\|_{2,w}^2
\leq\frac{\beta^2(t'-t)}{2}\|P''\|_{1,w}, U ( t ′ , y ) − U ( t , y ) ≤ ψ h ( y + 2 β 2 t ′ P ′′ ) − ψ h ( y + 2 β 2 t P ′′ ) − 4 β 2 ( t ′ − t ) ∥ P ′′ ∥ 2 , w 2 ≤ 2 β 2 ( t ′ − t ) ∥ P ′′ ∥ 1 , w , which is at most β 2 ( t ′ − t ) / 2 \beta^2(t'-t)/2 β 2 ( t ′ − t ) /2
The next lemma compares U ( 1 , 0 ) U(1,0) U ( 1 , 0 ) (41) is a finite-dimensional form of the representation of the Parisi formula by a Hopf–Lax formula, due to Mourrat [107 ] for h = 0 h=0 h = 0 [56 Theorem 6.7] . We use the Parisi equation (9) , the Parisi functional P β , h \mathcal{P}_{\beta,h} P β , h (10) for P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) μ r = ∑ j = 0 K w j δ r j \mu_r=\sum_{j=0}^K w_j\delta_{r_j} μ r = ∑ j = 0 K w j δ r j r ∈ C K ∩ [ 0 , 1 ] K + 1 r\in C_K\cap[0,1]^{K+1} r ∈ C K ∩ [ 0 , 1 ] K + 1
Lemma 3.10. Let β > 0 \beta>0 β > 0 h ∈ R h\in\mathbb{R} h ∈ R r ∈ C K ∩ [ 0 , 1 ] K + 1 r\in C_K\cap[0,1]^{K+1} r ∈ C K ∩ [ 0 , 1 ] K + 1
ψ h ( β 2 2 r ) − β 2 4 ∥ r ∥ 2 , w 2 = β 2 4 − P β , h ( μ r ) , (41) \psi_h\left(\frac{\beta^2}{2}r\right)-\frac{\beta^2}{4}\|r\|_{2,w}^2=\frac{\beta^2}{4}-\mathcal{P}_{\beta,h}(\mu_r), \tag*{(41)} ψ h ( 2 β 2 r ) − 4 β 2 ∥ r ∥ 2 , w 2 = 4 β 2 − P β , h ( μ r ) , (41) and U ( 1 , 0 ) ≥ β 2 / 4 − P β , h ( μ r ) U(1,0)\geq\beta^2/4-\mathcal{P}_{\beta,h}(\mu_r) U ( 1 , 0 ) ≥ β 2 /4 − P β , h ( μ r ) μ ↦ P β , h ( μ ) \mu\mapsto\mathcal{P}_{\beta,h}(\mu) μ ↦ P β , h ( μ ) [ 0 , 1 ] [0,1] [ 0 , 1 ]
Proof. We first prove (41) . Let Φ = Φ μ r \Phi=\Phi_{\mu_r} Φ = Φ μ r (9) . Since μ r \mu_r μ r Φ \Phi Φ [7 ] : on [ r K , 1 ] [r_K,1] [ r K , 1 ] μ r ( [ 0 , t ] ) = 1 \mu_r([0,t])=1 μ r ([ 0 , t ]) = 1 Φ ( t , x ) = log 2 cosh x + β 2 ( 1 − t ) / 2 \Phi(t,x)=\log2\cosh x+\beta^2(1-t)/2 Φ ( t , x ) = log 2 cosh x + β 2 ( 1 − t ) /2 [ r j , r j + 1 ) [r_j,r_{j+1}) [ r j , r j + 1 ) μ r ( [ 0 , t ] ) = ζ j \mu_r([0,t])=\zeta_j μ r ([ 0 , t ]) = ζ j Φ ( r j , x ) = ζ j − 1 log E exp ( ζ j Φ ( r j + 1 , x + β r j + 1 − r j Z ) ) \Phi(r_j,x)=\zeta_j^{-1}\log\mathbb{E}\exp(\zeta_j\Phi(r_{j+1},x+\beta\sqrt{r_{j+1}-r_j}Z)) Φ ( r j , x ) = ζ j − 1 log E exp ( ζ j Φ ( r j + 1 , x + β r j + 1 − r j Z )) Φ ( 0 , x ) = E Φ ( r 0 , x + β r 0 Z ) \Phi(0,x)=\mathbb{E}\Phi(r_0,x+\beta\sqrt{r_0}Z) Φ ( 0 , x ) = E Φ ( r 0 , x + β r 0 Z ) ϕ j \phi_j ϕ j v = β 2 r v=\beta^2r v = β 2 r Δ j + 1 = β 2 ( r j + 1 − r j ) \Delta_{j+1}=\beta^2(r_{j+1}-r_j) Δ j + 1 = β 2 ( r j + 1 − r j ) Φ ( r j , ⋅ ) = ϕ j + β 2 ( 1 − r K ) / 2 \Phi(r_j,\cdot)=\phi_j+\beta^2(1-r_K)/2 Φ ( r j , ⋅ ) = ϕ j + β 2 ( 1 − r K ) /2 j j j Φ ( 0 , h ) = E ϕ 0 ( h + β r 0 Z ) + β 2 ( 1 − r K ) / 2 \Phi(0,h)=\mathbb{E}\phi_0(h+\beta\sqrt{r_0}Z)+\beta^2(1-r_K)/2 Φ ( 0 , h ) = E ϕ 0 ( h + β r 0 Z ) + β 2 ( 1 − r K ) /2 X ∼ μ r X\sim\mu_r X ∼ μ r
∫ 0 1 t μ r ( [ 0 , t ] ) d t = E ∫ X 1 t d t = 1 − ∥ r ∥ 2 , w 2 2 . \int_0^1t\mu_r([0,t])\,dt=\mathbb{E}\int_X^1t\,dt=\frac{1-\|r\|_{2,w}^2}{2}. ∫ 0 1 t μ r ([ 0 , t ]) d t = E ∫ X 1 t d t = 2 1 − ∥ r ∥ 2 , w 2 . We conclude that
P β , h ( μ r ) = E ϕ 0 ( h + β r 0 Z ) + β 2 4 − β 2 2 r K + β 2 4 ∥ r ∥ 2 , w 2 . \mathcal{P}_{\beta,h}(\mu_r)=\mathbb{E}\phi_0(h+\beta\sqrt{r_0}Z)+\frac{\beta^2}{4}-\frac{\beta^2}{2}r_K+\frac{\beta^2}{4}\|r\|_{2,w}^2. P β , h ( μ r ) = E ϕ 0 ( h + β r 0 Z ) + 4 β 2 − 2 β 2 r K + 4 β 2 ∥ r ∥ 2 , w 2 . Since ψ h ( β 2 r / 2 ) = Ψ h ( β 2 r ) = β 2 r K / 2 − E ϕ 0 ( h + β r 0 Z ) \psi_h(\beta^2r/2)=\Psi_h(\beta^2r)=\beta^2r_K/2-\mathbb{E}\phi_0(h+\beta\sqrt{r_0}Z) ψ h ( β 2 r /2 ) = Ψ h ( β 2 r ) = β 2 r K /2 − E ϕ 0 ( h + β r 0 Z ) (41) . Taking y ′ = β 2 r / 2 y'=\beta^2r/2 y ′ = β 2 r /2 (39) with t = 1 t=1 t = 1 y = 0 y=0 y = 0 ∥ y ′ ∥ 2 , w 2 / β 2 = β 2 ∥ r ∥ 2 , w 2 / 4 \|y'\|_{2,w}^2/\beta^2=\beta^2\|r\|_{2,w}^2/4 ∥ y ′ ∥ 2 , w 2 / β 2 = β 2 ∥ r ∥ 2 , w 2 /4 U ( 1 , 0 ) U(1,0) U ( 1 , 0 )
Finally, Φ μ ( 0 , h ) \Phi_\mu(0,h) Φ μ ( 0 , h ) μ \mu μ
∫ 0 1 t μ ( [ 0 , t ] ) d t = ( 1 − ∫ x 2 μ ( d x ) ) / 2. \int_0^1t\mu([0,t])\,dt=\left(1-\int x^2\mu(dx)\right)/2. ∫ 0 1 t μ ([ 0 , t ]) d t = ( 1 − ∫ x 2 μ ( d x ) ) /2. This proves the continuity of P β , h \mathcal{P}_{\beta,h} P β , h
The upper bound on regular graphs
The goal of this section is to prove the following proposition, which states that the pressure of the Edwards–Anderson model on any D D D
Proposition 4.1. For every β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R
lim sup D → ∞ sup G ∈ G D p G ( β , h ) ≤ P ( β , h ) . \limsup_{D\to\infty}\sup_{G\in\mathcal{G}_D} p_G(\beta,h) \le\mathcal{P}(\beta,h). D → ∞ lim sup G ∈ G D sup p G ( β , h ) ≤ P ( β , h ) . The supremum is over all finite simple D D D G G G [106 , 108 ] , who proved upper bounds for mean-field models with nonconvex interactions by showing that an enriched free energy is an approximate viscosity supersolution of a Hamilton–Jacobi equation, with Ghirlanda–Guerra identities and synchronization enforced at contact points. Fix a finite cascade as in Section 3. On the cone of V V V t t t β 2 t / 4 − p G ( β t , h ) \beta^2t/4-p_G(\beta\sqrt{t},h) β 2 t /4 − p G ( β t , h ) § 4.3 . Its time derivative differs from a quadratic Hamiltonian evaluated at its gradient by β 2 ( 4 n ) − 1 ∑ j w j tr ( S C j ) \beta^2(4n)^{-1}\sum_j w_j\operatorname{tr}(SC_j) β 2 ( 4 n ) − 1 ∑ j w j tr ( S C j ) C j C_j C j σ x 2 σ x 2 \sigma_x^2\sigma_x^2 σ x 2 σ x 2 S S S η > 0 \eta>0 η > 0 ∣ tr ( S C j ) ∣ ≤ η tr C j + ( 4 η ) − 1 tr ( S 2 C j ) |\operatorname{tr}(SC_j)|\le\eta\operatorname{tr}C_j+(4\eta)^{-1}\operatorname{tr}(S^2C_j) ∣ tr ( S C j ) ∣ ≤ η tr C j + ( 4 η ) − 1 tr ( S 2 C j ) tr C j ≤ n \operatorname{tr}C_j\le n tr C j ≤ n tr ( S 2 C j ) \operatorname{tr}(S^2C_j) tr ( S 2 C j ) O ( n / D ) O(n/D) O ( n / D ) S S S 1 / n 1/n 1/ n § 4.6 , and Lemma 3.10 compares the value of the subsolution with the Parisi formula.
Section 4.1 introduces the Hamiltonians on the cone and proves the comparison principle. Section 4.2 constructs the subsolution. Section 4.3 defines the enriched free energy and records its properties. Section 4.4 states Mourrat’s bound on conditional variances, Section 4.5 introduces the Gaussian perturbations, and Section 4.6 bounds the drift term. Section 4.7 completes the proof of Proposition 4.1.
Throughout this section we fix D ≥ 1 D \ge1 D ≥ 1 G = ( V , E ) ∈ G D G = (V,E) \in\mathcal{G}_D G = ( V , E ) ∈ G D n = ∣ V ∣ n = |V| n = ∣ V ∣ S = A G / D S = A_G/D S = A G / D S 1 = 1 S\mathbf{1} = \mathbf{1} S 1 = 1 ∥ S ∥ ≤ 1 \|S\| \le1 ∥ S ∥ ≤ 1 tr S 2 = n / D \operatorname{tr} S^2 = n/D tr S 2 = n / D D D D G G G n n n
Z G ( β , h ) = ∑ σ ∈ { − 1 , 1 } V exp ( β H G ( σ ) + h ∑ x σ x ) , p G ( β , h ) = 1 n E log Z G ( β , h ) . Z_G(\beta,h) = \sum_{\sigma\in\{-1,1\}^V} \exp\left(\beta H_G(\sigma) + h\sum_x \sigma_x\right), \qquad p_G(\beta,h) = \frac{1}{n}\mathbb{E}\log Z_G(\beta,h). Z G ( β , h ) = σ ∈ { − 1 , 1 } V ∑ exp ( β H G ( σ ) + h x ∑ σ x ) , p G ( β , h ) = n 1 E log Z G ( β , h ) . Hamiltonians on the cone and comparison
Let V K = V × { 0 , … , K } V_K = V \times\{0,\ldots,K\} V K = V × { 0 , … , K } p ∈ R V K p \in\mathbb{R}^{V_K} p ∈ R V K p x = ( p x j ) j ≤ K p_x = (p_{xj})_{j \le K} p x = ( p x j ) j ≤ K p j = ( p x j ) x ∈ V ∈ R V p_j = (p_{xj})_{x \in V} \in\mathbb{R}^V p j = ( p x j ) x ∈ V ∈ R V
⟨ p , q ⟩ V = 1 n ∑ x ∈ V ⟨ p x , q x ⟩ w , ∥ p ∥ 1 = 1 n ∑ x ∈ V ∥ p x ∥ 1 , w , ∥ p ∥ 2 = ⟨ p , p ⟩ V 1 / 2 . \langle p,q\rangle_V = \frac{1}{n}\sum_{x\in V}\langle p_x,q_x\rangle_w,\qquad\|p\|_1 = \frac{1}{n}\sum_{x\in V}\|p_x\|_{1,w},\qquad\|p\|_2 = \langle p,p\rangle_V^{1/2}. ⟨ p , q ⟩ V = n 1 x ∈ V ∑ ⟨ p x , q x ⟩ w , ∥ p ∥ 1 = n 1 x ∈ V ∑ ∥ p x ∥ 1 , w , ∥ p ∥ 2 = ⟨ p , p ⟩ V 1/2 . so that ∥ p ∥ 1 ≤ ∥ p ∥ 2 \|p\|_1 \le\|p\|_2 ∥ p ∥ 1 ≤ ∥ p ∥ 2 w j / n w_j/n w j / n C K V = { q ∈ R V K : q x ∈ C K for every x } \mathcal{C}_K^V = \{q \in\mathbb{R}^{V_K}: q_x \in\mathcal{C}_K\text{ for every }x\} C K V = { q ∈ R V K : q x ∈ C K for every x } ⟨ ⋅ , ⋅ ⟩ V \langle\cdot,\cdot\rangle_V ⟨ ⋅ , ⋅ ⟩ V ( C K ∗ ) V (\mathcal{C}_K^*)^V ( C K ∗ ) V p ⪰ ∗ p ′ p \succeq_* p' p ⪰ ∗ p ′ p x ⪰ ∗ p x ′ p_x \succeq_* p'_x p x ⪰ ∗ p x ′ x x x u u u q ∈ R V K q \in\mathbb{R}^{V_K} q ∈ R V K D q u D_qu D q u ⟨ ⋅ , ⋅ ⟩ V \langle\cdot,\cdot\rangle_V ⟨ ⋅ , ⋅ ⟩ V ( D q u ) x j = n w j − 1 ∂ u / ∂ q x j (D_qu)_{xj} = nw_j^{-1}\partial u/\partial q_{xj} ( D q u ) x j = n w j − 1 ∂ u / ∂ q x j C K V \mathcal{C}_K^V C K V
Hamiltonians
For p ∈ R V K p \in\mathbb{R}^{V_K} p ∈ R V K
H S ( p ) = β 2 4 n ∑ j = 0 K w j p j ⋅ S p j (42) H_S(p) = \frac{\beta^2}{4n}\sum_{j=0}^{K} w_j\,p_j\cdot Sp_j \tag*{(42)} H S ( p ) = 4 n β 2 j = 0 ∑ K w j p j ⋅ S p j (42) and
H e x t ( p ) = inf { H S ( A ) : A ∈ C K V , A ⪰ ∗ p } . (43) H_{\mathrm{ext}}(p) = \inf\{H_S(A): A \in\mathcal{C}_K^V,\ A \succeq_* p\}. \tag*{(43)} H ext ( p ) = inf { H S ( A ) : A ∈ C K V , A ⪰ ∗ p } . (43) This is the monotone modification constructed in the proof of [36 Lemma 2.4] , applied to H S H_S H S z ∈ R K + 1 z \in\mathbb{R}^{K+1} z ∈ R K + 1 0 ≤ ℓ ≤ K + 1 0 \le\ell\le K+1 0 ≤ ℓ ≤ K + 1 τ ℓ ( z ) = ∑ j ≥ ℓ w j z j \tau_\ell(z) = \sum_{j\ge\ell}w_jz_j τ ℓ ( z ) = ∑ j ≥ ℓ w j z j τ ‾ ℓ ( z ) = max ℓ ≤ ℓ ′ ≤ K + 1 τ ℓ ′ ( z ) \overline{\tau}_\ell(z) = \max_{\ell\le\ell'\le K+1}\tau_{\ell'}(z) τ ℓ ( z ) = max ℓ ≤ ℓ ′ ≤ K + 1 τ ℓ ′ ( z ) W ℓ = τ ℓ ( 1 ) W_\ell= \tau_\ell(\mathbf{1}) W ℓ = τ ℓ ( 1 ) τ K + 1 ( z ) = 0 \tau_{K+1}(z) = 0 τ K + 1 ( z ) = 0 τ ‾ ℓ ( z ) ≥ 0 \overline{\tau}_\ell(z) \ge0 τ ℓ ( z ) ≥ 0 z ⪰ ∗ z ′ z \succeq_* z' z ⪰ ∗ z ′ τ ℓ ( z ) ≥ τ ℓ ( z ′ ) \tau_\ell(z) \ge\tau_\ell(z') τ ℓ ( z ) ≥ τ ℓ ( z ′ ) ℓ \ell ℓ
Lemma 4.2. The following hold.
(a) If A , B ∈ C K V A, B \in C^V_K A , B ∈ C K V B ⪰ ∗ A B \succeq_* A B ⪰ ∗ A H S ( B ) ≥ H S ( A ) H_S(B) \ge H_S(A) H S ( B ) ≥ H S ( A )
(b) Let p ∈ R K V p \in\mathbb{R}^V_K p ∈ R K V x x x C x \mathcal{C}_x C x [ 0 , 1 ] [0,1] [ 0 , 1 ] ( W ℓ , τ ˉ ℓ ( p x ) ) (W_\ell,\bar{\tau}_\ell(p_x)) ( W ℓ , τ ˉ ℓ ( p x )) 0 ≤ ℓ ≤ K + 1 0 \le\ell\le K+1 0 ≤ ℓ ≤ K + 1 A x j = ( C x ( W j ) − C x ( W j + 1 ) ) / w j A_{xj} = (\mathcal{C}_x(W_j)-\mathcal{C}_x(W_{j+1}))/w_j A x j = ( C x ( W j ) − C x ( W j + 1 )) / w j A ∈ C K V A \in C^V_K A ∈ C K V A ⪰ ∗ p A \succeq_* p A ⪰ ∗ p B ∈ C K V B \in C^V_K B ∈ C K V B ⪰ ∗ p B \succeq_* p B ⪰ ∗ p B ⪰ ∗ A B \succeq_* A B ⪰ ∗ A H e x t ( p ) = H S ( A ) H_{\mathrm{ext}}(p)=H_S(A) H ext ( p ) = H S ( A ) 0 ≤ w K A x K ≤ ∥ p x ∥ 1 , w 0 \le w_K A_{xK} \le\lVert p_x\rVert_{1,w} 0 ≤ w K A x K ≤ ∥ p x ∥ 1 , w x x x p ∈ ( 0 , ∞ ) K V p \in(0,\infty)^V_K p ∈ ( 0 , ∞ ) K V A ∈ ( 0 , ∞ ) K V A \in(0,\infty)^V_K A ∈ ( 0 , ∞ ) K V j 0 ≤ K j_0 \le K j 0 ≤ K { i , … , i ′ − 1 } ∋ j 0 \{i,\ldots,i'-1\} \ni j_0 { i , … , i ′ − 1 } ∋ j 0 A x A_x A x p x p_x p x ∑ j = ℓ i ′ − 1 w j p x j ≤ A x j 0 ∑ j = ℓ i ′ − 1 w j \sum_{j=\ell}^{i'-1}w_jp_{xj} \le A_{xj_0}\sum_{j=\ell}^{i'-1}w_j ∑ j = ℓ i ′ − 1 w j p x j ≤ A x j 0 ∑ j = ℓ i ′ − 1 w j i ≤ ℓ < i ′ i \le\ell< i' i ≤ ℓ < i ′
(c) The function H e x t H_{\mathrm{ext}} H ext ⪰ ∗ \succeq_* ⪰ ∗ H S H_S H S C K V C^V_K C K V R > 0 R>0 R > 0 p , p ′ ∈ R K V p,p' \in\mathbb{R}^V_K p , p ′ ∈ R K V ∥ p ∥ 2 , ∥ p ′ ∥ 2 ≤ R \lVert p\rVert_2,\lVert p'\rVert_2 \le R ∥ p ∥ 2 , ∥ p ′ ∥ 2 ≤ R ∣ H e x t ( p ) − H e x t ( p ′ ) ∣ ≤ β 2 R w K − 1 ∥ p − p ′ ∥ 2 \lvert H_{\mathrm{ext}}(p)-H_{\mathrm{ext}}(p')\rvert\le\beta^2 Rw_K^{-1}\lVert p-p'\rVert_2 ∣ H ext ( p ) − H ext ( p ′ )∣ ≤ β 2 R w K − 1 ∥ p − p ′ ∥ 2
Proof. (a) For P ∈ C K V P \in C^V_K P ∈ C K V ( D H S ( P ) ) x j = ( β 2 / 2 ) ∑ y S x y P y j (D H_S(P))_{xj}=(\beta^2/2)\sum_y S_{xy}P_{yj} ( D H S ( P ) ) x j = ( β 2 /2 ) ∑ y S x y P y j j j j S S S P y j P_{yj} P y j j j j D H S ( P ) ∈ C K V D H_S(P) \in C^V_K D H S ( P ) ∈ C K V A A A B B B C K V C^V_K C K V B − A ∈ ( C K ∗ ) V B-A \in(C^*_K)^V B − A ∈ ( C K ∗ ) V H S ( B ) − H S ( A ) = ∫ 0 1 ⟨ D H S ( A + θ ( B − A ) ) , B − A ⟩ V d θ ≥ 0 H_S(B)-H_S(A)=\int_0^1\langle D H_S(A+\theta(B-A)),B-A\rangle_V\,d\theta\ge0 H S ( B ) − H S ( A ) = ∫ 0 1 ⟨ D H S ( A + θ ( B − A )) , B − A ⟩ V d θ ≥ 0
(b) Fix x x x a ℓ = τ ˉ ℓ ( p x ) a_\ell=\bar{\tau}_\ell(p_x) a ℓ = τ ˉ ℓ ( p x ) ℓ \ell ℓ 0 0 0 K + 1 K+1 K + 1 W ℓ W_\ell W ℓ 1 1 1 0 0 0 a ℓ a_\ell a ℓ a K + 1 = 0 a_{K+1}=0 a K + 1 = 0 C x \mathcal{C}_x C x C x ( 0 ) = a K + 1 = 0 \mathcal{C}_x(0)=a_{K+1}=0 C x ( 0 ) = a K + 1 = 0 C x ( 1 ) = a 0 \mathcal{C}_x(1)=a_0 C x ( 1 ) = a 0 C x \mathcal{C}_x C x a 0 = max ℓ a ℓ a_0=\max_\ell a_\ell a 0 = max ℓ a ℓ 1 1 1 C x \mathcal{C}_x C x W W W A x ∈ C K A_x \in C_K A x ∈ C K C x ( 0 ) = 0 \mathcal{C}_x(0)=0 C x ( 0 ) = 0 τ ℓ ( A x ) = C x ( W ℓ ) ≥ a ℓ ≥ τ ℓ ( p x ) \tau_\ell(A_x)=\mathcal{C}_x(W_\ell)\ge a_\ell\ge\tau_\ell(p_x) τ ℓ ( A x ) = C x ( W ℓ ) ≥ a ℓ ≥ τ ℓ ( p x ) A x ⪰ ∗ p x A_x\succeq_*p_x A x ⪰ ∗ p x w K A x K = C x ( W K ) ∈ [ 0 , a 0 ] w_KA_{xK}=\mathcal{C}_x(W_K)\in[0,a_0] w K A x K = C x ( W K ) ∈ [ 0 , a 0 ] a 0 ≤ max ℓ ∣ τ ℓ ( p x ) ∣ ≤ ∥ p x ∥ 1 , w a_0\le\max_\ell\lvert\tau_\ell(p_x)\rvert\le\lVert p_x\rVert_{1,w} a 0 ≤ max ℓ ∣ τ ℓ ( p x )∣ ≤ ∥ p x ∥ 1 , w B x ∈ C K B_x\in C_K B x ∈ C K B x ⪰ ∗ p x B_x\succeq_*p_x B x ⪰ ∗ p x Φ \Phi Φ ( W ℓ , τ ℓ ( B x ) ) (W_\ell,\tau_\ell(B_x)) ( W ℓ , τ ℓ ( B x )) B x j B_{xj} B x j j j j ℓ ≤ ℓ ′ \ell\le\ell' ℓ ≤ ℓ ′ Φ ( W ℓ ) ≥ Φ ( W ℓ ′ ) = τ ℓ ′ ( B x ) ≥ τ ℓ ′ ( p x ) \Phi(W_\ell)\ge\Phi(W_{\ell'})=\tau_{\ell'}(B_x)\ge\tau_{\ell'}(p_x) Φ ( W ℓ ) ≥ Φ ( W ℓ ′ ) = τ ℓ ′ ( B x ) ≥ τ ℓ ′ ( p x ) Φ ( W ℓ ) ≥ a ℓ \Phi(W_\ell)\ge a_\ell Φ ( W ℓ ) ≥ a ℓ Φ ≥ C x \Phi\ge\mathcal{C}_x Φ ≥ C x B x ⪰ ∗ A x B_x\succeq_*A_x B x ⪰ ∗ A x H S ( B ) ≥ H S ( A ) H_S(B)\ge H_S(A) H S ( B ) ≥ H S ( A ) B B B (43) , and A A A H e x t ( p ) = H S ( A ) H_{\mathrm{ext}}(p)=H_S(A) H ext ( p ) = H S ( A )
Let now p ∈ ( 0 , ∞ ) K V p\in(0,\infty)^V_K p ∈ ( 0 , ∞ ) K V τ ℓ ( p x ) \tau_\ell(p_x) τ ℓ ( p x ) ℓ \ell ℓ a ℓ = τ ℓ ( p x ) a_\ell=\tau_\ell(p_x) a ℓ = τ ℓ ( p x ) i ≤ j 0 < i ′ i\le j_0<i' i ≤ j 0 < i ′ W i ′ W_{i'} W i ′ W i W_i W i C x \mathcal{C}_x C x [ W i ′ , W i ] [W_{i'},W_i] [ W i ′ , W i ] C x \mathcal{C}_x C x A x j = A x j 0 A_{xj}=A_{xj_0} A x j = A x j 0 i ≤ j < i ′ i\le j<i' i ≤ j < i ′ C x ( W i ) = τ i ( p x ) \mathcal{C}_x(W_i)=\tau_i(p_x) C x ( W i ) = τ i ( p x ) C x ( W i ′ ) = τ i ′ ( p x ) \mathcal{C}_x(W_{i'})=\tau_{i'}(p_x) C x ( W i ′ ) = τ i ′ ( p x ) p x p_x p x { i , … , i ′ − 1 } \{i,\ldots,i'-1\} { i , … , i ′ − 1 } i ≤ ℓ < i ′ i\le\ell<i' i ≤ ℓ < i ′ τ ℓ ( p x ) ≤ C x ( W ℓ ) = τ i ′ ( p x ) + A x j 0 ( W ℓ − W i ′ ) \tau_\ell(p_x)\le\mathcal{C}_x(W_\ell)=\tau_{i'}(p_x)+A_{xj_0}(W_\ell-W_{i'}) τ ℓ ( p x ) ≤ C x ( W ℓ ) = τ i ′ ( p x ) + A x j 0 ( W ℓ − W i ′ )
(c) By (b), H e x t H_{\mathrm{ext}} H ext p ′ ⪰ ∗ p p'\succeq_*p p ′ ⪰ ∗ p A A A p ′ p' p ′ p p p H e x t ( p ) ≤ H e x t ( p ′ ) H_{\mathrm{ext}}(p)\le H_{\mathrm{ext}}(p') H ext ( p ) ≤ H ext ( p ′ ) p ∈ C K V p\in C^V_K p ∈ C K V p p p A A A A ⪰ ∗ p A\succeq_*p A ⪰ ∗ p H e x t ( p ) = H S ( p ) H_{\mathrm{ext}}(p)=H_S(p) H ext ( p ) = H S ( p )
For the Lipschitz bound, let ∥ p ∥ 2 , ∥ p ′ ∥ 2 ≤ R \lVert p\rVert_2,\lVert p'\rVert_2\le R ∥ p ∥ 2 , ∥ p ′ ∥ 2 ≤ R A A A p p p λ x = τ ˉ 0 ( p x ′ − p x ) ≤ ∥ p x ′ − p x ∥ 1 , w \lambda_x=\bar{\tau}_0(p'_x-p_x)\le\lVert p'_x-p_x\rVert_{1,w} λ x = τ ˉ 0 ( p x ′ − p x ) ≤ ∥ p x ′ − p x ∥ 1 , w c x = λ x w K − 1 1 { K } ∈ C K c_x=\lambda_xw_K^{-1}\mathbf{1}_{\{K\}}\in C_K c x = λ x w K − 1 1 { K } ∈ C K τ ℓ ( c x ) = λ x ≥ τ ℓ ( p x ′ − p x ) \tau_\ell(c_x)=\lambda_x\ge\tau_\ell(p'_x-p_x) τ ℓ ( c x ) = λ x ≥ τ ℓ ( p x ′ − p x ) ℓ ≤ K \ell\le K ℓ ≤ K A + c ∈ C K V A+c \in C_K^V A + c ∈ C K V p ′ p' p ′ c j = 0 c_j=0 c j = 0 j < K j<K j < K ∥ S ∥ ≤ 1 \|S\|\le1 ∥ S ∥ ≤ 1
H e x t ( p ′ ) − H e x t ( p ) ≤ H S ( A + c ) − H S ( A ) ≤ β 2 w K 4 n ( 2 ∣ c K ∣ ∣ A K ∣ + ∣ c K ∣ 2 ) . H_{\mathrm{ext}}(p')-H_{\mathrm{ext}}(p)\le H_S(A+c)-H_S(A)\le\frac{\beta^2 w_K}{4n}\left(2|c_K||A_K|+|c_K|^2\right). H ext ( p ′ ) − H ext ( p ) ≤ H S ( A + c ) − H S ( A ) ≤ 4 n β 2 w K ( 2∣ c K ∣∣ A K ∣ + ∣ c K ∣ 2 ) . where ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣ R V \mathbb{R}^V R V ∥ z ∥ 1 , w ≤ ∥ z ∥ 2 , w \|z\|_{1,w}\le\|z\|_{2,w} ∥ z ∥ 1 , w ≤ ∥ z ∥ 2 , w w K 2 ∣ A K ∣ 2 ≤ ∑ x ∥ p x ∥ 1 , w 2 ≤ n ∥ p ∥ 2 w_K^2|A_K|^2\le\sum_x\|p_x\|_{1,w}^2\le n\|p\|^2 w K 2 ∣ A K ∣ 2 ≤ ∑ x ∥ p x ∥ 1 , w 2 ≤ n ∥ p ∥ 2 w K 2 ∣ c K ∣ 2 ≤ n ∥ p ′ − p ∥ 2 2 w_K^2|c_K|^2\le n\|p'-p\|_2^2 w K 2 ∣ c K ∣ 2 ≤ n ∥ p ′ − p ∥ 2 2 ∥ p ∥ 2 ≤ R \|p\|_2\le R ∥ p ∥ 2 ≤ R ∥ p ′ − p ∥ 2 ≤ 2 R \|p'-p\|_2\le2R ∥ p ′ − p ∥ 2 ≤ 2 R β 2 R w K − 1 ∥ p ′ − p ∥ 2 \beta^2Rw_K^{-1}\|p'-p\|_2 β 2 R w K − 1 ∥ p ′ − p ∥ 2 p p p p ′ p' p ′ □ \square □
If p x p_x p x A x A_x A x p x p_x p x ( w j ) (w_j) ( w j ) p x p_x p x ∥ ⋅ ∥ 2 , w \|\cdot\|_{2,w} ∥ ⋅ ∥ 2 , w [122 ] .
Lemma 4.3. Let P ∈ C K P\in C_K P ∈ C K r ∈ ( 0 , ∞ ) V K r\in(0,\infty)^{V_K} r ∈ ( 0 , ∞ ) V K M j = n − 1 ∑ x r x j M_j=n^{-1}\sum_x r_{xj} M j = n − 1 ∑ x r x j b x j = P j M j / r x j b_{xj}=P_jM_j/r_{xj} b x j = P j M j / r x j
H e x t ( b ) ≥ β 2 4 ∑ j = 0 K w j P j 2 . H_{\mathrm{ext}}(b)\ge\frac{\beta^2}{4}\sum_{j=0}^{K}w_jP_j^2. H ext ( b ) ≥ 4 β 2 j = 0 ∑ K w j P j 2 . Proof. Suppose first that P j > 0 P_j>0 P j > 0 j j j b ∈ ( 0 , ∞ ) V K b\in(0,\infty)^{V_K} b ∈ ( 0 , ∞ ) V K A A A p = b p=b p = b x x x { i , … , i ′ − 1 } \{i,\ldots,i'-1\} { i , … , i ′ − 1 } A x A_x A x a a a i ≤ ℓ < i ′ i\le\ell<i' i ≤ ℓ < i ′ b x j b_{xj} b x j { ℓ , … , i ′ − 1 } \{\ell,\ldots,i'-1\} { ℓ , … , i ′ − 1 } a a a 1 / y 1/y 1/ y ∑ j = ℓ i ′ − 1 w j ( a − 1 − b x j − 1 ) ≤ 0 \sum_{j=\ell}^{i'-1}w_j(a^{-1}-b_{xj}^{-1})\le0 ∑ j = ℓ i ′ − 1 w j ( a − 1 − b x j − 1 ) ≤ 0 { ℓ , … , K } \{\ell,\ldots,K\} { ℓ , … , K } τ ℓ ( A x − 1 − b x − 1 ) ≤ 0 \tau_\ell(A_x^{-1}-b_x^{-1})\le0 τ ℓ ( A x − 1 − b x − 1 ) ≤ 0 ℓ \ell ℓ v j = n − 1 ∑ x A x j − 1 v_j=n^{-1}\sum_xA_{xj}^{-1} v j = n − 1 ∑ x A x j − 1 n − 1 ∑ x b x j − 1 = P j − 1 n^{-1}\sum_xb_{xj}^{-1}=P_j^{-1} n − 1 ∑ x b x j − 1 = P j − 1 x x x τ ℓ ( v − P − 1 ) ≤ 0 \tau_\ell(v-P^{-1})\le0 τ ℓ ( v − P − 1 ) ≤ 0 ℓ \ell ℓ P − 1 − v ∈ C K ∗ P^{-1}-v\in C_K^* P − 1 − v ∈ C K ∗
The weights S x y / n S_{xy}/n S x y / n V × V V\times V V × V a , a ′ > 0 a,a'>0 a , a ′ > 0 a − 1 a^{-1} a − 1 a ′ − 1 a'^{-1} a ′ − 1 a a ′ ≥ ( ( a − 1 + a ′ − 1 ) / 2 ) − 2 aa'\ge((a^{-1}+a'^{-1})/2)^{-2} a a ′ ≥ (( a − 1 + a ′ − 1 ) /2 ) − 2 a ↦ a − 2 a\mapsto a^{-2} a ↦ a − 2 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ )
1 n ∑ x , y S x y A x j A y j ≥ 1 n ∑ x , y S x y ( A x j − 1 + A y j − 1 2 ) − 2 ≥ v j − 2 . \frac{1}{n}\sum_{x,y}S_{xy}A_{xj}A_{yj}\ge\frac{1}{n}\sum_{x,y}S_{xy}\left(\frac{A_{xj}^{-1}+A_{yj}^{-1}}{2}\right)^{-2}\ge v_j^{-2}. n 1 x , y ∑ S x y A x j A y j ≥ n 1 x , y ∑ S x y ( 2 A x j − 1 + A y j − 1 ) − 2 ≥ v j − 2 . The tangent line of the convex function a ↦ a − 2 a\mapsto a^{-2} a ↦ a − 2 P j − 1 P_j^{-1} P j − 1 v j − 2 ≥ P j 2 − 2 P j 3 ( v j − P j − 1 ) v_j^{-2}\ge P_j^2-2P_j^3(v_j-P_j^{-1}) v j − 2 ≥ P j 2 − 2 P j 3 ( v j − P j − 1 ) ( P j 3 ) j (P_j^3)_j ( P j 3 ) j C K C_K C K P − 1 − v ∈ C K ∗ P^{-1}-v\in C_K^* P − 1 − v ∈ C K ∗
∑ j w j v j − 2 ≥ ∑ j w j P j 2 + 2 ⟨ ( P j 3 ) j , P − 1 − v ⟩ w ≥ ∑ j w j P j 2 . \sum_jw_jv_j^{-2}\ge\sum_jw_jP_j^2+2\langle(P_j^3)_j,P^{-1}-v\rangle_w\ge\sum_jw_jP_j^2. j ∑ w j v j − 2 ≥ j ∑ w j P j 2 + 2 ⟨( P j 3 ) j , P − 1 − v ⟩ w ≥ j ∑ w j P j 2 . Combining the last two displays with the identity H e x t ( b ) = H S ( A ) H_{\mathrm{ext}}(b)=H_S(A) H ext ( b ) = H S ( A ) θ > 0 \theta>0 θ > 0 P ( θ ) = P + θ 1 P^{(\theta)}=P+\theta\mathbf{1} P ( θ ) = P + θ 1 C K ∩ ( 0 , ∞ ) K + 1 C_K \cap(0,\infty)^{K+1} C K ∩ ( 0 , ∞ ) K + 1 b ( θ ) b^{(\theta)} b ( θ ) b ( θ ) → b b^{(\theta)} \to b b ( θ ) → b θ → 0 \theta\to0 θ → 0 H e x t H_{\mathrm{ext}} H ext
Viscosity solutions and comparison
Viscosity solutions were introduced by Crandall and Lions [49 ] ; see also [50 ] . The following definition, in which test points on the boundary of the cone are allowed, follows [36 Definition 1.1] .
Definition 4.4. Let H : R V K → R H:\mathbb{R}^{V_K} \to\mathbb{R} H : R V K → R c ∈ R c \in\mathbb{R} c ∈ R u : [ 0 , 2 ] × C K V → R u:[0,2] \times C_K^V \to\mathbb{R} u : [ 0 , 2 ] × C K V → R u u u subsolution of ∂ t u − H ( D q u ) ≤ c \partial_tu - H(D_qu) \le c ∂ t u − H ( D q u ) ≤ c ( t 0 , q 0 ) ∈ ( 0 , 2 ) × C K V (t_0,q_0) \in(0,2) \times C_K^V ( t 0 , q 0 ) ∈ ( 0 , 2 ) × C K V φ \varphi φ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) R × R V K \mathbb{R} \times\mathbb{R}^{V_K} R × R V K u − φ u-\varphi u − φ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) ( 0 , 2 ) × C K V (0,2) \times C_K^V ( 0 , 2 ) × C K V ∂ t φ ( t 0 , q 0 ) − H ( D q φ ( t 0 , q 0 ) ) ≤ c \partial_t\varphi(t_0,q_0) - H(D_q\varphi(t_0,q_0)) \le c ∂ t φ ( t 0 , q 0 ) − H ( D q φ ( t 0 , q 0 )) ≤ c u u u supersolution of ∂ t u − H ( D q u ) ≥ c \partial_tu - H(D_qu) \ge c ∂ t u − H ( D q u ) ≥ c
The tests are relative to the closed cone: points on its boundary are included, and no boundary condition is imposed. We use the following elementary fact repeatedly. Let u u u t t t t 0 ∈ ( 0 , 2 ) t_0 \in(0,2) t 0 ∈ ( 0 , 2 ) C K V C_K^V C K V q q q u − φ u-\varphi u − φ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) ( 0 , 2 ) × C K V (0,2) \times C_K^V ( 0 , 2 ) × C K V q 0 + ϵ r q_0+\epsilon r q 0 + ϵr C K V C_K^V C K V r ∈ C K V r \in C_K^V r ∈ C K V ϵ ≥ 0 \epsilon\ge0 ϵ ≥ 0
∂ t u ( t 0 , q 0 ) = ∂ t φ ( t 0 , q 0 ) , D q u ( t 0 , q 0 ) − D q φ ( t 0 , q 0 ) ∈ ( C K ∗ ) V . (44) \partial_tu(t_0,q_0) = \partial_t\varphi(t_0,q_0), \qquad D_qu(t_0,q_0) - D_q\varphi(t_0,q_0) \in(C_K^*)^V. \tag*{(44)} ∂ t u ( t 0 , q 0 ) = ∂ t φ ( t 0 , q 0 ) , D q u ( t 0 , q 0 ) − D q φ ( t 0 , q 0 ) ∈ ( C K ∗ ) V . (44) Lemma 4.5. Let H : R V K → R H:\mathbb{R}^{V_K} \to\mathbb{R} H : R V K → R every R > 0 R>0 R > 0 L R L_R L R ∣ H ( p ) − H ( p ′ ) ∣ ≤ L R ∥ p − p ′ ∥ 2 |H(p)-H(p')| \le L_R\lVert p-p'\rVert_2 ∣ H ( p ) − H ( p ′ ) ∣ ≤ L R ∥ p − p ′ ∥ 2 ∥ p ∥ 2 , ∥ p ′ ∥ 2 ≤ R \lVert p\rVert_2,\lVert p'\rVert_2 \le R ∥ p ∥ 2 , ∥ p ′ ∥ 2 ≤ R τ ≥ 0 \tau\ge0 τ ≥ 0 c 0 ∈ R c_0 \in\mathbb{R} c 0 ∈ R V , E : [ 0 , 2 ] × C K V → R V,E:[0,2] \times C_K^V \to\mathbb{R} V , E : [ 0 , 2 ] × C K V → R L t , L q L_t,L_q L t , L q
(i) V V V ∂ t u − H ( D q u ) ≤ 0 \partial_tu-H(D_qu) \le0 ∂ t u − H ( D q u ) ≤ 0 E E E ∂ t u − H ( D q u ) ≥ − τ \partial_tu-H(D_qu) \ge-\tau ∂ t u − H ( D q u ) ≥ − τ
(ii) ∣ E ( t , q ) − E ( t ′ , q ′ ) ∣ ≤ L t ∣ t − t ′ ∣ + L q ∥ q − q ′ ∥ 1 |E(t,q)-E(t',q')| \le L_t|t-t'|+L_q\lVert q-q'\rVert_1 ∣ E ( t , q ) − E ( t ′ , q ′ ) ∣ ≤ L t ∣ t − t ′ ∣ + L q ∥ q − q ′ ∥ 1 ∣ V ( t , q ) − V ( t ′ , q ) ∣ ≤ L t ∣ t − t ′ ∣ |V(t,q)-V(t',q)| \le L_t|t-t'| ∣ V ( t , q ) − V ( t ′ , q ) ∣ ≤ L t ∣ t − t ′ ∣
(iii) V ( 0 , q ) ≤ E ( 0 , q ) + c 0 V(0,q) \le E(0,q)+c_0 V ( 0 , q ) ≤ E ( 0 , q ) + c 0 every q ∈ C K V q \in C_K^V q ∈ C K V
Then V ( t , q ) ≤ E ( t , q ) + c 0 + τ t V(t,q) \le E(t,q)+c_0+\tau t V ( t , q ) ≤ E ( t , q ) + c 0 + τ t every t ∈ [ 0 , 2 ) t \in[0,2) t ∈ [ 0 , 2 ) q ∈ C K V q \in C_K^V q ∈ C K V
The proof is the doubling-of-variables argument of [49 , 50 ] ; see [56 Theorem 3.5] and [36 Proposition 3.1] for comparison principles on R d \mathbb{R}^d R d V V V q q q
Proof. Suppose that V ( t ˉ , q ˉ ) − E ( t ˉ , q ˉ ) − c 0 − τ t ˉ = 4 θ > 0 V(\bar{t},\bar{q})-E(\bar{t},\bar{q})-c_0-\tau\bar{t}=4\theta>0 V ( t ˉ , q ˉ ) − E ( t ˉ , q ˉ ) − c 0 − τ t ˉ = 4 θ > 0 t ˉ < 2 \bar{t}<2 t ˉ < 2 t ˉ > 0 \bar{t}>0 t ˉ > 0 L = max { 1 , ( L t 2 + L q 2 ) 1 / 2 } L=\max\{1,(L_t^2+L_q^2)^{1/2}\} L = max { 1 , ( L t 2 + L q 2 ) 1/2 } T ′ ∈ ( t ˉ , 2 ) T'\in(\bar{t},2) T ′ ∈ ( t ˉ , 2 ) ρ > 0 \rho>0 ρ > 0 ρ t ˉ + ρ / ( T ′ − t ˉ ) ≤ θ \rho\bar{t}+\rho/(T'-\bar{t})\le\theta ρ t ˉ + ρ / ( T ′ − t ˉ ) ≤ θ γ ∈ ( 0 , 1 ] \gamma\in(0,1] γ ∈ ( 0 , 1 ] γ ( 1 + ∥ q ˉ ∥ 2 2 ) 1 / 2 ≤ θ \gamma(1+\lVert\bar{q}\rVert_2^2)^{1/2}\le\theta γ ( 1 + ∥ q ˉ ∥ 2 2 ) 1/2 ≤ θ γ L 2 L + 1 < ρ \gamma L_{2L+1}<\rho γ L 2 L + 1 < ρ b > 0 b>0 b > 0 ( t , q , s , r ) ∈ [ 0 , T ′ ) × C K V × [ 0 , 2 ] × C K V (t,q,s,r)\in[0,T')\times C_K^V\times[0,2]\times C_K^V ( t , q , s , r ) ∈ [ 0 , T ′ ) × C K V × [ 0 , 2 ] × C K V
Φ b ( t , q , s , r ) = V ( t , q ) − E ( s , r ) − c 0 − ( τ + ρ ) t − ρ T ′ − t − γ ⟨ q ⟩ − ∣ t − s ∣ 2 + ∥ q − r ∥ 2 2 2 b , \Phi_b(t,q,s,r)=V(t,q)-E(s,r)-c_0-(\tau+\rho)t-\frac{\rho}{T'-t}-\gamma\langle q\rangle-\frac{|t-s|^2+\lVert q-r\rVert_2^2}{2b}, Φ b ( t , q , s , r ) = V ( t , q ) − E ( s , r ) − c 0 − ( τ + ρ ) t − T ′ − t ρ − γ ⟨ q ⟩ − 2 b ∣ t − s ∣ 2 + ∥ q − r ∥ 2 2 , where ⟨ q ⟩ = ( 1 + ∥ q ∥ 2 2 ) 1 / 2 \langle q\rangle= (1 + \lVert q\rVert_2^2)^{1/2} ⟨ q ⟩ = ( 1 + ∥ q ∥ 2 2 ) 1/2 V ( t , q ) − E ( s , r ) ≤ c 0 + 2 L t t + L t ∣ t − s ∣ + L q ∥ q − r ∥ 2 V(t,q) - E(s,r) \le c_0 + 2L_t t + L_t|t-s| + L_q\lVert q-r\rVert_2 V ( t , q ) − E ( s , r ) ≤ c 0 + 2 L t t + L t ∣ t − s ∣ + L q ∥ q − r ∥ 2 Φ b \Phi_b Φ b − ∞ -\infty − ∞ ∥ q ∥ 2 → ∞ \lVert q\rVert_2 \to\infty ∥ q ∥ 2 → ∞ ∥ q − r ∥ 2 → ∞ \lVert q-r\rVert_2 \to\infty ∥ q − r ∥ 2 → ∞ t → T ′ t \to T' t → T ′ Φ b \Phi_b Φ b ( t b , q b , s b , r b ) (t_b,q_b,s_b,r_b) ( t b , q b , s b , r b ) Φ b ( t b , q b , s b , r b ) ≥ Φ b ( t ˉ , q ˉ , t ˉ , q ˉ ) ≥ 2 θ \Phi_b(t_b,q_b,s_b,r_b) \ge\Phi_b(\bar{t},\bar{q},\bar{t},\bar{q}) \ge2\theta Φ b ( t b , q b , s b , r b ) ≥ Φ b ( t ˉ , q ˉ , t ˉ , q ˉ ) ≥ 2 θ
Comparing with Φ b ( t b , q b , t b , q b ) \Phi_b(t_b,q_b,t_b,q_b) Φ b ( t b , q b , t b , q b ) X = ( ∣ t b − s b ∣ 2 + ∥ q b − r b ∥ 2 2 ) 1 / 2 X = (|t_b-s_b|^2 + \lVert q_b-r_b\rVert_2^2)^{1/2} X = ( ∣ t b − s b ∣ 2 + ∥ q b − r b ∥ 2 2 ) 1/2
X 2 2 b ≤ E ( t b , q b ) − E ( s b , r b ) ≤ L X , so X ≤ 2 b L . \frac{X^2}{2b} \le E(t_b,q_b) - E(s_b,r_b) \le LX,\qquad\text{so}\qquad X \le2bL. 2 b X 2 ≤ E ( t b , q b ) − E ( s b , r b ) ≤ L X , so X ≤ 2 b L . If t b = 0 t_b = 0 t b = 0 Φ b ≤ E ( 0 , q b ) − E ( s b , r b ) ≤ L X ≤ 2 b L 2 \Phi_b \le E(0,q_b) - E(s_b,r_b) \le LX \le2bL^2 Φ b ≤ E ( 0 , q b ) − E ( s b , r b ) ≤ L X ≤ 2 b L 2 s b = 0 s_b = 0 s b = 0 Φ b ≤ V ( t b , q b ) − V ( 0 , q b ) + E ( 0 , q b ) − E ( 0 , r b ) ≤ 2 L X ≤ 4 b L 2 \Phi_b \le V(t_b,q_b) - V(0,q_b) + E(0,q_b) - E(0,r_b) \le2LX \le4bL^2 Φ b ≤ V ( t b , q b ) − V ( 0 , q b ) + E ( 0 , q b ) − E ( 0 , r b ) ≤ 2 L X ≤ 4 b L 2 Φ b ≥ 2 θ \Phi_b \ge2\theta Φ b ≥ 2 θ b < θ / ( 2 L 2 ) b < \theta/(2L^2) b < θ / ( 2 L 2 ) s b ≤ t b + 2 b L < 2 s_b \le t_b + 2bL < 2 s b ≤ t b + 2 b L < 2 b b b t b ∈ ( 0 , T ′ ) t_b \in(0,T') t b ∈ ( 0 , T ′ ) s b ∈ ( 0 , 2 ) s_b \in(0,2) s b ∈ ( 0 , 2 ) b b b
The function ( t , q ) ↦ V ( t , q ) − φ 1 ( t , q ) (t,q) \mapsto V(t,q) - \varphi_1(t,q) ( t , q ) ↦ V ( t , q ) − φ 1 ( t , q )
φ 1 ( t , q ) = ( τ + ρ ) t + ρ T ′ − t + γ ⟨ q ⟩ + ∣ t − s b ∣ 2 + ∥ q − r b ∥ 2 2 2 b , \varphi_1(t,q) = (\tau+\rho)t + \frac{\rho}{T'-t} + \gamma\langle q\rangle+ \frac{|t-s_b|^2+\lVert q-r_b\rVert_2^2}{2b}, φ 1 ( t , q ) = ( τ + ρ ) t + T ′ − t ρ + γ ⟨ q ⟩ + 2 b ∣ t − s b ∣ 2 + ∥ q − r b ∥ 2 2 , has a local maximum at ( t b , q b ) (t_b,q_b) ( t b , q b ) ( 0 , 2 ) × C K V (0,2)\times C_K^V ( 0 , 2 ) × C K V ( s , r ) ↦ E ( s , r ) − φ 2 ( s , r ) (s,r) \mapsto E(s,r) - \varphi_2(s,r) ( s , r ) ↦ E ( s , r ) − φ 2 ( s , r ) φ 2 ( s , r ) = − ( ∣ t b − s ∣ 2 + ∥ q b − r ∥ 2 2 ) / ( 2 b ) \varphi_2(s,r) = -(|t_b-s|^2+\lVert q_b-r\rVert_2^2)/(2b) φ 2 ( s , r ) = − ( ∣ t b − s ∣ 2 + ∥ q b − r ∥ 2 2 ) / ( 2 b ) ( s b , r b ) (s_b,r_b) ( s b , r b ) p = ( q b − r b ) / b p=(q_b-r_b)/b p = ( q b − r b ) / b D q ⟨ q ⟩ = q / ⟨ q ⟩ D_q\langle q\rangle=q/\langle q\rangle D q ⟨ q ⟩ = q / ⟨ q ⟩
τ + ρ + ρ ( T ′ − t b ) 2 + t b − s b b − H ( p + γ q b ⟨ q b ⟩ ) ≤ 0 ≤ t b − s b b − H ( p ) + τ . \tau+\rho+\frac{\rho}{(T'-t_b)^2}+\frac{t_b-s_b}{b}-H\left(p+\frac{\gamma q_b}{\langle q_b\rangle}\right)\le0\le\frac{t_b-s_b}{b}-H(p)+\tau. τ + ρ + ( T ′ − t b ) 2 ρ + b t b − s b − H ( p + ⟨ q b ⟩ γ q b ) ≤ 0 ≤ b t b − s b − H ( p ) + τ . We have ∥ p ∥ 2 ≤ X / b ≤ 2 L \lVert p\rVert_2\le X/b\le2L ∥ p ∥ 2 ≤ X / b ≤ 2 L ∥ γ q b / ⟨ q b ⟩ ∥ 2 ≤ γ ≤ 1 \lVert\gamma q_b/\langle q_b\rangle\rVert_2\le\gamma\le1 ∥ γ q b / ⟨ q b ⟩ ∥ 2 ≤ γ ≤ 1 ρ ≤ H ( p + γ q b / ⟨ q b ⟩ ) − H ( p ) ≤ L 2 L + 1 γ < ρ \rho\le H(p+\gamma q_b/\langle q_b\rangle)-H(p)\le L_{2L+1}\gamma<\rho ρ ≤ H ( p + γ q b / ⟨ q b ⟩) − H ( p ) ≤ L 2 L + 1 γ < ρ
The subsolution
In this subsection β > 0 \beta>0 β > 0 h ∈ R h\in\mathbb{R} h ∈ R ψ h \psi_h ψ h (28) , and U U U (39) . We evaluate U U U q x q_x q x [37 Appendix B] , where the weighted mean of the paths is used and the initial conditions are compared through the convexity of ψ 0 \psi_0 ψ 0 [86 Section 7] . Here we average the square roots of the paths, which Proposition 3.8 allows for every h h h δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 ) q ∈ C K V q\in C_K^V q ∈ C K V
r x j ( q ) = ( q x j + δ ) 1 / 2 , M j ( q ) = 1 n ∑ x r x j ( q ) , g j ( q ) = M j ( q ) 2 − δ . (45) r_{xj}(q)=(q_{xj}+\delta)^{1/2},\qquad M_j(q)=\frac{1}{n}\sum_x r_{xj}(q),\qquad g_j(q)=M_j(q)^2-\delta. \tag*{(45)} r x j ( q ) = ( q x j + δ ) 1/2 , M j ( q ) = n 1 x ∑ r x j ( q ) , g j ( q ) = M j ( q ) 2 − δ . (45) Since r x j r_{xj} r x j j j j M j ≥ δ M_j \ge\sqrt{\delta} M j ≥ δ g ( q ) ∈ C K g(q) \in C_K g ( q ) ∈ C K q ∈ C K V q \in C_K^V q ∈ C K V q ~ \widetilde q q C K V C_K^V C K V q ~ x 0 = 0 \widetilde q_{x_0}=0 q x 0 = 0 q ~ x j = q x j \widetilde q_{x_j}=q_{x_j} q x j = q x j 1 ≤ j ≤ K 1 \le j \le K 1 ≤ j ≤ K
V δ ( t , q ) = U ( t , g ( q ~ ) ) − β 2 ζ 0 t 4 . (46) V_\delta(t,q)=U(t,g(\widetilde q))-\frac{\beta^2\zeta_0t}{4}. \tag*{(46)} V δ ( t , q ) = U ( t , g ( q )) − 4 β 2 ζ 0 t . (46) Lemma 4.6. The function V δ V_\delta V δ [ 0 , 2 ] × C K V [0,2]\times C_K^V [ 0 , 2 ] × C K V
(a) ∣ V δ ( t , q ) − V δ ( t ′ , q ) ∣ ≤ β 2 ∣ t − t ′ ∣ |V_\delta(t,q)-V_\delta(t',q)|\le\beta^2|t-t'| ∣ V δ ( t , q ) − V δ ( t ′ , q ) ∣ ≤ β 2 ∣ t − t ′ ∣
(b) V δ V_\delta V δ ∂ t u − H e x t ( D q u ) ≤ 0 \partial_tu-H_{\mathrm{ext}}(D_qu)\le0 ∂ t u − H ext ( D q u ) ≤ 0
(c) V δ ( 0 , q ) ≤ n − 1 ∑ x ψ h ( q ~ x ) + δ V_\delta(0,q)\le n^{-1}\sum_x\psi_h(\widetilde q_x)+\delta V δ ( 0 , q ) ≤ n − 1 ∑ x ψ h ( q x ) + δ q ∈ C K V q\in C_K^V q ∈ C K V
(d) V δ ( 1 , 0 ) = U ( 1 , 0 ) − β 2 ζ 0 / 4 V_\delta(1,0)=U(1,0)-\beta^2\zeta_0/4 V δ ( 1 , 0 ) = U ( 1 , 0 ) − β 2 ζ 0 /4
Proof. Continuity and (a) follow from Lemma 3.9 (b),(c), and (d) holds because g ( 0 ) = 0 g(0)=0 g ( 0 ) = 0
(b) Let φ \varphi φ ( t 0 , q 0 ) ∈ ( 0 , 2 ) × C K V (t_0,q_0)\in(0,2)\times C_K^V ( t 0 , q 0 ) ∈ ( 0 , 2 ) × C K V V δ − φ V_\delta-\varphi V δ − φ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) y ′ y' y ′ (39) for U ( t 0 , g ( q ~ 0 ) ) U(t_0,g(\widetilde q_0)) U ( t 0 , g ( q 0 )) P = 2 ( y ′ − g ( q ~ 0 ) ) / ( β 2 t 0 ) ∈ C K ∩ [ 0 , 1 ] K + 1 P=2(y'-g(\widetilde q_0))/(\beta^2t_0)\in C_K\cap[0,1]^{K+1} P = 2 ( y ′ − g ( q 0 )) / ( β 2 t 0 ) ∈ C K ∩ [ 0 , 1 ] K + 1 Lemma 3.9 (a). The function
B ( t , q ) = ψ h ( y ′ ) − ∥ y ′ − g ( q ~ ) ∥ 2 , w 2 β 2 t − β 2 ζ 0 t 4 B(t,q)=\psi_h(y')-\frac{\lVert y'-g(\widetilde q)\rVert_{2,w}^2}{\beta^2t}-\frac{\beta^2\zeta_0t}{4} B ( t , q ) = ψ h ( y ′ ) − β 2 t ∥ y ′ − g ( q ) ∥ 2 , w 2 − 4 β 2 ζ 0 t is smooth near ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) B ≤ V δ B\le V_\delta B ≤ V δ ( 0 , 2 ) × C K V (0,2)\times C_K^V ( 0 , 2 ) × C K V B ( t 0 , q 0 ) = V δ ( t 0 , q 0 ) B(t_0,q_0)=V_\delta(t_0,q_0) B ( t 0 , q 0 ) = V δ ( t 0 , q 0 ) B − φ ≤ V δ − φ ≤ 0 B-\varphi\le V_\delta-\varphi\le0 B − φ ≤ V δ − φ ≤ 0 ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) φ − B \varphi-B φ − B ∂ t φ = ∂ t B \partial_t\varphi=\partial_tB ∂ t φ = ∂ t B D q φ ⪰ ∗ D q B D_q\varphi\succeq_*D_qB D q φ ⪰ ∗ D q B ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) q ~ \widetilde q q q x 0 q_{x_0} q x 0 ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) ( D q B ) x 0 = 0 (D_qB)_{x_0}=0 ( D q B ) x 0 = 0
∂ t B = ∥ y ′ − g ( q ~ 0 ) ∥ 2 , w 2 β 2 t 0 2 − β 2 ζ 0 4 = β 2 4 ( ∥ P ∥ 2 , w 2 − ζ 0 ) , \partial_tB=\frac{\lVert y'-g(\widetilde q_0)\rVert_{2,w}^2}{\beta^2t_0^2}-\frac{\beta^2\zeta_0}{4}
=\frac{\beta^2}{4}\left(\lVert P\rVert_{2,w}^2-\zeta_0\right), ∂ t B = β 2 t 0 2 ∥ y ′ − g ( q 0 ) ∥ 2 , w 2 − 4 β 2 ζ 0 = 4 β 2 ( ∥ P ∥ 2 , w 2 − ζ 0 ) , ( D q B ) x j = P j M j ( q ~ 0 ) r x j ( q ~ 0 ) ( 1 ≤ j ≤ K ) . (D_qB)_{x_j}=\frac{P_jM_j(\widetilde q_0)}{r_{x_j}(\widetilde q_0)}
\qquad(1\le j\le K). ( D q B ) x j = r x j ( q 0 ) P j M j ( q 0 ) ( 1 ≤ j ≤ K ) . Let P ′ = ( 0 , P 1 , … , P K ) ∈ C K P'=(0,P_1,\ldots,P_K)\in C_K P ′ = ( 0 , P 1 , … , P K ) ∈ C K D q B D_qB D q B b b b P ′ P' P ′ r = r ( q ~ 0 ) r=r(\widetilde q_0) r = r ( q 0 ) H e x t ( D q B ) ≥ ( β 2 / 4 ) ∑ j ≥ 1 w j P j 2 H_{\mathrm{ext}}(D_qB)\ge(\beta^2/4)\sum_{j\ge1}w_jP_j^2 H ext ( D q B ) ≥ ( β 2 /4 ) ∑ j ≥ 1 w j P j 2 w 0 = ζ 0 w_0=\zeta_0 w 0 = ζ 0 P 0 ≤ 1 P_0\le1 P 0 ≤ 1 ∂ t B ≤ ( β 2 / 4 ) ∑ j ≥ 1 w j P j 2 \partial_tB\le(\beta^2/4)\sum_{j\ge1}w_jP_j^2 ∂ t B ≤ ( β 2 /4 ) ∑ j ≥ 1 w j P j 2 Lemma 4.2 (c), we conclude that ∂ t φ = ∂ t B ≤ H e x t ( D q B ) ≤ H e x t ( D q φ ) \partial_t\varphi=\partial_tB\le H_{\mathrm{ext}}(D_qB)\le H_{\mathrm{ext}}(D_q\varphi) ∂ t φ = ∂ t B ≤ H ext ( D q B ) ≤ H ext ( D q φ )
(c) We have V δ ( 0 , q ) = ψ h ( g ( q ~ ) ) V_\delta(0,q)=\psi_h(g(\widetilde q)) V δ ( 0 , q ) = ψ h ( g ( q )) g ( q ~ ) g(\widetilde q) g ( q ) M ( q ~ ) 2 = ( M j ( q ~ ) 2 ) j M(\widetilde q)^2=(M_j(\widetilde q)^2)_j M ( q ) 2 = ( M j ( q ) 2 ) j C K C_K C K g ( q ~ ) ≤ M ( q ~ ) 2 g(\widetilde q)\le M(\widetilde q)^2 g ( q ) ≤ M ( q ) 2 ψ h ( g ( q ~ ) ) ≤ ψ h ( M ( q ~ ) 2 ) \psi_h(g(\widetilde q))\le\psi_h(M(\widetilde q)^2) ψ h ( g ( q )) ≤ ψ h ( M ( q ) 2 ) r x ( q ~ ) r_x(\widetilde q) r x ( q ) { s : 0 ≤ s 0 ≤ ⋯ ≤ s K } \{s:0\le s_0\le\cdots\le s_K\} { s : 0 ≤ s 0 ≤ ⋯ ≤ s K } M ( q ~ ) M(\widetilde q) M ( q ) ψ h ( M ( q ~ ) 2 ) ≤ n − 1 ∑ x ψ h ( q ~ x + δ 1 ) \psi_h(M(\widetilde q)^2)\le n^{-1}\sum_x\psi_h(\widetilde q_x+\delta\mathbf{1}) ψ h ( M ( q ) 2 ) ≤ n − 1 ∑ x ψ h ( q x + δ 1 ) Proposition 3.8 and Jensen’s inequality. Finally ψ h ( q ~ x + δ 1 ) ≤ ψ h ( q ~ x ) + δ \psi_h(\widetilde q_x+\delta\mathbf{1})\le\psi_h(\widetilde q_x)+\delta ψ h ( q x + δ 1 ) ≤ ψ h ( q x ) + δ ∥ δ 1 ∥ 1 , w = δ \lVert\delta\mathbf{1}\rVert_{1,w}=\delta ∥ δ 1 ∥ 1 , w = δ
The enriched free energy
Ruelle cascades
Let A = ⋃ k = 0 K N k \mathcal{A} = \bigcup_{k=0}^{K} \mathbb{N}^{k} A = ⋃ k = 0 K N k K K K ∅ \varnothing ∅ N K \mathbb{N}^{K} N K α ∈ N K \alpha\in\mathbb{N}^{K} α ∈ N K k ≤ K k \le K k ≤ K α ∣ k \alpha|_{k} α ∣ k α \alpha α k k k α , α ′ \alpha,\alpha' α , α ′ α ∧ α ′ = max { k : α ∣ k = α ′ ∣ k } \alpha\wedge\alpha' = \max\{k : \alpha|_{k} = \alpha'|_{k}\} α ∧ α ′ = max { k : α ∣ k = α ′ ∣ k } γ ∈ N k \gamma\in\mathbb{N}^{k} γ ∈ N k k < K k < K k < K ( u γ i ) i ≥ 1 (u_{\gamma i})_{i \ge1} ( u γ i ) i ≥ 1 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) ζ k x − 1 − ζ k d x \zeta_{k}x^{-1-\zeta_{k}}\,\mathrm{d}x ζ k x − 1 − ζ k d x γ \gamma γ w α = ∏ k = 1 K u α ∣ k w_{\alpha} = \prod_{k=1}^{K}u_{\alpha|_{k}} w α = ∏ k = 1 K u α ∣ k α ∈ N K \alpha\in\mathbb{N}^{K} α ∈ N K ∑ α w α \sum_{\alpha} w_{\alpha} ∑ α w α [56 Lemma 5.23] , and the Ruelle cascade [123 ] with masses ( ζ k ) k < K (\zeta_{k})_{k<K} ( ζ k ) k < K v α = w α / ∑ α ′ w α ′ v_{\alpha} = w_{\alpha}/\sum_{\alpha'}w_{\alpha'} v α = w α / ∑ α ′ w α ′ [56 ] the mass ζ k \zeta_{k} ζ k ζ k + 1 \zeta_{k+1} ζ k + 1
Theorem 4.7. Let ( ω γ ) γ ∈ A ∖ { ∅ } (\omega_{\gamma})_{\gamma\in\mathcal{A}\setminus\{\varnothing\}} ( ω γ ) γ ∈ A ∖ { ∅ } ( u γ i ) (u_{\gamma i}) ( u γ i ) ω γ \omega_{\gamma} ω γ P k \mathbb{P}_{k} P k γ ∈ N k \gamma\in\mathbb{N}^{k} γ ∈ N k X : Ω 1 × ⋯ × Ω K → R X : \Omega_{1} \times\cdots\times\Omega_{K} \to\mathbb{R} X : Ω 1 × ⋯ × Ω K → R E exp ( c ∣ X ∣ ) < ∞ \mathbb{E}\exp(c|X|) < \infty E exp ( c ∣ X ∣ ) < ∞ c > 0 c > 0 c > 0 X K = X X_{K} = X X K = X X k − 1 = ζ k − 1 − 1 log E ω k exp ( ζ k − 1 X k ) X_{k-1} = \zeta_{k-1}^{-1}\log\mathbb{E}_{\omega_{k}}\exp(\zeta_{k-1}X_{k}) X k − 1 = ζ k − 1 − 1 log E ω k exp ( ζ k − 1 X k ) 1 ≤ k ≤ K 1 \le k \le K 1 ≤ k ≤ K E ω k \mathbb{E}_{\omega_{k}} E ω k ω k ∼ P k \omega_{k} \sim\mathbb{P}_{k} ω k ∼ P k
E log ∑ α ∈ N K v α exp X ( ω α ∣ 1 , … , ω α ∣ K ) = X 0 . \mathbb{E}\log\sum_{\alpha\in\mathbb{N}^{K}}v_{\alpha}\exp X(\omega_{\alpha|_{1}},\ldots,\omega_{\alpha|_{K}})=X_{0}. E log α ∈ N K ∑ v α exp X ( ω α ∣ 1 , … , ω α ∣ K ) = X 0 . Moreover, if ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ N K \mathbb{N}^{K} N K v α exp X ( ω α ∣ 1 , … , ω α ∣ K ) v_{\alpha}\exp X(\omega_{\alpha|_{1}},\ldots,\omega_{\alpha|_{K}}) v α exp X ( ω α ∣ 1 , … , ω α ∣ K ) α 1 , α 2 \alpha^{1},\alpha^{2} α 1 , α 2
E ⟨ 1 { α 1 ∧ α 2 = j } ⟩ = w j ( 0 ≤ j ≤ K ) . (47) \mathbb{E}\langle\mathbf{1}\{\alpha^{1}\wedge\alpha^{2}=j\}\rangle=w_{j}\qquad(0\le j\le K). \tag*{(47)} E ⟨ 1 { α 1 ∧ α 2 = j }⟩ = w j ( 0 ≤ j ≤ K ) . (47) The first statement is [56 Theorem 5.25] , applied with a trivial root variable; see also [114 Theorem 2.9] . For X = 0 X=0 X = 0 [56 (5.93) in Theorem 5.28] . For general X X X [56 Section 6.4, before Lemma 6.2] for the enriched free energy considered there, by observing that the proof of [56 Theorem 5.28] , which rests on the invariance property [56 Lemma 5.27] of Bolthausen and Sznitman [27 ] , remains valid when each w α w_{\alpha} w α α \alpha α ( w α ) (w_{\alpha}) ( w α ) N K \mathbb{N}^{K} N K ∧ \wedge ∧ w α exp X ( ω α ∣ 1 , … , ω α ∣ K ) w_{\alpha}\exp X(\omega_{\alpha|_{1}},\ldots,\omega_{\alpha|_{K}}) w α exp X ( ω α ∣ 1 , … , ω α ∣ K )
We also use Gaussian integration by parts for Gibbs measures [56 Theorem 4.6] . Let Σ \Sigma Σ μ \mu μ Σ \Sigma Σ ( y ( ς ) ) ς ∈ Σ , ( z ( ς ) ) ς ∈ Σ (y(\varsigma))_{\varsigma\in\Sigma},(z(\varsigma))_{\varsigma\in\Sigma} ( y ( ς ) ) ς ∈ Σ , ( z ( ς ) ) ς ∈ Σ ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ e y \mathrm{e}^{y} e y μ \mu μ C ( ς , ς ′ ) = E z ( ς ) y ( ς ′ ) C(\varsigma,\varsigma')=\mathbb{E}z(\varsigma)y(\varsigma') C ( ς , ς ′ ) = E z ( ς ) y ( ς ′ ) r ≥ 1 r\ge1 r ≥ 1 f : Σ r → R f:\Sigma^{r}\to\mathbb{R} f : Σ r → R
E ⟨ f ( ς 1 , … , ς r ) z ( ς 1 ) ⟩ = E ⟨ f ( ς 1 , … , ς r ) ( ∑ ℓ = 1 r C ( ς 1 , ς ℓ ) − r C ( ς 1 , ς r + 1 ) ) ⟩ . (48) \mathbb{E}\langle f(\varsigma^{1},\ldots,\varsigma^{r})z(\varsigma^{1})\rangle
=
\mathbb{E}\left\langle f(\varsigma^{1},\ldots,\varsigma^{r})\left(\sum_{\ell=1}^{r}C(\varsigma^{1},\varsigma^{\ell})-rC(\varsigma^{1},\varsigma^{r+1})\right)\right\rangle. \tag*{(48)} E ⟨ f ( ς 1 , … , ς r ) z ( ς 1 )⟩ = E ⟨ f ( ς 1 , … , ς r ) ( ℓ = 1 ∑ r C ( ς 1 , ς ℓ ) − r C ( ς 1 , ς r + 1 ) ) ⟩ . (48) Finally, we use the following invariance of Poisson point processes [56 Theorem 5.19 and Proposition 5.18] . Let ( u i ) (u_i) ( u i ) ζ x − 1 − ζ d x \zeta x^{-1-\zeta}\,\mathrm{d}x ζ x − 1 − ζ d x ζ ∈ ( 0 , 1 ) \zeta\in(0,1) ζ ∈ ( 0 , 1 ) E ( ∑ i u i ) a < ∞ \mathbb{E}(\sum_i u_i)^a < \infty E ( ∑ i u i ) a < ∞ 0 < a < ζ 0<a<\zeta 0 < a < ζ ( Y i ) (Y_i) ( Y i ) ( u i ) (u_i) ( u i ) E Y 1 ζ < ∞ \mathbb{E}Y_1^\zeta< \infty E Y 1 ζ < ∞
∑ i u i Y i has the law of ( E Y 1 ζ ) 1 / ζ ∑ i u i . (49) \sum_i u_iY_i \quad\text{has the law of} \quad(\mathbb{E}Y_1^\zeta)^{1/\zeta}\sum_i u_i. \tag*{(49)} i ∑ u i Y i has the law of ( E Y 1 ζ ) 1/ ζ i ∑ u i . (49) The model
Fix reference values z j = j / K z_j=j/K z j = j / K 0 ≤ j ≤ K 0\leq j\leq K 0 ≤ j ≤ K α , α ′ \alpha,\alpha' α , α ′ ϑ ( α , α ′ ) = z α ∧ α ′ \vartheta(\alpha,\alpha')=z_{\alpha\wedge\alpha'} ϑ ( α , α ′ ) = z α ∧ α ′ ∑ k = 1 K ( z k − z k − 1 ) 1 / 2 e α ∣ k \sum_{k=1}^K(z_k-z_{k-1})^{1/2}e_{\alpha|k} ∑ k = 1 K ( z k − z k − 1 ) 1/2 e α ∣ k α ′ \alpha' α ′ ( e γ ) (e_\gamma) ( e γ ) ϑ \vartheta ϑ A \mathcal{A} A
to each a ∈ N a\in\mathbb{N} a ∈ N ( g e ( a ) ) e ∈ E (g_{e}^{(a)})_{e\in E} ( g e ( a ) ) e ∈ E H ( a ) ( σ ) = ∑ x y ∈ E D − 1 / 2 g x y ( a ) σ x σ y H^{(a)}(\sigma)=\sum_{xy\in E}D^{-1/2}g_{xy}^{(a)}\sigma_x\sigma_y H ( a ) ( σ ) = ∑ x y ∈ E D − 1/2 g x y ( a ) σ x σ y
to each vertex γ ≠ ∅ \gamma\neq\varnothing γ = ∅ x ∈ V x\in V x ∈ V ξ x , γ \xi_{x,\gamma} ξ x , γ
to each vertex γ ≠ ∅ \gamma\neq\varnothing γ = ∅ G x , ι G_{x,\iota} G x , ι
There is no variable attached to the root. Let S ⊆ V \mathcal{S}\subseteq V S ⊆ V I I I G x , ι G_{x,\iota} G x , ι x ∈ S x\in\mathcal{S} x ∈ S ι ∈ I \iota\in I ι ∈ I s ≥ 0 s\geq0 s ≥ 0 t ≥ 0 t\geq0 t ≥ 0 q ∈ C K V q\in\mathbb{C}_K^V q ∈ C K V y ∈ Y = [ 1 / 2 , 3 / 2 ] S × I y\in\mathcal{Y}=[1/2,3/2]^{\mathcal{S}\times I} y ∈ Y = [ 1/2 , 3/2 ] S × I α ∈ N K \alpha\in\mathbb{N}^K α ∈ N K σ ∈ { − 1 , 1 } V \sigma\in\{-1,1\}^V σ ∈ { − 1 , 1 } V
H ( α , σ ) = β t H ( α 1 ) ( σ ) + h ∑ x σ x + ∑ x ( Y x ( α ) σ x − q x K ) + s ∑ x ∈ S , ι ∈ I y x ι G x , ι ( α , σ ) , (50) \mathcal{H}(\alpha,\sigma)=\beta\sqrt{t}\,H^{(\alpha_1)}(\sigma)+h\sum_x\sigma_x+\sum_x\bigl(Y_x(\alpha)\sigma_x-q_xK\bigr)+s\sum_{x\in\mathcal{S},\iota\in I}y_{x\iota}G_{x,\iota}(\alpha,\sigma), \tag*{(50)} H ( α , σ ) = β t H ( α 1 ) ( σ ) + h x ∑ σ x + x ∑ ( Y x ( α ) σ x − q x K ) + s x ∈ S , ι ∈ I ∑ y x ι G x , ι ( α , σ ) , (50) where
Y x ( α ) = ( 2 q x 1 ) 1 / 2 ξ x , α ∣ 1 + ∑ k = 2 K ( 2 ( q x k − q x , k − 1 ) ) 1 / 2 ξ x , α ∣ k . Y_x(\alpha)=(2q_{x1})^{1/2}\xi_{x,\alpha|1}+\sum_{k=2}^K\bigl(2(q_{xk}-q_{x,k-1})\bigr)^{1/2}\xi_{x,\alpha|k}. Y x ( α ) = ( 2 q x 1 ) 1/2 ξ x , α ∣1 + k = 2 ∑ K ( 2 ( q x k − q x , k − 1 ) ) 1/2 ξ x , α ∣ k . We have E Y x ( α ) Y x ( α ′ ) = 2 q x , α ∧ α ′ \mathbb{E}Y_x(\alpha)Y_x(\alpha')=2q_{x,\alpha\wedge\alpha'} E Y x ( α ) Y x ( α ′ ) = 2 q x , α ∧ α ′ α ∧ α ′ ≥ 1 \alpha\wedge\alpha'\geq1 α ∧ α ′ ≥ 1 0 0 0 q x 0 q_{x0} q x 0 Z = ∑ α v α ∑ σ exp H ( α , σ ) Z=\sum_\alpha v_\alpha\sum_\sigma\exp\mathcal{H}(\alpha,\sigma) Z = ∑ α v α ∑ σ exp H ( α , σ )
F ~ ( t , q , y ) = − 1 n log Z + β 2 t 4 , F ( t , q , y ) = E F ~ ( t , q , y ) , F 0 ( t , q ) = F ( t , q , y ) ∣ s = 0 . (51) \widetilde{F}(t,q,y)=-\frac{1}{n}\log Z+\frac{\beta^2t}{4},\qquad F(t,q,y)=\mathbb{E}\widetilde{F}(t,q,y),\qquad F_0(t,q)=F(t,q,y)\big|_{s=0}. \tag*{(51)} F ( t , q , y ) = − n 1 log Z + 4 β 2 t , F ( t , q , y ) = E F ( t , q , y ) , F 0 ( t , q ) = F ( t , q , y ) s = 0 . (51) We write ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ N K × { − 1 , 1 } V \mathbb{N}^{K}\times\{-1,1\}^{V} N K × { − 1 , 1 } V v α exp H ( α , σ ) v_{\alpha}\exp\mathcal{H}(\alpha,\sigma) v α exp H ( α , σ ) ( α ℓ , σ ℓ ) ℓ ≥ 1 (\alpha^{\ell},\sigma^{\ell})_{\ell\geq1} ( α ℓ , σ ℓ ) ℓ ≥ 1 ℓ , ℓ ′ \ell,\ell' ℓ , ℓ ′
ϑ ℓ ℓ ′ = z α ℓ ∧ α ℓ ′ , u ℓ ℓ ′ ∗ = 1 { α 1 ℓ = α 1 ℓ ′ } σ ℓ ⊙ σ ℓ ′ , \vartheta_{\ell\ell'}=z_{\alpha^{\ell}\wedge\alpha^{\ell'}},\qquad u^{*}_{\ell\ell'}=\mathbf{1}\{\alpha^{\ell}_{1}=\alpha^{\ell'}_{1}\}\sigma^{\ell}\odot\sigma^{\ell'}, ϑ ℓ ℓ ′ = z α ℓ ∧ α ℓ ′ , u ℓ ℓ ′ ∗ = 1 { α 1 ℓ = α 1 ℓ ′ } σ ℓ ⊙ σ ℓ ′ , ϱ x , ℓ ℓ ′ = ( S u ℓ ℓ ′ ∗ ) x = 1 { α 1 ℓ = α 1 ℓ ′ } ∑ y S x y σ y ℓ σ y ℓ ′ . \varrho_{x,\ell\ell'}=(Su^{*}_{\ell\ell'})_{x}=\mathbf{1}\{\alpha^{\ell}_{1}=\alpha^{\ell'}_{1}\}\sum_{y}S_{xy}\sigma^{\ell}_{y}\sigma^{\ell'}_{y}. ϱ x , ℓ ℓ ′ = ( S u ℓ ℓ ′ ∗ ) x = 1 { α 1 ℓ = α 1 ℓ ′ } y ∑ S x y σ y ℓ σ y ℓ ′ . and u ∗ = u 12 ∗ u^{*}=u^{*}_{12} u ∗ = u 12 ∗ u ∗ u^{*} u ∗ u = σ 1 ⊙ σ 2 u=\sigma^{1}\odot\sigma^{2} u = σ 1 ⊙ σ 2 f f f E ⟨ f ∣ ϑ 12 = z j ⟩ = w j − 1 E ⟨ f 1 { ϑ 12 = z j } ⟩ \mathbb{E}\langle f\mid\vartheta_{12}=z_{j}\rangle=w_{j}^{-1}\mathbb{E}\langle f\mathbf{1}\{\vartheta_{12}=z_{j}\}\rangle E ⟨ f ∣ ϑ 12 = z j ⟩ = w j − 1 E ⟨ f 1 { ϑ 12 = z j }⟩ Theorem 4.7 , applied with X = log ∑ σ exp H ( α , σ ) X=\log\sum_{\sigma}\exp\mathcal{H}(\alpha,\sigma) X = log ∑ σ exp H ( α , σ ) E ⟨ ⋅ ⟩ \mathbb{E}\langle\cdot\rangle E ⟨ ⋅ ⟩
Lemma 4.8. Let x ∈ V x\in V x ∈ V p ∈ N p\in\mathbb{N} p ∈ N p ≥ 1 p\geq1 p ≥ 1 c ∈ [ 0 , 1 ] c\in[0,1] c ∈ [ 0 , 1 ] k ( ϑ , ϱ ) = ( c ϑ + ( 1 − c ) ϱ ) p k(\vartheta,\varrho)=(c\vartheta+(1-c)\varrho)^{p} k ( ϑ , ϱ ) = ( c ϑ + ( 1 − c ) ϱ ) p ( G ( α , σ ) ) (G(\alpha,\sigma)) ( G ( α , σ )) γ ≠ ∅ \gamma\neq\varnothing γ = ∅ A \mathcal{A} A
E G ( α , σ ) G ( α ′ , σ ′ ) = k ( z α ∧ α ′ , 1 { α 1 = α 1 ′ } ∑ y S x y σ y σ y ′ ) . \mathbb{E}\,G(\alpha,\sigma)G(\alpha',\sigma')=k\left(z_{\alpha\wedge\alpha'},\mathbf{1}\{\alpha_{1}=\alpha'_{1}\}\sum_{y}S_{xy}\sigma_{y}\sigma'_{y}\right). E G ( α , σ ) G ( α ′ , σ ′ ) = k ( z α ∧ α ′ , 1 { α 1 = α 1 ′ } y ∑ S x y σ y σ y ′ ) . The covariance is at most one in absolute value, equals one on the diagonal, and vanishes if α 1 ≠ α 1 ′ \alpha_{1}\neq\alpha'_{1} α 1 = α 1 ′
Proof. Expanding the power, we have k = ∑ ℓ = 0 p ( p ℓ ) c p − ℓ ( 1 − c ) ℓ ϑ p − ℓ ϱ ℓ k=\sum_{\ell=0}^{p}\binom{p}{\ell}c^{p-\ell}(1-c)^{\ell}\vartheta^{p-\ell}\varrho^{\ell} k = ∑ ℓ = 0 p ( ℓ p ) c p − ℓ ( 1 − c ) ℓ ϑ p − ℓ ϱ ℓ 1 ≤ ℓ ≤ p 1\leq\ell\leq p 1 ≤ ℓ ≤ p ϑ p − ℓ ϱ ℓ \vartheta^{p-\ell}\varrho^{\ell} ϑ p − ℓ ϱ ℓ a ℓ ( α ∧ α ′ ) ⟨ F ( σ ) ⊗ ℓ , F ( σ ′ ) ⊗ ℓ ⟩ a_{\ell}(\alpha\wedge\alpha')\langle F(\sigma)^{\otimes\ell},F(\sigma')^{\otimes\ell}\rangle a ℓ ( α ∧ α ′ ) ⟨ F ( σ ) ⊗ ℓ , F ( σ ′ ) ⊗ ℓ ⟩ F ( σ ) = ( S x y 1 / 2 σ y ) y ∈ V F(\sigma)=(S_{xy}^{1/2}\sigma_{y})_{y\in V} F ( σ ) = ( S x y 1/2 σ y ) y ∈ V a ℓ ( k ) = z k p − ℓ 1 { k ≥ 1 } a_{\ell}(k)=z_{k}^{p-\ell}\mathbf{1}\{k\geq1\} a ℓ ( k ) = z k p − ℓ 1 { k ≥ 1 } ℓ = 0 \ell=0 ℓ = 0 a 0 ( α ∧ α ′ ) a_{0}(\alpha\wedge\alpha') a 0 ( α ∧ α ′ ) a 0 ( k ) = z k p a_{0}(k)=z_{k}^{p} a 0 ( k ) = z k p a ℓ a_{\ell} a ℓ { 0 , … , K } \{0,\ldots,K\} { 0 , … , K } 0 0 0 a a a F ′ F' F ′ { − 1 , 1 } V \{-1,1\}^{V} { − 1 , 1 } V ∑ k = 1 K ( a ( k ) − a ( k − 1 ) ) 1 / 2 ⟨ ξ α ∣ k , F ′ ( σ ) ⟩ \sum_{k=1}^{K}(a(k)-a(k-1))^{1/2}\langle\xi_{\alpha|k},F'(\sigma)\rangle ∑ k = 1 K ( a ( k ) − a ( k − 1 ) ) 1/2 ⟨ ξ α ∣ k , F ′ ( σ )⟩ ξ γ \xi_{\gamma} ξ γ a ( α ∧ α ′ ) ⟨ F ′ ( σ ) , F ′ ( σ ′ ) ⟩ a(\alpha\wedge\alpha')\langle F'(\sigma),F'(\sigma')\rangle a ( α ∧ α ′ ) ⟨ F ′ ( σ ) , F ′ ( σ ′ )⟩ G G G ∣ ϑ ∣ , ∣ ϱ ∣ ≤ 1 |\vartheta|,|\varrho|\leq1 ∣ ϑ ∣ , ∣ ϱ ∣ ≤ 1 ϑ = 1 \vartheta=1 ϑ = 1 ϱ = ∑ y S x y = 1 \varrho=\sum_{y}S_{xy}=1 ϱ = ∑ y S x y = 1 α 1 ≠ α 1 ′ \alpha_{1}\neq\alpha'_{1} α 1 = α 1 ′ ϑ = ϱ = 0 \vartheta=\varrho=0 ϑ = ϱ = 0 □ \square □
For ( x , ι ) ∈ S × I (x,\iota)\in S\times I ( x , ι ) ∈ S × I ι = ( c ι , p ι ) \iota=(c_{\iota},p_{\iota}) ι = ( c ι , p ι ) G x , ι G_{x,\iota} G x , ι x x x k ι ( ϑ , ϱ ) = ( c ι ϑ + ( 1 − c ι ) ϱ ) p ι k_{\iota}(\vartheta,\varrho)=(c_{\iota}\vartheta+(1-c_{\iota})\varrho)^{p_{\iota}} k ι ( ϑ , ϱ ) = ( c ι ϑ + ( 1 − c ι ) ϱ ) p ι ( x , ι ) (x,\iota) ( x , ι ) k x ι , ℓ ℓ ′ = k ι ( ϑ ℓ ℓ ′ , ϱ x , ℓ ℓ ′ ) k_{x\iota,\ell\ell'}=k_{\iota}(\vartheta_{\ell\ell'},\varrho_{x,\ell\ell'}) k x ι , ℓ ℓ ′ = k ι ( ϑ ℓ ℓ ′ , ϱ x , ℓ ℓ ′ ) [106 Section 4] , with the row overlap ϱ x \varrho_{x} ϱ x
Lemma 4.9. The following hold for t ≥ 0 t\geq0 t ≥ 0 q ∈ C K V q\in C_{K}^{V} q ∈ C K V y ∈ Y y\in Y y ∈ Y
(a) F F F t > 0 t>0 t > 0 C K V C_{K}^{V} C K V q q q y y y f f f E ⟨ f ⟩ \mathbb{E}\langle f\rangle E ⟨ f ⟩ ( t , q , y ) (t,q,y) ( t , q , y )
(b) For t > 0 t>0 t > 0 ∂ t F = β 2 4 n E ⟨ u ∗ ⋅ S u ∗ ⟩ \partial_tF=\frac{\beta^2}{4n}\mathbb{E}\langle u^*\cdot Su^*\rangle ∂ t F = 4 n β 2 E ⟨ u ∗ ⋅ S u ∗ ⟩ ∣ ∂ t F ∣ ≤ β 2 / 4 |\partial_tF|\le\beta^2/4 ∣ ∂ t F ∣ ≤ β 2 /4
(c) D q F = p D_qF=\mathbf{p} D q F = p p x 0 = 0 p_{x_0}=0 p x 0 = 0 p x j = E ⟨ u x ∗ ∣ ϑ 12 = z j ⟩ p_{x_j}=\mathbb{E}\langle u^*_{x}\mid\vartheta_{12}=z_j\rangle p x j = E ⟨ u x ∗ ∣ ϑ 12 = z j ⟩ 1 ≤ j ≤ K 1\le j\le K 1 ≤ j ≤ K p ∈ C K V ∩ [ 0 , 1 ] V K \mathbf{p}\in C^V_K\cap[0,1]^{V_K} p ∈ C K V ∩ [ 0 , 1 ] V K ∣ F ( t , q , y ) − F ( t , q ′ , y ) ∣ ≤ ∥ q − q ′ ∥ 1 |F(t,q,y)-F(t,q',y)|\le\|q-q'\|_1 ∣ F ( t , q , y ) − F ( t , q ′ , y ) ∣ ≤ ∥ q − q ′ ∥ 1
(d) F F F y y y ∂ y x ι F = − n − 1 s E ⟨ G x , ι ⟩ = − n − 1 s 2 y x ι ( 1 − E ⟨ k x ι , 12 ⟩ ) \partial_{y_{x_\iota}}F=-n^{-1}s\mathbb{E}\langle G_{x,\iota}\rangle=-n^{-1}s^2y_{x\iota}(1-\mathbb{E}\langle k_{x\iota,12}\rangle) ∂ y x ι F = − n − 1 s E ⟨ G x , ι ⟩ = − n − 1 s 2 y x ι ( 1 − E ⟨ k x ι , 12 ⟩) ∣ F ( t , q , y ) − F 0 ( t , q ) ∣ ≤ r s |F(t,q,y)-F_0(t,q)|\le r_s ∣ F ( t , q , y ) − F 0 ( t , q ) ∣ ≤ r s r s = 9 ∣ S ∣ ∣ I ∣ s 2 / ( 4 n ) r_s=9|\mathcal{S}||\mathcal{I}|s^2/(4n) r s = 9∣ S ∣∣ I ∣ s 2 / ( 4 n )
(e) We have
F 0 ( t , 0 ) = β 2 t 4 − 1 n ζ 0 log E Z G ( β t , h ) ζ 0 ≤ β 2 t 4 − p G ( β t , h ) . F_0(t,0)=\frac{\beta^2t}{4}-\frac{1}{n\zeta_0}\log\mathbb{E}Z_G(\beta\sqrt{t},h)^{\zeta_0}\le\frac{\beta^2t}{4}-p_G(\beta\sqrt{t},h). F 0 ( t , 0 ) = 4 β 2 t − n ζ 0 1 log E Z G ( β t , h ) ζ 0 ≤ 4 β 2 t − p G ( β t , h ) . (f) We have F 0 ( 0 , q ) = n − 1 ∑ x ψ h ( q ~ x ) F_0(0,q)=n^{-1}\sum_x\psi_h(\widetilde q_x) F 0 ( 0 , q ) = n − 1 ∑ x ψ h ( q x ) q ~ \widetilde q q
Proof. (a) We write every Gaussian variable in (4.9) as an amplitude times a fixed standard Gaussian variable; the amplitudes are β t \beta\sqrt{t} β t ( 2 q x 1 ) 1 / 2 (2q_{x_1})^{1/2} ( 2 q x 1 ) 1/2 ( 2 ( q x k − q x , k − 1 ) ) 1 / 2 (2(q_{x_k}-q_{x,k-1}))^{1/2} ( 2 ( q x k − q x , k − 1 ) ) 1/2 s y x ι sy_{x\iota} s y x ι X = log ∑ σ exp H ( α , σ ) X=\log\sum_\sigma\exp\mathcal{H}(\alpha,\sigma) X = log ∑ σ exp H ( α , σ ) F = − X 0 / n + β 2 t / 4 F=-X_0/n+\beta^2t/4 F = − X 0 / n + β 2 t /4 X 0 X_0 X 0 X X X K K K X 0 X_0 X 0 β t \beta\sqrt{t} β t y y y 2 q x 1 2q_{x_1} 2 q x 1 2 ( q x k − q x , k − 1 ) 2(q_{x_k}-q_{x,k-1}) 2 ( q x k − q x , k − 1 ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) F F F K K K Z ‾ α = sup K ∑ σ exp H ( α , σ ) \overline Z_\alpha=\sup_K\sum_\sigma\exp\mathcal{H}(\alpha,\sigma) Z α = sup K ∑ σ exp H ( α , σ ) α \alpha α ( v α ) (v_\alpha) ( v α ) E ∑ α v α Z ‾ α = E Z ‾ α < ∞ \mathbb{E}\sum_\alpha v_\alpha\overline Z_\alpha=\mathbb{E}\overline Z_\alpha<\infty E ∑ α v α Z α = E Z α < ∞ ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ K K K K K K ⟨ f ⟩ \langle f\rangle ⟨ f ⟩ E ⟨ f ⟩ \mathbb{E}\langle f\rangle E ⟨ f ⟩ a a a ∣ ∂ a H ∣ |\partial_a\mathcal{H}| ∣ ∂ a H ∣ ∣ ∂ a H ∣ 2 |\partial_a\mathcal{H}|^2 ∣ ∂ a H ∣ 2 a a a log Z \log Z log Z ⟨ ∂ a H ⟩ \langle\partial_a\mathcal{H}\rangle ⟨ ∂ a H ⟩ H + ∑ x q x K \mathcal{H}+\sum_xq_{xK} H + ∑ x q x K log Z + ∑ x q x K \log Z+\sum_xq_{xK} log Z + ∑ x q x K ⟨ ( ∂ a H − ⟨ ∂ a H ⟩ ) 2 ⟩ \langle(\partial_a\mathcal{H}-\langle\partial_a\mathcal{H}\rangle)^2\rangle ⟨( ∂ a H − ⟨ ∂ a H ⟩ ) 2 ⟩
The random function log Z + ∑ x q x K \log Z+\sum_xq_{xK} log Z + ∑ x q x K E ∣ log Z ∣ < ∞ \mathbb{E}|\log Z|<\infty E ∣ log Z ∣ < ∞ F F F F ~ \widetilde F F ( v α ) (v_\alpha) ( v α ) Σ = N K × { − 1 , 1 } V \Sigma=\mathbb{N}^K\times\{-1,1\}^V Σ = N K × { − 1 , 1 } V
(b) The process H ( α 1 ) ( σ ) \mathcal{H}^{(\alpha_1)}(\sigma) H ( α 1 ) ( σ ) 1 { α 1 = α 1 ′ } ( σ ⊙ σ ′ ) ⋅ S ( σ ⊙ σ ′ ) / 2 \mathbf{1}\{\alpha_1=\alpha'_1\}(\sigma\odot\sigma')\cdot S(\sigma\odot\sigma')/2 1 { α 1 = α 1 ′ } ( σ ⊙ σ ′ ) ⋅ S ( σ ⊙ σ ′ ) /2 n / 2 n/2 n /2 r = 1 r=1 r = 1
∂ t E log Z = β 2 t E ⟨ H ( α 1 ) ( σ ) ⟩ = β 2 2 ( n 2 − 1 2 E ⟨ u ∗ ⋅ S u ∗ ⟩ ) , \partial_t\mathbb{E}\log Z=\frac{\beta}{2\sqrt{t}}\mathbb{E}\langle\mathcal{H}^{(\alpha_1)}(\sigma)\rangle=\frac{\beta^2}{2}\left(\frac{n}{2}-\frac{1}{2}\mathbb{E}\langle u^*\cdot Su^*\rangle\right), ∂ t E log Z = 2 t β E ⟨ H ( α 1 ) ( σ )⟩ = 2 β 2 ( 2 n − 2 1 E ⟨ u ∗ ⋅ S u ∗ ⟩ ) , which gives the formula. Since ∥ S ∥ ≤ 1 \lVert S\rVert\le1 ∥ S ∥ ≤ 1 ∣ u x ∗ ∣ ≤ 1 |u_x^*|\le1 ∣ u x ∗ ∣ ≤ 1 ∣ u ∗ ⋅ S u ∗ ∣ ≤ n |u^*\cdot Su^*|\le n ∣ u ∗ ⋅ S u ∗ ∣ ≤ n
(c) Suppose first that all the increments q x 1 , q x k − q x , k − 1 q_{x1}, q_{xk} - q_{x,k-1} q x 1 , q x k − q x , k − 1 1 ≤ j ≤ K 1 \le j \le K 1 ≤ j ≤ K (48) the amplitude derivatives give, exactly as in [56 proof of Lemma 6.2] ,
∂ q x j E log Z = E ⟨ 1 − 1 { α 1 ∧ α 2 ≥ j } u x ⟩ − E ⟨ 1 − 1 { α 1 ∧ α 2 ≥ j + 1 } u x ⟩ 1 { j < K } − 1 { j = K } = − E ⟨ 1 { α 1 ∧ α 2 = j } u x ⟩ . \partial_{q_{xj}}\mathbb{E}\log Z = \mathbb{E}\langle1-\mathbf{1}\{\alpha^1 \wedge\alpha^2 \ge j\}u_x\rangle
-\mathbb{E}\langle1-\mathbf{1}\{\alpha^1 \wedge\alpha^2 \ge j+1\}u_x\rangle\mathbf{1}\{j<K\}-\mathbf{1}\{j=K\}
= -\mathbb{E}\langle\mathbf{1}\{\alpha^1 \wedge\alpha^2=j\}u_x\rangle. ∂ q x j E log Z = E ⟨ 1 − 1 { α 1 ∧ α 2 ≥ j } u x ⟩ − E ⟨ 1 − 1 { α 1 ∧ α 2 ≥ j + 1 } u x ⟩ 1 { j < K } − 1 { j = K } = − E ⟨ 1 { α 1 ∧ α 2 = j } u x ⟩ . where u = σ 1 ⊙ σ 2 u=\sigma^1\odot\sigma^2 u = σ 1 ⊙ σ 2 C K V C_K^V C K V j ≥ 1 j\ge1 j ≥ 1 1 { α 1 ∧ α 2 = j } u x = 1 { ϑ 12 = z j } u x ∗ \mathbf{1}\{\alpha^1\wedge\alpha^2=j\}u_x=\mathbf{1}\{\vartheta_{12}=z_j\}u_x^* 1 { α 1 ∧ α 2 = j } u x = 1 { ϑ 12 = z j } u x ∗ F F F q x 0 q_{x0} q x 0 (47) , this proves D q F = p D_qF=p D q F = p
It remains to show that p x ∈ C K ∩ [ 0 , 1 ] K + 1 p_x\in C_K\cap[0,1]^{K+1} p x ∈ C K ∩ [ 0 , 1 ] K + 1 [108 Proposition 3.6] , and we follow its proof through the cascade recursion. Fix x x x X = L ( Y x , ω ) X=L(Y_x,\omega) X = L ( Y x , ω ) Y x = γ x ( α ) Y_x=\gamma_x(\alpha) Y x = γ x ( α )
L ( s , ω ) = log ∑ σ exp ( s σ x + H ′ ( σ ) ) , L(s,\omega)=\log\sum_{\sigma}\exp\left(s\sigma_x+\mathcal{H}'(\sigma)\right), L ( s , ω ) = log σ ∑ exp ( s σ x + H ′ ( σ ) ) , H ′ \mathcal{H}' H ′ H \mathcal{H} H γ x ( α ) σ x \gamma_x(\alpha)\sigma_x γ x ( α ) σ x ω \omega ω α \alpha α ξ x , α ∣ k \xi_{x,\alpha|k} ξ x , α ∣ k L L L (22) . The integrations defining X 0 X_0 X 0 Theorem 4.7 are then exactly the recursion (23) with Δ 1 = 2 q x 1 \Delta_1=2q_{x1} Δ 1 = 2 q x 1 Δ k = 2 ( q x k − q x , k − 1 ) \Delta_k=2(q_{xk}-q_{x,k-1}) Δ k = 2 ( q x k − q x , k − 1 ) k ≥ 2 k\ge2 k ≥ 2 0 0 0 X 0 = L 0 ( 0 ) X_0=L_0(0) X 0 = L 0 ( 0 ) Lemma 3.1 (c), for 1 ≤ j < K 1\le j<K 1 ≤ j < K
∂ q x j X 0 = 2 ( ∂ Δ j − ∂ Δ j + 1 ) L 0 ( 0 ) = − w j Λ 0 , j ( 0 ) , \partial_{q_{xj}}X_0=2(\partial_{\Delta_j}-\partial_{\Delta_{j+1}})L_0(0)=-w_j\Lambda_{0,j}(0), ∂ q x j X 0 = 2 ( ∂ Δ j − ∂ Δ j + 1 ) L 0 ( 0 ) = − w j Λ 0 , j ( 0 ) , and ∂ q x K X 0 = 2 ∂ Δ K L 0 ( 0 ) − 1 = − w K Λ 0 , K ( 0 ) \partial_{q_{xK}}X_0=2\partial_{\Delta_K}L_0(0)-1=-w_K\Lambda_{0,K}(0) ∂ q x K X 0 = 2 ∂ Δ K L 0 ( 0 ) − 1 = − w K Λ 0 , K ( 0 ) − 1 -1 − 1 − q x K -q_{xK} − q x K H ′ \mathcal{H}' H ′ p x j = Λ 0 , j ( 0 ) p_{xj}=\Lambda_{0,j}(0) p x j = Λ 0 , j ( 0 ) 1 ≤ j ≤ K 1\le j\le K 1 ≤ j ≤ K Lemma 3.1 (a) shows that 0 ≤ p x 1 ≤ ⋯ ≤ p x K ≤ 1 0\le p_{x1}\le\cdots\le p_{xK}\le1 0 ≤ p x 1 ≤ ⋯ ≤ p x K ≤ 1 p x 0 = 0 p_{x0}=0 p x 0 = 0 p x ∈ C K ∩ [ 0 , 1 ] K + 1 p_x\in C_K\cap[0,1]^{K+1} p x ∈ C K ∩ [ 0 , 1 ] K + 1 (20) .
(d) Concavity holds because log Z \log Z log Z y y y (48) and Lemma 4.8 , the covariance of G x , ι G_{x,\iota} G x , ι H \mathcal{H} H s y x ι κ ι s y_{x\iota} \kappa_{\iota} s y x ι κ ι ∂ y x ι F \partial_{y_{x\iota}}F ∂ y x ι F s s s θ s \theta s θ s
∣ ∂ θ F ∣ ≤ 2 θ n ∑ x , ι s 2 y x ι 2 ≤ 9 θ 2 n ∣ S ∣ ∣ I ∣ s 2 , |\partial_\theta F|\le\frac{2\theta}{n}\sum_{x,\iota}s^2y_{x\iota}^2\le\frac{9\theta}{2n}|S||I|s^2, ∣ ∂ θ F ∣ ≤ n 2 θ x , ι ∑ s 2 y x ι 2 ≤ 2 n 9 θ ∣ S ∣∣ I ∣ s 2 , and integrating over θ ∈ [ 0 , 1 ] \theta\in[0,1] θ ∈ [ 0 , 1 ]
(e) If q = 0 q=0 q = 0 s = 0 s=0 s = 0 X = log Z G ( α 1 ) X=\log Z_G^{(\alpha_1)} X = log Z G ( α 1 ) Z G ( a ) Z_G^{(a)} Z G ( a ) Z G ( β t , h ) Z_G(\beta\sqrt{t},h) Z G ( β t , h ) g ( a ) g^{(a)} g ( a ) X k = X X_k=X X k = X k ≥ 1 k\ge1 k ≥ 1 X 0 = ζ 0 − 1 log E Z G ( β t , h ) ζ 0 X_0=\zeta_0^{-1}\log\mathbb{E}Z_G(\beta\sqrt{t},h)^{\zeta_0} X 0 = ζ 0 − 1 log E Z G ( β t , h ) ζ 0 Theorem 4.7 . By Jensen’s inequality, log E Z G ζ 0 ≥ ζ 0 E log Z G \log\mathbb{E}Z_G^{\zeta_0}\ge\zeta_0\mathbb{E}\log Z_G log E Z G ζ 0 ≥ ζ 0 E log Z G
(f) If t = 0 t=0 t = 0 s = 0 s=0 s = 0 X = ∑ x ( log 2 cosh ( h + Y x ( α ) ) − q x K ) X=\sum_x\left(\log2\cosh(h+Y_x(\alpha))-q_xK\right) X = ∑ x ( log 2 cosh ( h + Y x ( α )) − q x K ) x x x [56 Corollary 5.26] , so that X 0 = ∑ x X 0 , x X_0=\sum_x X_{0,x} X 0 = ∑ x X 0 , x X 0 , x X_{0,x} X 0 , x § 4.3 with v = 2 q ~ x = ( 0 , 2 q x 1 , … , 2 q x K ) v=2\widetilde q_x=(0,2q_{x1},\ldots,2q_{xK}) v = 2 q x = ( 0 , 2 q x 1 , … , 2 q x K ) h h h q x K q_{xK} q x K (28) , X 0 , x = − ψ h ( q ~ x ) X_{0,x}=-\psi_h(\widetilde q_x) X 0 , x = − ψ h ( q x )
The next lemma is the reason for attaching independent copies of the couplings to the first-level branches: the fluctuations of log Z \log Z log Z n n n
Lemma 4.10. There exists C f l C_{\mathrm{fl}} C fl ζ 0 \zeta_0 ζ 0 Var ( log Z ) ≤ C f l 2 \operatorname{Var}(\log Z)\le C_{\mathrm{fl}}^2 Var ( log Z ) ≤ C fl 2 ( t , q , y ) (t,q,y) ( t , q , y )
E ∣ F ~ ( t , q , y ) − F ( t , q , y ) ∣ ≤ C f l n . \mathbb{E}\left|\widetilde F(t,q,y)-F(t,q,y)\right|\le\frac{C_{\mathrm{fl}}}{n}. E F ( t , q , y ) − F ( t , q , y ) ≤ n C fl . Proof. Let Z α = ∑ σ exp H ( α , σ ) Z_\alpha=\sum_\sigma\exp\mathcal{H}(\alpha,\sigma) Z α = ∑ σ exp H ( α , σ ) γ ∈ N k \gamma\in\mathbb{N}^k γ ∈ N k 1 ≤ k ≤ K 1\le k\le K 1 ≤ k ≤ K
U γ = ∑ α : α ∣ k = γ ( ∏ i = k + 1 K u α ∣ i ) Z α . U_\gamma=\sum_{\alpha:\alpha|_k=\gamma}\left(\prod_{i=k+1}^{K}u_{\alpha|_i}\right)Z_\alpha. U γ = α : α ∣ k = γ ∑ ( i = k + 1 ∏ K u α ∣ i ) Z α . and let T γ T_\gamma T γ Z α Z_\alpha Z α log Z = log ∑ a ≥ 1 u ∅ a U a − log ∑ a ≥ 1 u ∅ a T a \log Z=\log\sum_{a\ge1}u_{\varnothing a}U_a-\log\sum_{a\ge1}u_{\varnothing a}T_a log Z = log ∑ a ≥ 1 u ∅ a U a − log ∑ a ≥ 1 u ∅ a T a ( U a , T a ) (U_a,T_a) ( U a , T a ) a ≥ 1 a\ge1 a ≥ 1 ( u ∅ a ) a (u_{\varnothing a})_a ( u ∅ a ) a
We claim that E U γ ζ k − 1 < ∞ \mathbb{E}U_\gamma^{\zeta_k-1}<\infty E U γ ζ k − 1 < ∞ γ ∈ N k \gamma\in\mathbb{N}^k γ ∈ N k k = K k=K k = K Z α Z_\alpha Z α k + 1 k+1 k + 1 γ ∈ N k \gamma\in\mathbb{N}^k γ ∈ N k U γ = ∑ i u γ i U γ i U_\gamma=\sum_i u_{\gamma i}U_{\gamma i} U γ = ∑ i u γ i U γ i γ \gamma γ γ \gamma γ U γ i U_{\gamma i} U γ i m γ m_\gamma m γ ζ k \zeta_k ζ k (49) , E U γ ζ k − 1 = E [ m γ ζ k − 1 / ζ k ] E ( ∑ i u γ i ) ζ k − 1 \mathbb{E}U_\gamma^{\zeta_k-1}=\mathbb{E}\left[m_\gamma^{\zeta_k-1/\zeta_k}\right]\mathbb{E}\left(\sum_i u_{\gamma i}\right)^{\zeta_k-1} E U γ ζ k − 1 = E [ m γ ζ k − 1/ ζ k ] E ( ∑ i u γ i ) ζ k − 1 (49) . The same holds for T γ T_\gamma T γ
Let S = ∑ a u ∅ a S=\sum_a u_{\varnothing a} S = ∑ a u ∅ a (49) , log ∑ a u ∅ a U a \log\sum_a u_{\varnothing a}U_a log ∑ a u ∅ a U a log ∑ a u ∅ a T a \log\sum_a u_{\varnothing a}T_a log ∑ a u ∅ a T a log S \log S log S Var ( log Z ) ≤ 4 Var ( log S ) ≤ 4 E ( log S ) 2 \operatorname{Var}(\log Z)\le4\operatorname{Var}(\log S)\le4\mathbb{E}(\log S)^2 Var ( log Z ) ≤ 4 Var ( log S ) ≤ 4 E ( log S ) 2 E e − λ S = exp ( − Γ ( 1 − ζ 0 ) λ ζ 0 ) \mathbb{E}e^{-\lambda S}=\exp\left(-\Gamma(1-\zeta_0)\lambda^{\zeta_0}\right) E e − λ S = exp ( − Γ ( 1 − ζ 0 ) λ ζ 0 ) λ ≥ 0 \lambda\ge0 λ ≥ 0 λ ∈ ( 0 , ∞ ) \lambda\in(0,\infty) λ ∈ ( 0 , ∞ )
E S − 1 = ∫ 0 ∞ exp ( − Γ ( 1 − ζ 0 ) λ ζ 0 ) d λ < ∞ . \mathbb{E}S^{-1}=\int_0^\infty\exp\left(-\Gamma(1-\zeta_0)\lambda^{\zeta_0}\right)\,d\lambda<\infty. E S − 1 = ∫ 0 ∞ exp ( − Γ ( 1 − ζ 0 ) λ ζ 0 ) d λ < ∞. Further, E S ζ 0 / 2 < ∞ \mathbb{E}S^{\zeta_0/2}<\infty E S ζ 0 /2 < ∞ (49) . Since ( log s ) 2 ≤ 16 ζ 0 − 2 ( s ζ 0 / 2 + s − 1 ) (\log s)^2\le16\zeta_0^{-2}(s^{\zeta_0/2}+s^{-1}) ( log s ) 2 ≤ 16 ζ 0 − 2 ( s ζ 0 /2 + s − 1 ) s > 0 s>0 s > 0 log S \log S log S ζ 0 \zeta_0 ζ 0 Var ( log Z ) \operatorname{Var}(\log Z) Var ( log Z ) E ∣ F ~ − F ∣ \mathbb{E}|\widetilde F-F| E ∣ F − F ∣ F ~ − F = − n − 1 ( log Z − E log Z ) \widetilde F-F=-n^{-1}(\log Z-\mathbb{E}\log Z) F − F = − n − 1 ( log Z − E log Z )
Synchronization
We use the following bound of Mourrat on conditional variances, which is [106 Proposition 5.5] , with the hypothesis of [106 Theorem 5.3] . It is a finitary form of the synchronization theorem of Panchenko [115 Theorem 4] , which was extended to Potts and vector spins in [116 , 117 ] . Its proof uses Panchenko’s ultrametricity theorem [113 ] for measures that satisfy the Ghirlanda–Guerra identities [74 ] . Let ( c i ) i ≥ 1 (c_i)_{i\ge1} ( c i ) i ≥ 1 Q ∩ [ 0 , 1 ] \mathbb{Q} \cap[0,1] Q ∩ [ 0 , 1 ]
Proposition 4.11. For every ε > 0 \varepsilon> 0 ε > 0 there exists δ > 0 \delta> 0 δ > 0 such that the following holds. Let G \mathcal{G} G ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ G ⊗ N \mathcal{G}^{\otimes\mathbb{N}} G ⊗ N ( σ ℓ 1 , σ ℓ 2 ) ℓ ≥ 1 (\sigma_\ell^1,\sigma_\ell^2)_{\ell\ge1} ( σ ℓ 1 , σ ℓ 2 ) ℓ ≥ 1 E \mathbb{E} E G \mathcal{G} G a ∈ { 1 , 2 } a\in\{1,2\} a ∈ { 1 , 2 } ℓ , ℓ ′ , r ≥ 1 \ell,\ell',r\ge1 ℓ , ℓ ′ , r ≥ 1 O a ℓ ℓ ′ = σ a ℓ ⋅ σ a ℓ ′ O_a^{\ell\ell'}=\sigma_a^\ell\cdot\sigma_a^{\ell'} O a ℓ ℓ ′ = σ a ℓ ⋅ σ a ℓ ′ O ≤ r = ( O a ℓ ℓ ′ ) a ∈ { 1 , 2 } , ℓ , ℓ ′ ≤ r O^{\le r}=(O_a^{\ell\ell'})_{a\in\{1,2\},\,\ell,\ell'\le r} O ≤ r = ( O a ℓ ℓ ′ ) a ∈ { 1 , 2 } , ℓ , ℓ ′ ≤ r r , i 1 , i 2 , p ∈ { 1 , … , ⌊ δ − 1 ⌋ } r,i_1,i_2,p\in\{1,\ldots,\lfloor\delta^{-1}\rfloor\} r , i 1 , i 2 , p ∈ { 1 , … , ⌊ δ − 1 ⌋} f : R 2 × r × r → R f:\mathbb{R}^{2\times r\times r}\to\mathbb{R} f : R 2 × r × r → R ∣ f ∣ ≤ 1 |f|\le1 ∣ f ∣ ≤ 1 k ℓ ℓ ′ = ( c i 1 O 1 ℓ ℓ ′ + c i 2 O 2 ℓ ℓ ′ ) p k_{\ell\ell'}=(c_{i_1}O_1^{\ell\ell'}+c_{i_2}O_2^{\ell\ell'})^p k ℓ ℓ ′ = ( c i 1 O 1 ℓ ℓ ′ + c i 2 O 2 ℓ ℓ ′ ) p
∣ E ⟨ f ( O ≤ r ) k 1 , r + 1 ⟩ − 1 r E ⟨ f ( O ≤ r ) ⟩ E ⟨ k 12 ⟩ − 1 r ∑ ℓ = 2 r E ⟨ f ( O ≤ r ) k 1 ℓ ⟩ ∣ ≤ δ . (52) \left|\mathbb{E}\left\langle f(O^{\le r})k_{1,r+1}\right\rangle-\frac{1}{r}\mathbb{E}\left\langle f(O^{\le r})\right\rangle\mathbb{E}\langle k_{12}\rangle-\frac{1}{r}\sum_{\ell=2}^{r}\mathbb{E}\left\langle f(O^{\le r})k_{1\ell}\right\rangle\right|\le\delta. \tag*{(52)} E ⟨ f ( O ≤ r ) k 1 , r + 1 ⟩ − r 1 E ⟨ f ( O ≤ r ) ⟩ E ⟨ k 12 ⟩ − r 1 ℓ = 2 ∑ r E ⟨ f ( O ≤ r ) k 1 ℓ ⟩ ≤ δ . (52) Suppose further that the law of O 2 12 O_2^{12} O 2 12 k − 1 ∑ ℓ = 1 k δ q ℓ k^{-1}\sum_{\ell=1}^{k}\delta_{q_\ell} k − 1 ∑ ℓ = 1 k δ q ℓ k ≥ 1 k\ge1 k ≥ 1 − 1 = q 0 < q 1 < ⋯ < q k ≤ 1 -1=q_0<q_1<\cdots<q_k\le1 − 1 = q 0 < q 1 < ⋯ < q k ≤ 1
E ⟨ ( O 1 12 − E ⟨ O 1 12 ∣ O 2 12 ⟩ ) 2 ⟩ ≤ 12 k + ε k 2 max 0 ≤ ℓ < k ( q ℓ + 1 − q ℓ ) − 1 . \mathbb{E}\left\langle\left(O_1^{12}-\mathbb{E}\left\langle O_1^{12}\mid O_2^{12}\right\rangle\right)^2\right\rangle\le\frac{12}{k}+\varepsilon k^2\max_{0\le\ell<k}(q_{\ell+1}-q_\ell)^{-1}. E ⟨ ( O 1 12 − E ⟨ O 1 12 ∣ O 2 12 ⟩ ) 2 ⟩ ≤ k 12 + ε k 2 0 ≤ ℓ < k max ( q ℓ + 1 − q ℓ ) − 1 . The law of O 2 12 O_2^{12} O 2 12 E ⟨ ⋅ ⟩ \mathbb{E}\langle\cdot\rangle E ⟨ ⋅ ⟩ [106 ] the number of replicas is denoted by n n n ( λ h ) (\lambda_h) ( λ h ) O a ℓ ℓ ′ O_a^{\ell\ell'} O a ℓ ℓ ′ R a ℓ , ℓ ′ R_a^{\ell,\ell'} R a ℓ , ℓ ′ G G G δ > 0 \delta>0 δ > 0
I δ = { ( c i 2 c i 1 + c i 2 , p ) : 1 ≤ i 1 , i 2 , p ≤ ⌊ δ − 1 ⌋ , c i 1 + c i 2 > 0 } ⊆ [ 0 , 1 ] × N . (53) I_\delta=\left\{\left(\frac{c_{i_2}}{c_{i_1}+c_{i_2}},p\right):1\le i_1,i_2,p\le\lfloor\delta^{-1}\rfloor,\ c_{i_1}+c_{i_2}>0\right\}\subseteq[0,1]\times\mathbb{N}. \tag*{(53)} I δ = { ( c i 1 + c i 2 c i 2 , p ) : 1 ≤ i 1 , i 2 , p ≤ ⌊ δ − 1 ⌋ , c i 1 + c i 2 > 0 } ⊆ [ 0 , 1 ] × N . (53) For x ∈ V x\in V x ∈ V
V x = ∑ j = 0 K w j Var ( q x , 12 ∣ ϑ 12 = z j ) , \mathcal{V}_x=\sum_{j=0}^{K}w_j\operatorname{Var}\left(q_{x,12}\mid\vartheta_{12}=z_j\right), V x = j = 0 ∑ K w j Var ( q x , 12 ∣ ϑ 12 = z j ) , where the variance is computed under E ⟨ ⋅ ∣ ϑ 12 = z j ⟩ \mathbb{E}\langle\cdot\mid\vartheta_{12}=z_j\rangle E ⟨ ⋅ ∣ ϑ 12 = z j ⟩
Gaussian perturbations and the envelope In this subsection S \mathcal{S} S I I I s s s ε ∈ ( 0 , 1 / 8 ] \varepsilon\in(0,1/8] ε ∈ ( 0 , 1/8 ]
F ⋆ ( t , q ) = min y ∈ Y { F ( t , q , y ) + s 2 n ε ∑ x ∈ S , ι ∈ I ( y x ι − 1 ) 2 } . (54) F_\star(t,q)=\min_{y\in\mathcal{Y}}\left\{F(t,q,y)+\frac{s^2}{n\varepsilon}\sum_{x\in\mathcal{S},\iota\in I}(y_{x\iota}-1)^2\right\}. \tag*{(54)} F ⋆ ( t , q ) = y ∈ Y min ⎩ ⎨ ⎧ F ( t , q , y ) + n ε s 2 x ∈ S , ι ∈ I ∑ ( y x ι − 1 ) 2 ⎭ ⎬ ⎫ . (54) By Lemma 4.9 (a), the minimum is attained. By Lemma 4.9 , since the penalty does not depend on ( t , q ) (t,q) ( t , q ) y = 1 y=1 y = 1
∣ F ⋆ ( t , q ) − F ⋆ ( t ′ , q ′ ) ∣ ≤ β 2 4 ∣ t − t ′ ∣ + ∥ q − q ′ ∥ 1 , ∣ F ⋆ ( t , q ) − F 0 ( t , q ) ∣ ≤ r s . (55) \left|F_\star(t,q)-F_\star(t',q')\right| \le\frac{\beta^2}{4}\left|t-t'\right|+\lVert q-q'\rVert_1,\qquad\left|F_\star(t,q)-F_0(t,q)\right|\le r_s. \tag*{(55)} ∣ F ⋆ ( t , q ) − F ⋆ ( t ′ , q ′ ) ∣ ≤ 4 β 2 ∣ t − t ′ ∣ + ∥ q − q ′ ∥ 1 , ∣ F ⋆ ( t , q ) − F 0 ( t , q ) ∣ ≤ r s . (55) Small Gaussian perturbations that enforce the Ghirlanda–Guerrera identities [74 ] through the concentration of the free energy are a standard tool; see [133 ] and [115 Section 3] . The proof of Lemma 4.12 follows Steps 1–3 of the proof of [106 Theorem 4.1] . There, the lower bound on the second derivative of the free energy in the perturbation parameters comes from a test function that touches the free energy; here it comes from the minimality of y ⋆ y_\star y ⋆ (54) . Because of Lemma 4.10 , the strength s s s n n n
Lemma 4.12. Suppose that s ≥ 2 s\ge2 s ≥ 2 ( t , q ) ∈ ( 0 , 2 ) × C K V (t,q)\in(0,2)\times C_K^V ( t , q ) ∈ ( 0 , 2 ) × C K V y ⋆ y_\star y ⋆ (54) . Then ∣ y ⋆ x ι − 1 ∣ ≤ 3 ϵ / 2 \left|y_{\star x\iota}-1\right|\le3\epsilon/2 ∣ y ⋆ x ι − 1 ∣ ≤ 3 ϵ /2 ( x , ι ) (x,\iota) ( x , ι ) ( t , q , y ⋆ ) (t,q,y_\star) ( t , q , y ⋆ ) ( x , ι ) ∈ S × I (x,\iota)\in S\times I ( x , ι ) ∈ S × I r ≥ 1 r\ge1 r ≥ 1 f f f r r r ∣ f ∣ ≤ 1 \left|f\right|\le1 ∣ f ∣ ≤ 1
∣ E ⟨ f k x ι , 1 , r + 1 ⟩ − 1 r E ⟨ f ⟩ E ⟨ k x ι , 12 ⟩ − 1 r ∑ ℓ = 2 r E ⟨ f k x ι , 1 ℓ ⟩ ∣ ≤ C G G s ϵ , (56) \left|\mathbb{E}\langle f k_{x\iota,1,r+1}\rangle-\frac{1}{r}\mathbb{E}\langle f\rangle\mathbb{E}\langle k_{x\iota,12}\rangle-\frac{1}{r}\sum_{\ell=2}^{r}\mathbb{E}\langle f k_{x\iota,1\ell}\rangle\right|\le\frac{C_{\mathrm{GG}}}{s\sqrt{\epsilon}}, \tag*{(56)} E ⟨ f k x ι , 1 , r + 1 ⟩ − r 1 E ⟨ f ⟩ E ⟨ k x ι , 12 ⟩ − r 1 ℓ = 2 ∑ r E ⟨ f k x ι , 1 ℓ ⟩ ≤ s ϵ C GG , (56) where C G G C_{\mathrm{GG}} C GG ζ 0 \zeta_0 ζ 0
Proof. Let λ = s 2 / ( n ϵ ) \lambda=s^2/(n\epsilon) λ = s 2 / ( n ϵ ) Lemma 4.9 (d) and Lemma 4.8 , ∣ ∂ y x ι F ∣ ≤ 2 s 2 y x ι / n ≤ 3 s 2 / n \left|\partial_{y_{x\iota}}F\right|\le2s^2y_{x\iota}/n\le3s^2/n ∣ ∂ y x ι F ∣ ≤ 2 s 2 y x ι / n ≤ 3 s 2 / n y ⋆ x ι = 3 / 2 y_{\star x\iota}=3/2 y ⋆ x ι = 3/2 ∂ y x ι F + λ ≤ 0 \partial_{y_{x\iota}}F+\lambda\le0 ∂ y x ι F + λ ≤ 0 λ ≤ 3 s 2 / n \lambda\le3s^2/n λ ≤ 3 s 2 / n ϵ ≤ 1 / 8 \epsilon\le1/8 ϵ ≤ 1/8 y ⋆ x ι = 1 / 2 y_{\star x\iota}=1/2 y ⋆ x ι = 1/2 y ⋆ y_\star y ⋆ ∣ y ⋆ x ι − 1 ∣ = ∣ ∂ y x ι F ∣ / ( 2 λ ) ≤ 3 ϵ / 2 ≤ 3 / 16 \left|y_{\star x\iota}-1\right|=\left|\partial_{y_{x\iota}}F\right|/(2\lambda)\le3\epsilon/2\le3/16 ∣ y ⋆ x ι − 1 ∣ = ∣ ∂ y x ι F ∣ / ( 2 λ ) ≤ 3 ϵ /2 ≤ 3/16 y ⋆ ± z e x ι ∈ Y y_\star\pm z e_{x\iota}\in Y y ⋆ ± z e x ι ∈ Y z ∈ [ 0 , 1 / 4 ] z\in[0,1/4] z ∈ [ 0 , 1/4 ]
Fix ( x , ι ) (x,\iota) ( x , ι ) g ( y ) = F ( t , q , y ) g(y)=F(t,q,y) g ( y ) = F ( t , q , y ) g ~ ( y ) = F ~ ( t , q , y ) \widetilde g(y)=\widetilde F(t,q,y) g ( y ) = F ( t , q , y ) y y y y ⋆ y_\star y ⋆ y x ι = y y_{x\iota}=y y x ι = y y ⋆ y_\star y ⋆ ∣ z ∣ ≤ 1 / 4 \left|z\right|\le1/4 ∣ z ∣ ≤ 1/4
g ( y ⋆ + z ) ≥ g ( y ⋆ ) + g ′ ( y ⋆ ) z − λ z 2 , (57) g(y_\star+z)\ge g(y_\star)+g'(y_\star)z-\lambda z^2, \tag*{(57)} g ( y ⋆ + z ) ≥ g ( y ⋆ ) + g ′ ( y ⋆ ) z − λ z 2 , (57) since the penalty changes by 2 λ ( y ⋆ − 1 ) z + λ z 2 2\lambda(y_\star-1)z+\lambda z^2 2 λ ( y ⋆ − 1 ) z + λ z 2 g ′ ( y ⋆ ) = − 2 λ ( y ⋆ − 1 ) g'(y_\star)=-2\lambda(y_\star-1) g ′ ( y ⋆ ) = − 2 λ ( y ⋆ − 1 ) g g g (57) that g ′ ′ ( y ⋆ ) ≥ − 2 λ g''(y_\star)\ge-2\lambda g ′′ ( y ⋆ ) ≥ − 2 λ g ~ \widetilde g g Lemma 4.9 (a), almost surely twice differentiable, with g ~ ′ = − n − 1 s ⟨ G x , ι ⟩ \widetilde g'=-n^{-1}s\langle G_{x,\iota}\rangle g ′ = − n − 1 s ⟨ G x , ι ⟩ g ~ ′ ′ = − n − 1 s 2 ⟨ ( G x , ι − ⟨ G x , ι ⟩ ) 2 ⟩ \widetilde g''=-n^{-1}s^2\langle(G_{x,\iota}-\langle G_{x,\iota}\rangle)^2\rangle g ′′ = − n − 1 s 2 ⟨( G x , ι − ⟨ G x , ι ⟩ ) 2 ⟩ E g ~ ′ = g ′ \mathbb{E}\widetilde g'=g' E g ′ = g ′ Lemma 4.9 (a). By Fatou’s lemma applied to the nonnegative quotients ( g ~ ′ ( y ⋆ ) − g ~ ′ ( y ⋆ + ϵ ′ ) ) / ϵ ′ (\widetilde g'(y_\star)-\widetilde g'(y_\star+\epsilon'))/\epsilon' ( g ′ ( y ⋆ ) − g ′ ( y ⋆ + ϵ ′ )) / ϵ ′ ϵ ′ ↓ 0 \epsilon'\downarrow0 ϵ ′ ↓ 0
s 2 n E ⟨ ( G x , ι − ⟨ G x , ι ⟩ ) 2 ⟩ ≤ − g ′ ′ ( y ⋆ ) ≤ 2 λ , so E ⟨ ( G x , ι − ⟨ G x , ι ⟩ ) 2 ⟩ ≤ 2 ϵ . (58) \frac{s^2}{n}\mathbb{E}\left\langle\left(G_{x,\iota}-\langle G_{x,\iota}\rangle\right)^2\right\rangle\le-g''(y_\star)\le2\lambda,\qquad\text{so}\qquad\mathbb{E}\left\langle\left(G_{x,\iota}-\langle G_{x,\iota}\rangle\right)^2\right\rangle\le\frac{2}{\epsilon}. \tag*{(58)} n s 2 E ⟨ ( G x , ι − ⟨ G x , ι ⟩ ) 2 ⟩ ≤ − g ′′ ( y ⋆ ) ≤ 2 λ , so E ⟨ ( G x , ι − ⟨ G x , ι ⟩ ) 2 ⟩ ≤ ϵ 2 . (58) Let z ∈ ( 0 , 1 / 4 ] z \in(0,1/4] z ∈ ( 0 , 1/4 ] Δ = g ~ − g \Delta= \widetilde{g} - g Δ = g − g g ~ \widetilde{g} g ± z \pm z ± z
g ~ ′ ( y ∗ ) − g ′ ( y ∗ ) ≤ Δ ( y ∗ ) − Δ ( y ∗ − z ) z + λ z , g ~ ′ ( y ∗ ) − g ′ ( y ∗ ) ≥ Δ ( y ∗ + z ) − Δ ( y ∗ ) z − λ z . \widetilde{g}'(y_*) - g'(y_*) \le\frac{\Delta(y_*)-\Delta(y_*-z)}{z}+\lambda z,\qquad\widetilde{g}'(y_*) - g'(y_*) \ge\frac{\Delta(y_*+z)-\Delta(y_*)}{z}-\lambda z. g ′ ( y ∗ ) − g ′ ( y ∗ ) ≤ z Δ ( y ∗ ) − Δ ( y ∗ − z ) + λ z , g ′ ( y ∗ ) − g ′ ( y ∗ ) ≥ z Δ ( y ∗ + z ) − Δ ( y ∗ ) − λ z . Since the three points y ∗ y_* y ∗ y ∗ ± z y_* \pm z y ∗ ± z E ∣ g ~ ′ ( y ∗ ) − g ′ ( y ∗ ) ∣ ≤ 3 C f l / ( n z ) + λ z \mathbb{E}|\widetilde{g}'(y_*) - g'(y_*)| \le3C_{\mathrm{fl}}/(nz) + \lambda z E ∣ g ′ ( y ∗ ) − g ′ ( y ∗ ) ∣ ≤ 3 C fl / ( n z ) + λ z
E ∣ ⟨ G x , ι ⟩ − E ⟨ G x , ι ⟩ ∣ ≤ 3 C f l s z + s z ϵ = 3 C f l + 1 ϵ . \mathbb{E}\left|\langle G_{x,\iota}\rangle-\mathbb{E}\langle G_{x,\iota}\rangle\right|\le\frac{3C_{\mathrm{fl}}}{sz}+\frac{sz}{\epsilon}=\frac{3C_{\mathrm{fl}}+1}{\sqrt{\epsilon}}. E ∣ ⟨ G x , ι ⟩ − E ⟨ G x , ι ⟩ ∣ ≤ sz 3 C fl + ϵ sz = ϵ 3 C fl + 1 . for the choice z = s − 1 ϵ z=s^{-1}\sqrt{\epsilon} z = s − 1 ϵ 1 / 4 1/4 1/4 s ≥ 2 s \ge2 s ≥ 2 ϵ ≤ 1 / 8 \epsilon\le1/8 ϵ ≤ 1/8
E ∣ ⟨ G x , ι ⟩ − E ⟨ G x , ι ⟩ ∣ ≤ 3 C f l + 1 + 2 ϵ . (59) \mathbb{E}\left|\langle G_{x,\iota}\rangle-\mathbb{E}\langle G_{x,\iota}\rangle\right|\le\frac{3C_{\mathrm{fl}}+1+\sqrt{2}}{\sqrt{\epsilon}}. \tag*{(59)} E ∣ ⟨ G x , ι ⟩ − E ⟨ G x , ι ⟩ ∣ ≤ ϵ 3 C fl + 1 + 2 . (59) Let f f f r r r ∣ f ∣ ≤ 1 |f| \le1 ∣ f ∣ ≤ 1 k x ι , 11 = 1 k_{x\iota,11}=1 k x ι , 11 = 1
E ⟨ G x , ι ( α 1 , σ 1 ) f ⟩ = s y ∗ x ι E ⟨ f ( ∑ ℓ = 1 r k x ι , 1 ℓ − r k x ι , 1 , r + 1 ) ⟩ , E ⟨ G x , ι ⟩ = s y ∗ x ι ( 1 − E ⟨ k x ι , 12 ⟩ ) . \mathbb{E}\langle G_{x,\iota}(\alpha^1,\sigma^1)f\rangle=s y_{*x\iota}\mathbb{E}\left\langle f\left(\sum_{\ell=1}^{r}k_{x\iota,1\ell}-r k_{x\iota,1,r+1}\right)\right\rangle,\qquad\mathbb{E}\langle G_{x,\iota}\rangle=s y_{*x\iota}\left(1-\mathbb{E}\langle k_{x\iota,12}\rangle\right). E ⟨ G x , ι ( α 1 , σ 1 ) f ⟩ = s y ∗ x ι E ⟨ f ( ℓ = 1 ∑ r k x ι , 1 ℓ − r k x ι , 1 , r + 1 ) ⟩ , E ⟨ G x , ι ⟩ = s y ∗ x ι ( 1 − E ⟨ k x ι , 12 ⟩ ) . Subtracting E ⟨ G x , ι ⟩ E ⟨ f ⟩ \mathbb{E}\langle G_{x,\iota}\rangle\mathbb{E}\langle f\rangle E ⟨ G x , ι ⟩ E ⟨ f ⟩ − r s y ∗ x ι -r s y_{*x\iota} − r s y ∗ x ι ∣ E ⟨ ( G x , ι ( α 1 , σ 1 ) − E ⟨ G x , ι ⟩ ) f ⟩ ∣ / ( r s y ∗ x ι ) |\mathbb{E}\langle(G_{x,\iota}(\alpha^1,\sigma^1)-\mathbb{E}\langle G_{x,\iota}\rangle)f\rangle|/(r s y_{*x\iota}) ∣ E ⟨( G x , ι ( α 1 , σ 1 ) − E ⟨ G x , ι ⟩) f ⟩ ∣/ ( r s y ∗ x ι ) y ∗ x ι ≥ 13 / 16 y_{*x\iota}\ge13/16 y ∗ x ι ≥ 13/16 C G G = 16 ( 3 C f l + 1 + 2 ) / 13 C_{\mathrm{GG}}=16(3C_{\mathrm{fl}}+1+\sqrt{2})/13 C GG = 16 ( 3 C fl + 1 + 2 ) /13
Sparsification and the drift
We use the following theorem of Cohen, Nelson, and Woodruff on row sparsification [42 Theorem 5] .
Theorem 4.13. There exists an absolute constant C 0 ≥ 1 C_0 \ge1 C 0 ≥ 1 such that the following holds. Let B B B n × d n \times d n × d ∥ B ∥ ≤ 1 \lVert B\rVert\le1 ∥ B ∥ ≤ 1 ∥ B ∥ F 2 ≤ r \lVert B\rVert_F^2 \le r ∥ B ∥ F 2 ≤ r r > 0 r>0 r > 0 λ ∈ ( 0 , 1 ) \lambda\in(0,1) λ ∈ ( 0 , 1 ) n × n n \times n n × n Θ \Theta Θ C 0 ( 1 + r / λ 2 ) C_0(1+r/\lambda^2) C 0 ( 1 + r / λ 2 ) ∥ B T Θ 2 B − B T B ∥ ≤ λ \lVert B^{\mathsf T}\Theta^2B-B^{\mathsf T}B\rVert\le\lambda ∥ B T Θ 2 B − B T B ∥ ≤ λ
In [42 Theorem 5] the diagonal matrix has O ( r / λ 2 ) O(r/\lambda^2) O ( r / λ 2 ) B T Θ 2 B B^{\mathsf T}\Theta^2B B T Θ 2 B [42 Theorem 5] adapts the deterministic construction of Batson, Spielman, and Srivastava [20 ] , in which the number of nonzero entries is bounded in terms of the rank of B B B B = S B=S B = S n n n ∥ S ∥ F 2 = n / D \lVert S\rVert_F^2=n/D ∥ S ∥ F 2 = n / D B = S B=S B = S r = tr S 2 = n / D ≥ 1 r=\operatorname{tr}S^2=n/D\ge1 r = tr S 2 = n / D ≥ 1 Θ \Theta Θ
∥ S Θ 2 S − S 2 ∥ ≤ λ , ∣ { x : Θ x x ≠ 0 } ∣ ≤ 2 C 0 n D λ 2 . (60) \lVert S\Theta^2S-S^2\rVert\le\lambda,\qquad\left|\{x:\Theta_{xx}\ne0\}\right|\le\frac{2C_0n}{D\lambda^2}. \tag*{(60)} ∥ S Θ 2 S − S 2 ∥ ≤ λ , ∣ { x : Θ xx = 0 } ∣ ≤ D λ 2 2 C 0 n . (60) Lemma 4.14. Let Θ \Theta Θ { x : Θ x x ≠ 0 } ⊆ S \{x:\Theta_{xx}\ne0\}\subseteq S { x : Θ xx = 0 } ⊆ S t > 0 t>0 t > 0 q ∈ C K V q\in C_K^V q ∈ C K V y ∈ Y y\in Y y ∈ Y ε 0 > 0 \varepsilon_0>0 ε 0 > 0 V x ≤ ε 0 V_x\le\varepsilon_0 V x ≤ ε 0 ( t , q , y ) (t,q,y) ( t , q , y ) x ∈ S x\in S x ∈ S η > 0 \eta>0 η > 0
∣ ∂ t F ( t , q , y ) − H S ( D q F ( t , q , y ) ) ∣ ≤ τ , τ = β 2 4 [ η + λ + ( 1 + λ ) ε 0 4 η ] . \left|\partial_tF(t,q,y)-H_S(D_qF(t,q,y))\right|\le\tau,\qquad\tau=\frac{\beta^2}{4}\left[\eta+\frac{\lambda+(1+\lambda)\varepsilon_0}{4\eta}\right]. ∣ ∂ t F ( t , q , y ) − H S ( D q F ( t , q , y )) ∣ ≤ τ , τ = 4 β 2 [ η + 4 η λ + ( 1 + λ ) ε 0 ] . Proof. For 1 ≤ j ≤ K 1\le j\le K 1 ≤ j ≤ K p j = E ⟨ u ∗ ∣ ϑ 12 = z j ⟩ ∈ R V \mathsf{p}_j=\mathbb{E}\langle u^*\mid\vartheta_{12}=z_j\rangle\in\mathbb{R}^V p j = E ⟨ u ∗ ∣ ϑ 12 = z j ⟩ ∈ R V C j C_j C j u ∗ u^* u ∗ E ⟨ ⋅ ∣ ϑ 12 = z j ⟩ \mathbb{E}\langle\cdot\mid\vartheta_{12}=z_j\rangle E ⟨ ⋅ ∣ ϑ 12 = z j ⟩ tr C j ≤ n \operatorname{tr}C_j\le n tr C j ≤ n u ∗ = 0 u^*=0 u ∗ = 0 { ϑ 12 = z 0 } \{\vartheta_{12}=z_0\} { ϑ 12 = z 0 }
∂ t F = β 2 4 n ∑ j = 1 K w j E ⟨ u ∗ ⋅ S u ∗ ∣ ϑ 12 = z j ⟩ = H S ( p ) + β 2 4 n ∑ j = 1 K w j tr ( S C j ) , \partial_tF=\frac{\beta^2}{4n}\sum_{j=1}^K w_j\mathbb{E}\langle u^*\cdot Su^*\mid\vartheta_{12}=z_j\rangle=H_S(\mathsf{p})+\frac{\beta^2}{4n}\sum_{j=1}^K w_j\operatorname{tr}(SC_j), ∂ t F = 4 n β 2 j = 1 ∑ K w j E ⟨ u ∗ ⋅ S u ∗ ∣ ϑ 12 = z j ⟩ = H S ( p ) + 4 n β 2 j = 1 ∑ K w j tr ( S C j ) , with p = D q F \mathsf{p}=D_qF p = D q F [56 Lemma 6.2] ; there the error term is the conditional variance of the overlap given the cascade overlap, which is nonnegative. For each eigenvalue μ \mu μ S S S ∣ μ ∣ ≤ η + μ 2 / ( 4 η ) |\mu|\le\eta+\mu^2/(4\eta) ∣ μ ∣ ≤ η + μ 2 / ( 4 η ) C j C_j C j ∣ tr ( S C j ) ∣ ≤ η tr C j + tr ( S 2 C j ) / ( 4 η ) |\operatorname{tr}(SC_j)|\le\eta\operatorname{tr}C_j+\operatorname{tr}(S^2C_j)/(4\eta) ∣ tr ( S C j ) ∣ ≤ η tr C j + tr ( S 2 C j ) / ( 4 η ) tr ( S 2 C j ) ≤ λ tr C j + tr ( S Θ 2 S C j ) \operatorname{tr}(S^2C_j)\le\lambda\operatorname{tr}C_j+\operatorname{tr}(S\Theta^2SC_j) tr ( S 2 C j ) ≤ λ tr C j + tr ( S Θ 2 S C j )
tr ( S Θ 2 S C j ) = ∑ x Θ x x 2 ( S C j S ) x x = ∑ x Θ x x 2 Var ( q x , 12 ∣ ϑ 12 = z j ) . \operatorname{tr}(S\Theta^2SC_j)=\sum_x\Theta_{xx}^2(SC_jS)_{xx}=\sum_x\Theta_{xx}^2\operatorname{Var}(q_{x,12}\mid\vartheta_{12}=z_j). tr ( S Θ 2 S C j ) = x ∑ Θ xx 2 ( S C j S ) xx = x ∑ Θ xx 2 Var ( q x , 12 ∣ ϑ 12 = z j ) . Let ω x = Θ x x 2 / n \omega_x=\Theta_{xx}^2/n ω x = Θ xx 2 / n 1 \mathbf{1} 1 S 1 = 1 S\mathbf{1}=\mathbf{1} S 1 = 1 ∑ x ω x ≤ 1 + λ \sum_x\omega_x\le1+\lambda ∑ x ω x ≤ 1 + λ
∑ j w j tr ( S 2 C j ) n ≤ λ + ∑ x ∈ S ω x V x ≤ λ + ( 1 + λ ) ε 0 , \sum_jw_j\frac{\operatorname{tr}(S^2C_j)}{n}\le\lambda+\sum_{x\in S}\omega_xV_x\le\lambda+(1+\lambda)\varepsilon_0, j ∑ w j n tr ( S 2 C j ) ≤ λ + x ∈ S ∑ ω x V x ≤ λ + ( 1 + λ ) ε 0 , and the claim follows.
Lemma 4.15. Suppose that ζ j = ( j + 1 ) / ( K + 1 ) \zeta_j=(j+1)/(K+1) ζ j = ( j + 1 ) / ( K + 1 ) 0 ≤ j ≤ K 0\le j\le K 0 ≤ j ≤ K δ M \delta_M δ M ε = K − 1 ( K + 1 ) − 3 \varepsilon=K^{-1}(K+1)^{-3} ε = K − 1 ( K + 1 ) − 3 I = I δ M I=I_{\delta_M} I = I δ M Θ \Theta Θ S = { x : Θ x x ≠ 0 } S=\{x:\Theta_{xx}\ne0\} S = { x : Θ xx = 0 }
s ≥ 2 and 2 ⌈ δ M − 1 ⌉ C G G s ε ≤ δ M . s\ge2\qquad\text{and}\qquad\frac{2^{\lceil\delta_M^{-1}\rceil}C_{GG}}{s\sqrt{\varepsilon}}\le\delta_M. s ≥ 2 and s ε 2 ⌈ δ M − 1 ⌉ C GG ≤ δ M .
Then F ⋆ F_\star F ⋆ ∂ t u − H e x t ( D q u ) ≥ − τ \partial_tu-H_{\mathrm{ext}}(D_qu)\ge-\tau ∂ t u − H ext ( D q u ) ≥ − τ τ \tau τ ε 0 = 13 / ( K + 1 ) \varepsilon_0=13/(K+1) ε 0 = 13/ ( K + 1 )
Proof. Let φ \varphi φ ( t 0 , q 0 ) ∈ ( 0 , 2 ) × C K V (t_0,q_0)\in(0,2)\times C_K^V ( t 0 , q 0 ) ∈ ( 0 , 2 ) × C K V F ⋆ − φ F_\star-\varphi F ⋆ − φ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) y ⋆ y_\star y ⋆ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) ( t , q ) ↦ F ( t , q , y ⋆ ) + s 2 ( n ε ) − 1 ∑ x , ι ( y x ι ⋆ − 1 ) 2 (t,q)\mapsto F(t,q,y_\star)+s^2(n\varepsilon)^{-1}\sum_{x,\iota}(y^\star_{x\iota}-1)^2 ( t , q ) ↦ F ( t , q , y ⋆ ) + s 2 ( n ε ) − 1 ∑ x , ι ( y x ι ⋆ − 1 ) 2 F ⋆ F_\star F ⋆ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) F ( ⋅ , ⋅ , y ⋆ ) − φ F(\cdot,\cdot,y_\star)-\varphi F ( ⋅ , ⋅ , y ⋆ ) − φ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) ∂ t φ = ∂ t F \partial_t\varphi=\partial_tF ∂ t φ = ∂ t F D q F ⪰ ∗ D q φ D_qF\succeq_*D_q\varphi D q F ⪰ ∗ D q φ ( t 0 , q 0 , y ⋆ ) (t_0,q_0,y_\star) ( t 0 , q 0 , y ⋆ ) x ∈ S x \in S x ∈ S G \mathbb{G} G ( t 0 , q 0 , y ∗ ) (t_0,q_0,y_*) ( t 0 , q 0 , y ∗ ) ( α , σ ) ↦ ( F x ( α , σ ) , e ( α ) ) (\alpha,\sigma) \mapsto(F_x(\alpha,\sigma),e(\alpha)) ( α , σ ) ↦ ( F x ( α , σ ) , e ( α ))
F x ( α , σ ) = ( S x y 1 / 2 σ y e α 1 ) y ∈ V ′ , e ( α ) = ∑ k = 1 K ( z k − z k − 1 ) 1 / 2 e α ∣ k . F_x(\alpha,\sigma)=\left(S_{xy}^{1/2}\sigma_y e_{\alpha_1}\right)_{y\in V'},\qquad e(\alpha)=\sum_{k=1}^{K}(z_k-z_{k-1})^{1/2}e_{\alpha|k}. F x ( α , σ ) = ( S x y 1/2 σ y e α 1 ) y ∈ V ′ , e ( α ) = k = 1 ∑ K ( z k − z k − 1 ) 1/2 e α ∣ k . for orthonormal families ( e a ) a ∈ N (e_a)_{a\in\mathbb{N}} ( e a ) a ∈ N ( e γ ) γ ∈ A ∖ { ∅ } (e_\gamma)_{\gamma\in A\setminus\{\varnothing\}} ( e γ ) γ ∈ A ∖ { ∅ } ∑ y S x y = 1 \sum_y S_{xy}=1 ∑ y S x y = 1 z K − z 0 = 1 z_K-z_0=1 z K − z 0 = 1 ( α ℓ , σ ℓ ) (\alpha^\ell,\sigma^\ell) ( α ℓ , σ ℓ ) O 1 ℓ ℓ ′ = ϱ x , ℓ ℓ ′ O_1^{\ell\ell'}=\varrho_{x,\ell\ell'} O 1 ℓ ℓ ′ = ϱ x , ℓ ℓ ′ O 2 ℓ ℓ ′ = ϑ ℓ ℓ ′ O_2^{\ell\ell'}=\vartheta_{\ell\ell'} O 2 ℓ ℓ ′ = ϑ ℓ ℓ ′ δ = δ M \delta=\delta_M δ = δ M r , i 1 , i 2 , p ≤ ⌊ δ M − 1 ⌋ r,i_1,i_2,p\leq\lfloor\delta_M^{-1}\rfloor r , i 1 , i 2 , p ≤ ⌊ δ M − 1 ⌋ f f f c i 1 + c i 2 = 0 c_{i_1}+c_{i_2}=0 c i 1 + c i 2 = 0 k ℓ ℓ ′ = 0 k_{\ell\ell'}=0 k ℓ ℓ ′ = 0 ℓ , ℓ ′ \ell,\ell' ℓ , ℓ ′
( c i 1 ϱ + c i 2 ϑ ) p = ( c i 1 + c i 2 ) p κ ℓ ( ϑ , ϱ ) , ι = ( c i 2 c i 1 + c i 2 , p ) ∈ I , (c_{i_1}\varrho+c_{i_2}\vartheta)^p=(c_{i_1}+c_{i_2})^p\kappa_\ell(\vartheta,\varrho),\qquad\iota=\left(\frac{c_{i_2}}{c_{i_1}+c_{i_2}},p\right)\in I, ( c i 1 ϱ + c i 2 ϑ ) p = ( c i 1 + c i 2 ) p κ ℓ ( ϑ , ϱ ) , ι = ( c i 1 + c i 2 c i 2 , p ) ∈ I , and f ( O ≤ r ) f(O^{\leq r}) f ( O ≤ r ) r r r ∣ f ( O ≤ r ) ∣ ≤ 1 \lvert f(O^{\leq r})\rvert\leq1 ∣ f ( O ≤ r )∣ ≤ 1 ( c i 1 + c i 2 ) p ≤ 2 ⌊ δ M − 1 ⌋ (c_{i_1}+c_{i_2})^p\leq2^{\lfloor\delta_M^{-1}\rfloor} ( c i 1 + c i 2 ) p ≤ 2 ⌊ δ M − 1 ⌋ δ M \delta_M δ M ϑ 12 \vartheta_{12} ϑ 12 ( K + 1 ) − 1 ∑ j = 0 K δ z j (K+1)^{-1}\sum_{j=0}^{K}\delta_{z_j} ( K + 1 ) − 1 ∑ j = 0 K δ z j k = K + 1 k=K+1 k = K + 1 q ℓ = z ℓ − 1 q_\ell=z_{\ell-1} q ℓ = z ℓ − 1 1 ≤ ℓ ≤ k 1\leq\ell\leq k 1 ≤ ℓ ≤ k q ℓ + 1 − q ℓ q_{\ell+1}-q_\ell q ℓ + 1 − q ℓ 1 1 1 ℓ = 0 \ell=0 ℓ = 0 1 / K 1/K 1/ K { ϑ 12 = z j } \{\vartheta_{12}=z_j\} { ϑ 12 = z j } w j > 0 w_j>0 w j > 0 V x V_x V x
V x ≤ 12 K + 1 + ε ( K + 1 ) 2 K = 13 K + 1 . V_x\leq\frac{12}{K+1}+\varepsilon(K+1)^2K=\frac{13}{K+1}. V x ≤ K + 1 12 + ε ( K + 1 ) 2 K = K + 1 13 . This holds for every x ∈ S x\in S x ∈ S ∂ t F ≥ H S ( D q F ) − τ \partial_tF\geq H_S(D_qF)-\tau ∂ t F ≥ H S ( D q F ) − τ ( t 0 , q 0 , y ∗ ) (t_0,q_0,y_*) ( t 0 , q 0 , y ∗ ) D q F ∈ C K V D_qF\in C_K^V D q F ∈ C K V H S ( D q F ) = H e x t ( D q F ) ≥ H e x t ( D q φ ) H_S(D_qF)=H_{\mathrm{ext}}(D_qF)\geq H_{\mathrm{ext}}(D_q\varphi) H S ( D q F ) = H ext ( D q F ) ≥ H ext ( D q φ ) ∂ t φ − H e x t ( D q φ ) ≥ − τ \partial_t\varphi-H_{\mathrm{ext}}(D_q\varphi)\geq-\tau ∂ t φ − H ext ( D q φ ) ≥ − τ ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 )
Proof of the upper bound
Proof of Proposition 4.1
If β = 0 \beta=0 β = 0 p G ( 0 , h ) = log 2 cosh h p_G(0,h)=\log2\cosh h p G ( 0 , h ) = log 2 cosh h P ( 0 , h ) = log 2 cosh h P(0,h)=\log2\cosh h P ( 0 , h ) = log 2 cosh h β > 0 \beta>0 β > 0 h ∈ R h\in\mathbb{R} h ∈ R υ ∈ ( 0 , 1 ) \upsilon\in(0,1) υ ∈ ( 0 , 1 ) D D D G G G ν = ν β , h \nu=\nu_{\beta,h} ν = ν β , h Q ( θ ) = min { s ∈ [ 0 , 1 ] : ν ( [ 0 , s ] ) ≥ θ } Q(\theta)=\min\{s\in[0,1]:\nu([0,s])\geq\theta\} Q ( θ ) = min { s ∈ [ 0 , 1 ] : ν ([ 0 , s ]) ≥ θ } θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) Q Q Q [ 0 , 1 ] [0,1] [ 0 , 1 ] Q ( 0 ) = min supp ν Q(0)=\min\operatorname{supp}\nu Q ( 0 ) = min supp ν Q ( 1 ) = max supp ν Q(1)=\max\operatorname{supp}\nu Q ( 1 ) = max supp ν Q Q Q ( 0 , 1 ) (0,1) ( 0 , 1 ) Q Q Q ν \nu ν K ≥ 1 K\geq1 K ≥ 1 r j ( K ) = Q ( ( j + 1 / 2 ) / ( K + 1 ) ) r_j^{(K)}=Q((j+1/2)/(K+1)) r j ( K ) = Q (( j + 1/2 ) / ( K + 1 )) 0 ≤ j ≤ K 0\leq j\leq K 0 ≤ j ≤ K r ( K ) ∈ C K ∩ [ 0 , 1 ] K + 1 r^{(K)}\in C_K\cap[0,1]^{K+1} r ( K ) ∈ C K ∩ [ 0 , 1 ] K + 1 μ ( K ) = ( K + 1 ) − 1 ∑ j = 0 K δ r j ( K ) \mu^{(K)}=(K+1)^{-1}\sum_{j=0}^{K}\delta_{r_j^{(K)}} μ ( K ) = ( K + 1 ) − 1 ∑ j = 0 K δ r j ( K ) f ∈ C ( [ 0 , 1 ] ) f\in C([0,1]) f ∈ C ([ 0 , 1 ]) ∫ f d μ ( K ) \int f\,d\mu^{(K)} ∫ f d μ ( K ) f ∘ Q f\circ Q f ∘ Q [ 0 , 1 ] [0,1] [ 0 , 1 ] K + 1 K+1 K + 1 f ∘ Q f\circ Q f ∘ Q Q Q Q ∫ 0 1 f ( Q ( θ ) ) d θ = ∫ f d ν \int_{0}^{1} f(Q(\theta))\,\mathrm{d}\theta= \int f\,\mathrm{d}\nu ∫ 0 1 f ( Q ( θ )) d θ = ∫ f d ν K → ∞ K \to\infty K → ∞ μ ( K ) → ν \mu^{(K)} \to\nu μ ( K ) → ν P β , h ( μ ( K ) ) → P ( β , h ) \mathcal{P}_{\beta,h}(\mu^{(K)}) \to\mathcal{P}(\beta,h) P β , h ( μ ( K ) ) → P ( β , h )
Let η = 2 v / ( 3 β 2 ) \eta= 2v/(3\beta^{2}) η = 2 v / ( 3 β 2 ) β 2 η / 4 = v / 6 \beta^{2}\eta/4 = v/6 β 2 η /4 = v /6 K ≥ 1 K \ge1 K ≥ 1 ζ j = ( j + 1 ) / ( K + 1 ) \zeta_j = (j+1)/(K+1) ζ j = ( j + 1 ) / ( K + 1 ) w j = w ˉ = ζ 0 = 1 / ( K + 1 ) ≤ 1 / 2 w_j = \bar{w} = \zeta_0 = 1/(K+1) \le1/2 w j = w ˉ = ζ 0 = 1/ ( K + 1 ) ≤ 1/2 μ r ( K ) = μ ( K ) \mu_{r^{(K)}} = \mu^{(K)} μ r ( K ) = μ ( K )
P β , h ( μ ( K ) ) ≤ P ( β , h ) + v 6 , β 2 ζ 0 4 ≤ v 6 and β 2 16 η ⋅ 21 w ˉ ≤ v 6 . \mathcal{P}_{\beta,h}(\mu^{(K)}) \le\mathcal{P}(\beta,h) + \frac{v}{6}, \qquad\frac{\beta^{2}\zeta_0}{4} \le\frac{v}{6} \qquad\text{and} \qquad\frac{\beta^{2}}{16\eta}\cdot21\bar{w} \le\frac{v}{6}. P β , h ( μ ( K ) ) ≤ P ( β , h ) + 6 v , 4 β 2 ζ 0 ≤ 6 v and 16 η β 2 ⋅ 21 w ˉ ≤ 6 v . Let λ = w ˉ \lambda= \bar{w} λ = w ˉ ϵ 0 = 13 w ˉ \epsilon_0 = 13\bar{w} ϵ 0 = 13 w ˉ λ + ( 1 + λ ) ϵ 0 ≤ 21 w ˉ \lambda+ (1+\lambda)\epsilon_0 \le21\bar{w} λ + ( 1 + λ ) ϵ 0 ≤ 21 w ˉ τ \tau τ τ ≤ v / 3 \tau\le v/3 τ ≤ v /3 δ = v / 6 \delta= v/6 δ = v /6 ϵ = 1 / 8 \epsilon= 1/8 ϵ = 1/8 δ M \delta_M δ M I = I δ M I = I_{\delta_M} I = I δ M C G G C_{\mathrm{GG}} C GG β \beta β h h h v v v
Now let D ≥ 1 D \ge1 D ≥ 1 s = D 1 / 4 s = D^{1/4} s = D 1/4 D 0 D_0 D 0 D ≥ D 0 D \ge D_0 D ≥ D 0 D ≥ D 0 D \ge D_0 D ≥ D 0 G ∈ G D G \in\mathcal{G}_D G ∈ G D Θ \Theta Θ S = { x : Θ x x ≠ 0 } S = \{x : \Theta_{xx} \ne0\} S = { x : Θ xx = 0 } S S S I I I s s s
r s = 9 4 ∣ S ∣ ∣ I ∣ s 2 n ≤ 9 C 0 ∣ I ∣ 2 λ 2 D . (61) r_s = \frac{9}{4}\frac{|S||I|s^2}{n} \le\frac{9C_0|I|}{2\lambda^2\sqrt{D}}. \tag*{(61)} r s = 4 9 n ∣ S ∣∣ I ∣ s 2 ≤ 2 λ 2 D 9 C 0 ∣ I ∣ . (61) We apply Lemma 4.5 with H = H e x t H = H_{\mathrm{ext}} H = H ext V = V δ V = V_{\delta} V = V δ E = F ⋆ E = F_{\star} E = F ⋆ [ 0 , 2 ] × C K V [0,2] \times C_K^V [ 0 , 2 ] × C K V H e x t H_{\mathrm{ext}} H ext L t = β 2 L_t = \beta^2 L t = β 2 L q = 1 L_q = 1 L q = 1
V δ ( 0 , q ) ≤ 1 n ∑ x ψ h ( q ~ x ) + δ = F 0 ( 0 , q ) + δ ≤ F ⋆ ( 0 , q ) + δ + r s . V_{\delta}(0,q) \le\frac{1}{n}\sum_x \psi_h(\widetilde{q}_x) + \delta= F_0(0,q) + \delta\le F_{\star}(0,q) + \delta+ r_s. V δ ( 0 , q ) ≤ n 1 x ∑ ψ h ( q x ) + δ = F 0 ( 0 , q ) + δ ≤ F ⋆ ( 0 , q ) + δ + r s . The lemma then gives V δ ( 1 , 0 ) ≤ F ⋆ ( 1 , 0 ) + δ + r s + τ V_{\delta}(1,0) \le F_{\star}(1,0) + \delta+ r_s + \tau V δ ( 1 , 0 ) ≤ F ⋆ ( 1 , 0 ) + δ + r s + τ
U ( 1 , 0 ) − β 2 ζ 0 4 ≤ F 0 ( 1 , 0 ) + δ + 2 r s + τ ≤ β 2 4 − p G ( β , h ) + δ + 2 r s + τ . U(1,0) - \frac{\beta^2\zeta_0}{4} \le F_0(1,0) + \delta+ 2r_s + \tau\le\frac{\beta^2}{4} - p_G(\beta,h) + \delta+ 2r_s + \tau. U ( 1 , 0 ) − 4 β 2 ζ 0 ≤ F 0 ( 1 , 0 ) + δ + 2 r s + τ ≤ 4 β 2 − p G ( β , h ) + δ + 2 r s + τ . By Lemma 3.10 with r = r ( K ) r = r^{(K)} r = r ( K ) K K K U ( 1 , 0 ) ≥ β 2 / 4 − P β , h ( μ ( K ) ) ≥ β 2 / 4 − P ( β , h ) − v / 6 U(1,0) \ge\beta^2/4 - \mathcal{P}_{\beta,h}(\mu^{(K)}) \ge\beta^2/4 - \mathcal{P}(\beta,h) - v/6 U ( 1 , 0 ) ≥ β 2 /4 − P β , h ( μ ( K ) ) ≥ β 2 /4 − P ( β , h ) − v /6
p G ( β , h ) ≤ P ( β , h ) + v 6 + β 2 ζ 0 4 + δ + τ + 2 r s ≤ P ( β , h ) + 5 v 6 + 9 C 0 ∣ I ∣ λ 2 D . p_G(\beta,h) \le\mathcal{P}(\beta,h) + \frac{v}{6} + \frac{\beta^2\zeta_0}{4} + \delta+ \tau+ 2r_s \le\mathcal{P}(\beta,h) + \frac{5v}{6} + \frac{9C_0|I|}{\lambda^2\sqrt{D}}. p G ( β , h ) ≤ P ( β , h ) + 6 v + 4 β 2 ζ 0 + δ + τ + 2 r s ≤ P ( β , h ) + 6 5 v + λ 2 D 9 C 0 ∣ I ∣ . The right side does not depend on G ∈ G D G \in\mathcal{G}_D G ∈ G D n n n G ∈ G D G \in\mathcal{G}_D G ∈ G D D → ∞ D \to\infty D → ∞ lim sup D → ∞ sup G ∈ G D p G ( β , h ) ≤ P ( β , h ) + 5 v / 6 \limsup_{D \to\infty} \sup_{G \in\mathcal{G}_D} p_G(\beta,h) \le\mathcal{P}(\beta,h) + 5v/6 lim sup D → ∞ sup G ∈ G D p G ( β , h ) ≤ P ( β , h ) + 5 v /6 v v v
Message-passing centers on tori and hypercubes
The goal of this section is to prove Proposition 5.2, which constructs the centers used in the lower bound on tori and hypercubes. A center is a vector m ∈ ( − 1 , 1 ) V m \in(-1,1)^V m ∈ ( − 1 , 1 ) V W W W
Definition 5.1. Let G ∈ G D G \in\mathcal{G}_D G ∈ G D T ≥ 1 T \ge1 T ≥ 1 query history of depth T T T on G G G U U U ( g e ) e ∈ E (g_e)_{e\in E} ( g e ) e ∈ E x 1 , … , x T ∈ R V x^1,\ldots,x^T \in\mathbb{R}^V x 1 , … , x T ∈ R V t ≤ T t \le T t ≤ T x t x^t x t σ \sigma σ U U U W x 1 , … , W x t − 1 Wx^1,\ldots,Wx^{t-1} W x 1 , … , W x t − 1 H \mathcal{H} H σ \sigma σ U U U W x 1 , … , W x T Wx^1,\ldots,Wx^T W x 1 , … , W x T center of the history is an H \mathcal{H} H m ∈ ( − 1 , 1 ) V m \in(-1,1)^V m ∈ ( − 1 , 1 ) V m = x t m=x^t m = x t t ≤ T t\le T t ≤ T
Throughout this section, ( G d ) d ≥ 1 (G_d)_{d\ge1} ( G d ) d ≥ 1 G d = T L d G_d=\mathbb{T}_L^d G d = T L d L ≥ 3 L\ge3 L ≥ 3 D = 2 d D=2d D = 2 d n = L d n=L^d n = L d G d = Q d G_d=Q_d G d = Q d D = d D=d D = d n = 2 d n=2^d n = 2 d ( Z / L Z ) d (\mathbb{Z}/L\mathbb{Z})^d ( Z / L Z ) d ( Z / 2 Z ) d (\mathbb{Z}/2\mathbb{Z})^d ( Z /2 Z ) d G d G_d G d ∥ w ∥ n = ( n − 1 ∑ x w x 2 ) 1 / 2 \lVert w\rVert_n=(n^{-1}\sum_x w_x^2)^{1/2} ∥ w ∥ n = ( n − 1 ∑ x w x 2 ) 1/2 w ∈ R V w\in\mathbb{R}^V w ∈ R V ν = ν β , h \nu=\nu_{\beta,h} ν = ν β , h q − ≤ q + q_-\le q_+ q − ≤ q + a a a v v v X t X_t X t M t = ∂ x Φ ( t , X t ) M_t=\partial_x\Phi(t,X_t) M t = ∂ x Φ ( t , X t ) μ + = Law ( M q + ) \mu_+=\operatorname{Law}(M_{q_+}) μ + = Law ( M q + )
Proposition 5.2. Let β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 ( G d ) d ≥ 1 (G_d)_{d\ge1} ( G d ) d ≥ 1 T L d \mathbb{T}_L^d T L d L ≥ 3 L\ge3 L ≥ 3 D = 2 d D=2d D = 2 d Q d Q_d Q d D = d D=d D = d ϵ > 0 \epsilon>0 ϵ > 0 T ≥ 1 T\ge1 T ≥ 1 μ ϵ \mu_\epsilon μ ϵ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] d d d T T T G d G_d G d m = m ( d ) m=m^{(d)} m = m ( d )
(a) With κ x = 1 − m x 2 \kappa_x=1-m_x^2 κ x = 1 − m x 2
lim inf d → ∞ 1 n E [ β H G d ( m ) + h ∑ x m x + ∑ x ent ( m x ) + β 2 4 ⟨ κ , S κ ⟩ ] ≥ P ( β , h ) − ϵ . \liminf_{d\to\infty}\frac{1}{n}\mathbb{E}\left[\beta H_{G_d}(m)+h\sum_x m_x+\sum_x\operatorname{ent}(m_x)+\frac{\beta^2}{4}\langle\kappa,S\kappa\rangle\right]\ge\mathcal{P}(\beta,h)-\epsilon. d → ∞ lim inf n 1 E [ β H G d ( m ) + h x ∑ m x + x ∑ ent ( m x ) + 4 β 2 ⟨ κ , S κ ⟩ ] ≥ P ( β , h ) − ϵ . (b) We have W 1 ( μ ϵ , μ + ) ≤ ϵ W_1(\mu_\epsilon,\mu_+)\le\epsilon W 1 ( μ ϵ , μ + ) ≤ ϵ W 1 W_1 W 1 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
(c) Let Υ : [ − 1 , 1 ] × [ 0 , ∞ ) → R \Upsilon\colon[-1,1]\times[0,\infty)\to\mathbb{R} Υ : [ − 1 , 1 ] × [ 0 , ∞ ) → R L Υ ≥ 0 L_{\Upsilon}\ge0 L Υ ≥ 0 ∣ Υ ( m ′ , r ) − Υ ( m ′ , r ′ ) ∣ ≤ L Υ ∣ r − r ′ ∣ |\Upsilon(m',r)-\Upsilon(m',r')|\le L_{\Upsilon}|r-r'| ∣Υ ( m ′ , r ) − Υ ( m ′ , r ′ ) ∣ ≤ L Υ ∣ r − r ′ ∣ m ′ ∈ [ − 1 , 1 ] m'\in[-1,1] m ′ ∈ [ − 1 , 1 ] r , r ′ ≥ 0 r,r'\ge0 r , r ′ ≥ 0 a , c , B ≥ 0 a,c,B\ge0 a , c , B ≥ 0
lim d → ∞ E sup r ∈ a 1 + c S [ 0 , B ] V ∣ 1 n ∑ x Υ ( m x , r x ) − 1 n ∑ x ∫ Υ ( m ′ , r x ) μ ϵ ( d m ′ ) ∣ = 0. \lim_{d\to\infty}\mathbb{E}\sup_{r\in a\mathbf{1}+cS[0,B]^{V}}\left|\frac{1}{n}\sum_x\Upsilon(m_x,r_x)-\frac{1}{n}\sum_x\int\Upsilon(m',r_x)\,\mu_{\epsilon}(\mathrm{d}m')\right|=0. d → ∞ lim E r ∈ a 1 + c S [ 0 , B ] V sup n 1 x ∑ Υ ( m x , r x ) − n 1 x ∑ ∫ Υ ( m ′ , r x ) μ ϵ ( d m ′ ) = 0. Remark 5.3. We record three comments on the statement.
The integer T T T μ ϵ \mu_{\epsilon} μ ϵ β \beta β h h h ϵ \epsilon ϵ L L L d d d d d d
All terms in (a) are integrable, since ∣ H G d ( m ) ∣ ≤ ∑ e ∣ J e ∣ |H_{G_d}(m)|\le\sum_e|J_e| ∣ H G d ( m ) ∣ ≤ ∑ e ∣ J e ∣
The supremum in (c) is a random variable: since a 1 + c S [ 0 , B ] V a\mathbf{1}+cS[0,B]^{V} a 1 + c S [ 0 , B ] V r r r
The centers are produced by approximate message passing with full memory and deterministic Onsager coefficients. The Gaussian state evolution of message passing was first proved by Bolthausen [25 ] and by Bayati and Montanari [21 ] . Section 5.1 introduces this recursion and states its state evolution, Proposition 5.6. We deduce Proposition 5.6 from a theorem of Hachem [80 ] on a recursion whose Onsager coefficients are computed from the disorder. Section 5.2 states this theorem and a bound on ∥ W ∥ \lVert W\rVert ∥ W ∥ [125 ] . Its first phase is an iteration of the type used by Bolthausen [25 ] , and its second phase is the incremental message passing of Montanari [103 ] and El Alaoui, Montanari, and Sellke [60 ] , whose value is described by the stochastic control problem of [60 Section 4] . For the SK model without field, Montanari showed that the incremental algorithm constructs approximate solutions of the TAP equations for all large β \beta β [ 0 , q ∗ ] [0,q_*] [ 0 , q ∗ ] β \beta β [103 Theorem 5] . Section 5.8 proves the homogenization estimate, and Section 5.9 completes the proof of Proposition 5.2.
The Onsager coefficients of our recursion are real numbers computed from the state evolution. Using Hachem’s coefficients, which explicitly involve squared individual couplings, would require information beyond the earlier products and would not in general satisfy Definition 5.1. Deterministic Onsager coefficients for sparse variance profiles also appear in Theorems 2 and 4 of [80 ] , which treat exact and approximate iterates, respectively, under a nondegeneracy assumption. At high temperature, [81 Theorem 5] treats an exact recursion with the additional sparsity condition K n ≥ log n K_n\ge\log n K n ≥ log n [80 Proposition 8] and prove the replacement by deterministic coefficients in Section 5.4.
Throughout this section, Z Z Z ∥ ξ ∥ p = ( E ∣ ξ ∣ p ) 1 / p \lVert\xi\rVert_p = (\mathbb{E}|\xi|^p)^{1/p} ∥ ξ ∥ p = ( E ∣ ξ ∣ p ) 1/ p ξ \xi ξ R k \mathbb{R}^k R k polynomial growth if it is bounded by C ( 1 + ∣ z ∣ p ) C(1+|z|^p) C ( 1 + ∣ z ∣ p ) C , p ≥ 0 C,p \ge0 C , p ≥ 0 f : R t → R f:\mathbb{R}^t \to\mathbb{R} f : R t → R u 1 , … , u t ∈ R V u^1,\ldots,u^t \in\mathbb{R}^V u 1 , … , u t ∈ R V f ( u 1 , … , u t ) ∈ R V f(u^1,\ldots,u^t) \in\mathbb{R}^V f ( u 1 , … , u t ) ∈ R V f ( u x 1 , … , u x t ) f(u_x^1,\ldots,u_x^t) f ( u x 1 , … , u x t ) w ⊙ w ′ w \odot w' w ⊙ w ′
Message passing with deterministic Onsager coefficients
Fix an integer T ≥ 1 T \ge1 T ≥ 1 f 0 ∈ R f_0 \in\mathbb{R} f 0 ∈ R f t : R t → R f_t:\mathbb{R}^t \to\mathbb{R} f t : R t → R 1 ≤ t ≤ T − 1 1 \le t \le T-1 1 ≤ t ≤ T − 1 u = ( u 1 , u 2 , … ) u=(u^1,u^2,\ldots) u = ( u 1 , u 2 , … ) u ˉ t = ( u 1 , … , u t ) \bar{u}^t=(u^1,\ldots,u^t) u ˉ t = ( u 1 , … , u t ) u ˉ 0 \bar{u}^0 u ˉ 0 f 0 ( u ˉ 0 ) = f 0 f_0(\bar{u}^0)=f_0 f 0 ( u ˉ 0 ) = f 0 state evolution is the centered Gaussian vector U = ( U 1 , … , U T ) U=(U_1,\ldots,U_T) U = ( U 1 , … , U T )
E U r + 1 U s + 1 = E f r ( U ˉ r ) f s ( U ˉ s ) ( 0 ≤ r , s ≤ T − 1 ) . (62) \mathbb{E} U_{r+1}U_{s+1}=\mathbb{E} f_r(\bar{U}^r)f_s(\bar{U}^s) \qquad(0\le r,s\le T-1). \tag*{(62)} E U r + 1 U s + 1 = E f r ( U ˉ r ) f s ( U ˉ s ) ( 0 ≤ r , s ≤ T − 1 ) . (62) The recursion (5.1) defines the law of U U U U ˉ t + 1 \bar{U}^{t+1} U ˉ t + 1 U ˉ t \bar{U}^t U ˉ t Onsager coefficients are the real numbers
b t , s = E ∂ s f t ( U ˉ t ) ( 1 ≤ s ≤ t ≤ T − 1 ) . (63) b_{t,s}=\mathbb{E}\,\partial_s f_t(\bar{U}^t) \qquad(1\le s\le t\le T-1). \tag*{(63)} b t , s = E ∂ s f t ( U ˉ t ) ( 1 ≤ s ≤ t ≤ T − 1 ) . (63) On a graph G ∈ G D G\in\mathcal{G}_D G ∈ G D u 1 , … , u T ∈ R V u^1,\ldots,u^T\in\mathbb{R}^V u 1 , … , u T ∈ R V u 1 = f 0 W 1 u^1=f_0W\mathbf{1} u 1 = f 0 W 1
u t + 1 = W f t ( u ˉ t ) − ∑ s = 1 t b t , s f s − 1 ( u ˉ s − 1 ) ( 1 ≤ t ≤ T − 1 ) . (64) u^{t+1}=Wf_t(\bar{u}^t)-\sum_{s=1}^{t}b_{t,s}f_{s-1}(\bar{u}^{s-1}) \qquad(1\le t\le T-1). \tag*{(64)} u t + 1 = W f t ( u ˉ t ) − s = 1 ∑ t b t , s f s − 1 ( u ˉ s − 1 ) ( 1 ≤ t ≤ T − 1 ) . (64) where f 0 ( u ˉ 0 ) = f 0 1 f_0(\bar{u}^0)=f_0\mathbf{1} f 0 ( u ˉ 0 ) = f 0 1 b t , s b_{t,s} b t , s f 0 , … , f T − 1 f_0,\ldots,f_{T-1} f 0 , … , f T − 1 G G G b t , s b_{t,s} b t , s n − 1 ∑ x ∂ s f t ( u ˉ x t ) n^{-1}\sum_x \partial_s f_t(\bar{u}_x^t) n − 1 ∑ x ∂ s f t ( u ˉ x t ) [103 (2.1)] , whose state evolution is (5.1).
Lemma 5.4. Let G ∈ G D G\in\mathcal{G}_D G ∈ G D 1 ≤ K ≤ T 1\le K\le T 1 ≤ K ≤ T θ : R K → ( − 1 , 1 ) \theta:\mathbb{R}^K\to(-1,1) θ : R K → ( − 1 , 1 ) x t = f t − 1 ( u ˉ t − 1 ) x^t=f_{t-1}(\bar{u}^{t-1}) x t = f t − 1 ( u ˉ t − 1 ) 1 ≤ t ≤ K 1\le t\le K 1 ≤ t ≤ K x K + 1 = θ ( u ˉ K ) x^{K+1}=\theta(\bar{u}^K) x K + 1 = θ ( u ˉ K ) x 1 , … , x K + 1 x^1,\ldots,x^{K+1} x 1 , … , x K + 1 U U U K + 1 K+1 K + 1 G G G m = θ ( u ˉ K ) m=\theta(\bar{u}^K) m = θ ( u ˉ K )
Proof. We show by induction on t ≤ K t\le K t ≤ K u ˉ t \bar{u}^t u ˉ t σ ( W x 1 , … , W x t ) \sigma(Wx^1,\ldots,Wx^t) σ ( W x 1 , … , W x t ) x 1 = f 0 1 x^1=f_0\mathbf{1} x 1 = f 0 1 u 1 = W x 1 u^1=Wx^1 u 1 = W x 1 t < K t<K t < K x t + 1 = f t ( u ˉ t ) x^{t+1}=f_t(\bar{u}^t) x t + 1 = f t ( u ˉ t ) σ ( W x 1 , … , W x t ) \sigma(Wx^1,\ldots,Wx^t) σ ( W x 1 , … , W x t ) u t + 1 = W x t + 1 − ∑ s ≤ t b t , s x s u^{t+1}=Wx^{t+1}-\sum_{s\le t}b_{t,s}x^s u t + 1 = W x t + 1 − ∑ s ≤ t b t , s x s σ ( W x 1 , … , W x t + 1 ) \sigma(Wx^1,\ldots,Wx^{t+1}) σ ( W x 1 , … , W x t + 1 ) x 1 x^1 x 1 2 ≤ t ≤ K + 1 2\le t\le K+1 2 ≤ t ≤ K + 1 x t x^t x t u ˉ t − 1 \bar{u}^{t-1} u ˉ t − 1 σ ( W x 1 , … , W x t − 1 ) \sigma(Wx^1,\ldots,Wx^{t-1}) σ ( W x 1 , … , W x t − 1 ) m = x K + 1 m=x^{K+1} m = x K + 1 ( − 1 , 1 ) V (-1,1)^V ( − 1 , 1 ) V H \mathcal{H} H
The next lemma is the covariance isometry behind the state evolution.
Lemma 5.5. For 1 ≤ k ≤ T 1 \le k \le T 1 ≤ k ≤ T F k \mathcal{F}_k F k f 0 , f 1 ( U ‾ 1 ) , … , f k − 1 ( U ‾ k − 1 ) f_0,f_1(\overline{U}^{1}),\ldots,f_{k-1}(\overline{U}^{k-1}) f 0 , f 1 ( U 1 ) , … , f k − 1 ( U k − 1 ) L 2 L^2 L 2 U k \mathcal{U}_k U k U 1 , … , U k U_1,\ldots,U_k U 1 , … , U k ι k : F k → U k \iota_k:\mathcal{F}_k \to\mathcal{U}_k ι k : F k → U k ι k ( f j ( U ‾ j ) ) = U j + 1 \iota_k(f_j(\overline{U}^{j}))=U_{j+1} ι k ( f j ( U j )) = U j + 1 U k \mathcal{U}_k U k ι k + 1 \iota_{k+1} ι k + 1 ι k \iota_k ι k k < T k<T k < T k < T k<T k < T L 2 L^2 L 2 U k + 1 U_{k+1} U k + 1 U k \mathcal{U}_k U k L 2 L^2 L 2 f k ( U ‾ k ) f_k(\overline{U}^{k}) f k ( U k ) F k \mathcal{F}_k F k ι k + 1 \iota_{k+1} ι k + 1 ξ ∈ F k + 1 \xi\in\mathcal{F}_{k+1} ξ ∈ F k + 1 F k \mathcal{F}_k F k U k + 1 \mathcal{U}_{k+1} U k + 1 U k \mathcal{U}_k U k
Proof. By (62) , for all real coefficients c j , c j ′ c_j,c'_j c j , c j ′
E [ ( ∑ j c j f j ( U ‾ j ) ) ( ∑ j c j ′ f j ( U ‾ j ) ) ] = E [ ( ∑ j c j U j + 1 ) ( ∑ j c j ′ U j + 1 ) ] . \mathbb{E}\left[\left(\sum_j c_j f_j(\overline{U}^{j})\right)\left(\sum_j c'_j f_j(\overline{U}^{j})\right)\right]=\mathbb{E}\left[\left(\sum_j c_j U_{j+1}\right)\left(\sum_j c'_j U_{j+1}\right)\right]. E [ ( j ∑ c j f j ( U j ) ) ( j ∑ c j ′ f j ( U j ) ) ] = E [ ( j ∑ c j U j + 1 ) ( j ∑ c j ′ U j + 1 ) ] . If ∑ j c j f j ( U ‾ j ) = 0 \sum_j c_j f_j(\overline{U}^{j})=0 ∑ j c j f j ( U j ) = 0 L 2 L^2 L 2 c ′ = c c'=c c ′ = c ∑ j c j U j + 1 = 0 \sum_j c_j U_{j+1}=0 ∑ j c j U j + 1 = 0 L 2 L^2 L 2 ι k \iota_k ι k ι k + 1 \iota_{k+1} ι k + 1 ι k \iota_k ι k ι k + 1 \iota_{k+1} ι k + 1 F k + 1 \mathcal{F}_{k+1} F k + 1 U k + 1 \mathcal{U}_{k+1} U k + 1 F k \mathcal{F}_k F k U k \mathcal{U}_k U k F k \mathcal{F}_k F k F k + 1 \mathcal{F}_{k+1} F k + 1 U k \mathcal{U}_k U k U k + 1 \mathcal{U}_{k+1} U k + 1 f k ( U ‾ k ) f_k(\overline{U}^{k}) f k ( U k ) F k \mathcal{F}_k F k U k + 1 U_{k+1} U k + 1 U k \mathcal{U}_k U k
The following proposition is the state evolution of (64) on tori and hypercubes. It is proved in § 6.4 .
Proposition 5.6. Let ( G d ) (G_d) ( G d ) T L d \mathbb{T}_L^d T L d L ≥ 3 L \ge3 L ≥ 3 Q d Q_d Q d (64) on G d G_d G d T T T f 0 f_0 f 0 ( f t ) 1 ≤ t ≤ T − 1 (f_t)_{1 \le t \le T-1} ( f t ) 1 ≤ t ≤ T − 1 1 ≤ t ≤ T 1 \le t \le T 1 ≤ t ≤ T
(a) For every polynomial ψ : R t → R \psi:\mathbb{R}^t \to\mathbb{R} ψ : R t → R sup d max x ∈ V E ∣ ψ ( u ‾ x t ) ∣ < ∞ \sup_d \max_{x \in V}\mathbb{E}|\psi(\overline{u}_x^t)|<\infty sup d max x ∈ V E ∣ ψ ( u x t ) ∣ < ∞
(b) For every continuous function ψ : R t → R \psi:\mathbb{R}^t \to\mathbb{R} ψ : R t → R x = x d ∈ V x=x_d \in V x = x d ∈ V d → ∞ d \to\infty d → ∞
E ψ ( u ‾ x t ) → E ψ ( U ‾ t ) , \mathbb{E}\psi(\overline{u}_x^t) \to\mathbb{E}\psi(\overline{U}^t), E ψ ( u x t ) → E ψ ( U t ) , E ∣ 1 n ∑ y ∈ V ψ ( u ‾ y t ) − E ψ ( U ‾ t ) ∣ → 0 , E ∣ 1 D ∑ y ∼ x ψ ( u ‾ y t ) − E ψ ( U ‾ t ) ∣ → 0. \mathbb{E}\left|\frac{1}{n}\sum_{y \in V}\psi(\overline{u}_y^t)-\mathbb{E}\psi(\overline{U}^t)\right| \to0,\qquad
\mathbb{E}\left|\frac{1}{D}\sum_{y \sim x}\psi(\overline{u}_y^t)-\mathbb{E}\psi(\overline{U}^t)\right| \to0. E n 1 y ∈ V ∑ ψ ( u y t ) − E ψ ( U t ) → 0 , E D 1 y ∼ x ∑ ψ ( u y t ) − E ψ ( U t ) → 0. By Lemma 5.17 below, the quantities in (b) have laws that do not depend on x x x x x x
Two imported results
We use the following special case of a theorem of Hachem [80, Proposition 8], which extends the polynomial state evolution of Bayati, Lelarge, and Montanari [22 ] to sparse variance profiles. In Hachem’s recursion, the Onsager term is computed from the squared entries of the matrix and from the current iterates. State evolution for matrices with a variance profile was also proved by Wang, Zhong, and Fan [140 ] and by Bao, Han, and Xu [13 ] , for entries of variance O ( 1 / n ) O(1/n) O ( 1/ n ) W W W 1 / D 1/D 1/ D D / n → 0 D/n \to0 D / n → 0
Theorem 5.7. Let ( n k ) k ≥ 1 (n_k)_{k\geq1} ( n k ) k ≥ 1 ( D k ) k ≥ 1 (D_k)_{k\geq1} ( D k ) k ≥ 1 D k ≤ n k D_k \leq n_k D k ≤ n k D k → ∞ D_k \to\infty D k → ∞ k k k S ( k ) S^{(k)} S ( k ) n k × n k n_k \times n_k n k × n k D k D_k D k D k − 1 D_k^{-1} D k − 1 ( g x y ) 1 ≤ x < y ≤ n k (g_{xy})_{1\leq x<y\leq n_k} ( g x y ) 1 ≤ x < y ≤ n k W ( k ) W^{(k)} W ( k ) W x y = ( S x y ( k ) ) 1 / 2 g x y W_{xy}=(S_{xy}^{(k)})^{1/2}g_{xy} W x y = ( S x y ( k ) ) 1/2 g x y x < y x<y x < y q , T ≥ 1 q,T\geq1 q , T ≥ 1 f t = ( f t , 1 , … , f t , q ) : R q → R q f_t=(f_{t,1},\ldots,f_{t,q}):\mathbb{R}^q\to\mathbb{R}^q f t = ( f t , 1 , … , f t , q ) : R q → R q 0 ≤ t ≤ T − 1 0\leq t\leq T-1 0 ≤ t ≤ T − 1 k k k f − 1 = 0 f_{-1}=0 f − 1 = 0 z x t ∈ R q z_x^t\in\mathbb{R}^q z x t ∈ R q x ∈ [ n k ] x\in[n_k] x ∈ [ n k ] 0 ≤ t ≤ T 0\leq t\leq T 0 ≤ t ≤ T z x − 1 = z x 0 = 0 z_x^{-1}=z_x^0=0 z x − 1 = z x 0 = 0
z x t + 1 ( r ) = ∑ y W x y f t , r ( z y t ) − ∑ s = 1 q f t − 1 , s ( z x t − 1 ) ∑ y W x y 2 ∂ s f t , r ( z y t ) ( r ∈ [ q ] ) . (65) z_x^{t+1}(r)=\sum_y W_{xy}f_{t,r}(z_y^t)-\sum_{s=1}^q f_{t-1,s}(z_x^{t-1})\sum_y W_{xy}^2\partial_s f_{t,r}(z_y^t)\qquad(r\in[q]). \tag*{(65)} z x t + 1 ( r ) = y ∑ W x y f t , r ( z y t ) − s = 1 ∑ q f t − 1 , s ( z x t − 1 ) y ∑ W x y 2 ∂ s f t , r ( z y t ) ( r ∈ [ q ]) . (65) Define positive-semidefinite q × q q\times q q × q
Ξ x 1 = ∑ y S x y ( k ) f 0 ( 0 ) f 0 ( 0 ) T , Ξ x t + 1 = ∑ y S x y ( k ) E [ f t ( ζ y t ) f t ( ζ y t ) T ] , \Xi_x^1=\sum_y S_{xy}^{(k)}f_0(0)f_0(0)^{\mathsf T},\qquad\Xi_x^{t+1}=\sum_y S_{xy}^{(k)}\mathbb{E}\left[f_t(\zeta_y^t)f_t(\zeta_y^t)^{\mathsf T}\right], Ξ x 1 = y ∑ S x y ( k ) f 0 ( 0 ) f 0 ( 0 ) T , Ξ x t + 1 = y ∑ S x y ( k ) E [ f t ( ζ y t ) f t ( ζ y t ) T ] , where ζ y t ∼ N ( 0 , Ξ y t ) \zeta_y^t\sim N(0,\Xi_y^t) ζ y t ∼ N ( 0 , Ξ y t ) 1 ≤ t ≤ T 1\leq t\leq T 1 ≤ t ≤ T ψ : R q → R \psi:\mathbb{R}^q\to\mathbb{R} ψ : R q → R
(i) We have sup k max x ∥ Ξ x t ∥ < ∞ \sup_k\max_x\|\Xi_x^t\|<\infty sup k max x ∥ Ξ x t ∥ < ∞ sup k max x E ∣ ψ ( z x t ) ∣ < ∞ \sup_k\max_x\mathbb{E}|\psi(z_x^t)|<\infty sup k max x E ∣ ψ ( z x t ) ∣ < ∞
(ii) For every C > 0 C>0 C > 0 I k ⊂ [ n k ] I_k\subset[n_k] I k ⊂ [ n k ] ∣ I k ∣ ≤ C D k |I_k|\leq CD_k ∣ I k ∣ ≤ C D k k → ∞ k\to\infty k → ∞
1 D k ∑ x ∈ I k ( ψ ( z x t ) − E ψ ( ζ x t ) ) → 0 and 1 n k ∑ x ∈ [ n k ] ( ψ ( z x t ) − E ψ ( ζ x t ) ) → 0 \frac{1}{D_k}\sum_{x\in I_k}\left(\psi(z_x^t)-\mathbb{E}\psi(\zeta_x^t)\right)\to0
\qquad\text{and}\qquad
\frac{1}{n_k}\sum_{x\in[n_k]}\left(\psi(z_x^t)-\mathbb{E}\psi(\zeta_x^t)\right)\to0 D k 1 x ∈ I k ∑ ( ψ ( z x t ) − E ψ ( ζ x t ) ) → 0 and n k 1 x ∈ [ n k ] ∑ ( ψ ( z x t ) − E ψ ( ζ x t ) ) → 0 in probability.
In the notation of [80 ] , Theorem 5.7 is Proposition 8, equations (19)–(21), with sparsity parameter K n = D k K_n=D_k K n = D k C c a r d = C S = 1 C_{\mathrm{card}}=C_S=1 C card = C S = 1 n n n [80 ] requires only K n → ∞ K_n\to\infty K n → ∞ K n ≤ n K_n\leq n K n ≤ n K n K_n K n log n \log n log n D k − 1 D_k^{-1} D k − 1 D k D_k D k ∥ W ∥ \lVert W\rVert ∥ W ∥
The second imported result is a bound of Bandeira and van Handel [12 Theorem 1.1] on the norm of a Gaussian matrix with a variance profile.
Theorem 5.8. Let X X X n × n n \times n n × n X x y = b x y g x y X_{xy}=b_{xy}g_{xy} X x y = b x y g x y { g x y : x ≥ y } \{g_{xy}:x\ge y\} { g x y : x ≥ y } { b x y : x ≥ y } \{b_{xy}:x\ge y\} { b x y : x ≥ y } b x y = b y x b_{xy}=b_{yx} b x y = b y x σ = max x ( ∑ y b x y 2 ) 1 / 2 \sigma=\max_x\left(\sum_y b_{xy}^2\right)^{1/2} σ = max x ( ∑ y b x y 2 ) 1/2 σ ∗ = max x , y ∣ b x y ∣ \sigma_*=\max_{x,y}|b_{xy}| σ ∗ = max x , y ∣ b x y ∣ 0 < ϵ ′ ≤ 1 / 2 0<\epsilon'\le1/2 0 < ϵ ′ ≤ 1/2
E ∥ X ∥ ≤ ( 1 + ϵ ′ ) ( 2 σ + 6 log ( 1 + ϵ ′ ) σ ∗ log n ) . \mathbb{E}\lVert X\rVert\le(1+\epsilon')\left(2\sigma+\frac{6}{\sqrt{\log(1+\epsilon')}}\sigma_*\sqrt{\log n}\right). E ∥ X ∥ ≤ ( 1 + ϵ ′ ) ( 2 σ + log ( 1 + ϵ ′ ) 6 σ ∗ log n ) . Lemma 5.9. Let G ∈ G D G\in\mathcal{G}_D G ∈ G D n ≥ 2 n\ge2 n ≥ 2 E ∥ W ∥ ≤ 3 + 15 ( D − 1 log n ) 1 / 2 \mathbb{E}\lVert W\rVert\le3+15(D^{-1}\log n)^{1/2} E ∥ W ∥ ≤ 3 + 15 ( D − 1 log n ) 1/2 T L d \mathbb{T}_L^d T L d L ≥ 3 L\ge3 L ≥ 3 Q d Q_d Q d sup d E ∥ W ∥ p < ∞ \sup_d\mathbb{E}\lVert W\rVert^p<\infty sup d E ∥ W ∥ p < ∞ p ≥ 1 p\ge1 p ≥ 1
Proof. We apply Theorem 5.8 with b x y = S x y 1 / 2 b_{xy}=S_{xy}^{1/2} b x y = S x y 1/2 b x x = 0 b_{xx}=0 b xx = 0 σ = 1 \sigma=1 σ = 1 σ ∗ = D − 1 / 2 \sigma_*=D^{-1/2} σ ∗ = D − 1/2 ϵ ′ = 1 / 2 \epsilon'=1/2 ϵ ′ = 1/2 9 ( log ( 3 / 2 ) ) − 1 / 2 ≤ 15 9(\log(3/2))^{-1/2}\le15 9 ( log ( 3/2 ) ) − 1/2 ≤ 15 ∥ W ∥ \lVert W\rVert ∥ W ∥ g = ( g e ) e ∈ E g=(g_e)_{e\in E} g = ( g e ) e ∈ E g , g ′ g,g' g , g ′
∣ ∥ W ( g ) ∥ − ∥ W ( g ′ ) ∥ ∣ ≤ ∥ W ( g ) − W ( g ′ ) ∥ H S = ( 2 D ∑ e ∈ E ( g e − g e ′ ) 2 ) 1 / 2 , \left|\lVert W(g)\rVert-\lVert W(g')\rVert\right|\le\lVert W(g)-W(g')\rVert_{\mathrm{HS}}=\left(\frac{2}{D}\sum_{e\in E}(g_e-g'_e)^2\right)^{1/2}, ∣ ∥ W ( g )∥ − ∥ W ( g ′ )∥ ∣ ≤ ∥ W ( g ) − W ( g ′ ) ∥ HS = ( D 2 e ∈ E ∑ ( g e − g e ′ ) 2 ) 1/2 , that is, ∥ W ∥ \lVert W\rVert ∥ W ∥ ( 2 / D ) 1 / 2 (2/D)^{1/2} ( 2/ D ) 1/2 ∥ ∥ W ∥ − E ∥ W ∥ ∥ p ≤ C p ( 2 / D ) 1 / 2 \left\|\lVert W\rVert-\mathbb{E}\lVert W\rVert\right\|_p\le C_p(2/D)^{1/2} ∥ ∥ W ∥ − E ∥ W ∥ ∥ p ≤ C p ( 2/ D ) 1/2 C p C_p C p p p p D − 1 log n = ( log L ) / 2 D^{-1}\log n=(\log L)/2 D − 1 log n = ( log L ) /2 D − 1 log n = log 2 D^{-1}\log n=\log2 D − 1 log n = log 2 sup d E ∥ W ∥ < ∞ \sup_d\mathbb{E}\lVert W\rVert<\infty sup d E ∥ W ∥ < ∞ □ \square □
Tame polynomials and coefficient replacement
In this subsection G ∈ G D G\in\mathcal{G}_D G ∈ G D W e = J e W_e=J_e W e = J e e ∈ E e\in E e ∈ E N ( 0 , D − 1 ) N(0,D^{-1}) N ( 0 , D − 1 ) D D D x x x x x x C D s CD^s C D s s s s C / D C/D C / D [22 ] . The letter C C C D D D G G G
A monomial is a product M = ∏ e ∈ E W e k e M=\prod_{e\in E}W_e^{k_e} M = ∏ e ∈ E W e k e k e ≥ 0 k_e\ge0 k e ≥ 0 deg M = ∑ e k e \deg M=\sum_e k_e deg M = ∑ e k e supp M = { e : k e ≥ 1 } \operatorname{supp}M=\{e:k_e\ge1\} supp M = { e : k e ≥ 1 } connected if the graph formed by these edges and their endpoints is connected.
Definition 5.10. Let x ∈ V x \in V x ∈ V k ≥ 0 k \ge0 k ≥ 0 C 0 ≥ 0 C_0 \ge0 C 0 ≥ 0 Y = ∑ M c M M Y = \sum_M c_M M Y = ∑ M c M M tame rooted at x x x of type ( k , C 0 ) (k,C_0) ( k , C 0 ) M M M c M ≠ 0 c_M \ne0 c M = 0 deg M ≤ k \deg M \le k deg M ≤ k ∣ c M ∣ ≤ C 0 |c_M| \le C_0 ∣ c M ∣ ≤ C 0 supp M \operatorname{supp} M supp M x x x
Lemma 5.11. Let x ∈ V x \in V x ∈ V
(a) If Y Y Y Y ′ Y' Y ′ x x x ( k , C 0 ) (k,C_0) ( k , C 0 ) ( k ′ , C 0 ′ ) (k',C_0') ( k ′ , C 0 ′ ) Y + Y ′ Y+Y' Y + Y ′ x x x ( max ( k , k ′ ) , C 0 + C 0 ′ ) (\max(k,k'), C_0+C_0') ( max ( k , k ′ ) , C 0 + C 0 ′ ) Y Y ′ YY' Y Y ′ x x x ( k + k ′ , 2 k + k ′ C 0 C 0 ′ ) (k+k',2^{k+k'}C_0C_0') ( k + k ′ , 2 k + k ′ C 0 C 0 ′ ) m m m C 1 C_1 C 1 j j j x x x ( k , C 0 ) (k,C_0) ( k , C 0 ) x x x k k k C 0 C_0 C 0 m m m C 1 C_1 C 1 j j j
(b) If Y y Y_y Y y y y y ( k , C 0 ) (k,C_0) ( k , C 0 ) y y y x x x ∑ y ∼ x W x y Y y \sum_{y\sim x} W_{xy}Y_y ∑ y ∼ x W x y Y y ∑ y ∼ x W x y 2 Y y \sum_{y\sim x} W_{xy}^2Y_y ∑ y ∼ x W x y 2 Y y x x x ( k + 1 , ( k + 1 ) C 0 ) (k+1,(k+1)C_0) ( k + 1 , ( k + 1 ) C 0 ) ( k + 2 , ( k + 2 ) C 0 ) (k+2,(k+2)C_0) ( k + 2 , ( k + 2 ) C 0 )
Proof. (a) The statement on sums is immediate. The coefficient of M M M Y Y ′ YY' Y Y ′ ∑ M 1 M 2 = M c M 1 c M 2 ′ \sum_{M_1M_2=M}c_{M_1}c'_{M_2} ∑ M 1 M 2 = M c M 1 c M 2 ′ M M M ∏ e ( k e + 1 ) ≤ 2 deg M \prod_e(k_e+1)\le2^{\deg M} ∏ e ( k e + 1 ) ≤ 2 d e g M supp ( M 1 M 2 ) = supp M 1 ∪ supp M 2 \operatorname{supp}(M_1M_2)=\operatorname{supp}M_1\cup\operatorname{supp}M_2 supp ( M 1 M 2 ) = supp M 1 ∪ supp M 2 x x x x x x x x x ( 0 , ∣ c ∣ ) (0,|c|) ( 0 , ∣ c ∣ )
(b) Let M y M_y M y Y y Y_y Y y W x y M y W_{xy}M_y W x y M y supp M y ∪ { x y } \operatorname{supp}M_y\cup\{xy\} supp M y ∪ { x y } x y xy x y x x x supp M y \operatorname{supp}M_y supp M y y y y y y y x y xy x y M M M W x y M y W_{xy}M_y W x y M y y y y x y ∈ supp M xy\in\operatorname{supp}M x y ∈ supp M deg M ≤ k + 1 \deg M\le k+1 deg M ≤ k + 1 M y = M / W x y M_y=M/W_{xy} M y = M / W x y M M M ( k + 1 ) C 0 (k+1)C_0 ( k + 1 ) C 0 W x y 2 M y W_{xy}^2M_y W x y 2 M y
:::
Lemma 5.12. Let x ∈ V x \in V x ∈ V v ≥ 2 v \ge2 v ≥ 2 x x x v v v 4 v 2 v 2 D v − 1 4^v2^{v^2}D^{v-1} 4 v 2 v 2 D v − 1
Proof. Such a set S S S x x x 4 v 4^v 4 v v v v D v − 1 D^{v-1} D v − 1 x x x S S S v v v 2 v 2 2^{v^2} 2 v 2
Lemma 5.13. If Y 1 , … , Y j Y_1,\ldots,Y_j Y 1 , … , Y j x x x ( k , C 0 ) (k,C_0) ( k , C 0 ) ∣ E [ Y 1 ⋯ Y j ] ∣ ≤ C |\mathbb{E}[Y_1\cdots Y_j]|\le C ∣ E [ Y 1 ⋯ Y j ] ∣ ≤ C C C C j j j k k k C 0 C_0 C 0 Y Y Y x x x ( k , C 0 ) (k,C_0) ( k , C 0 ) E ∣ Y ∣ p ≤ C \mathbb{E}|Y|^p\le C E ∣ Y ∣ p ≤ C p ≥ 1 p\ge1 p ≥ 1 C C C p p p k k k C 0 C_0 C 0
Proof. We expand E ∏ i Y i = ∑ ∏ i c M i E ∏ i M i \mathbb{E}\prod_iY_i=\sum\prod_i c_{M_i}\mathbb{E}\prod_iM_i E ∏ i Y i = ∑ ∏ i c M i E ∏ i M i j j j S = ⋃ i supp M i S=\bigcup_i\operatorname{supp}M_i S = ⋃ i supp M i μ e = ∑ i k e ( M i ) \mu_e=\sum_i k_e(M_i) μ e = ∑ i k e ( M i ) e ∈ S e\in S e ∈ S E ∏ i M i = ∏ e ∈ S E W e μ e \mathbb{E}\prod_iM_i=\prod_{e\in S}\mathbb{E}W_e^{\mu_e} E ∏ i M i = ∏ e ∈ S E W e μ e μ e \mu_e μ e
∣ E ∏ i M i ∣ = ∏ e ∈ S ( μ e − 1 ) ! ! D − μ e / 2 ≤ ( j k ) ! D − ∣ S ∣ , \left|\mathbb{E}\prod_i M_i\right|=\prod_{e\in S}(\mu_e-1)!!\,D^{-\mu_e/2}\le(jk)!\,D^{-|S|}, E i ∏ M i = e ∈ S ∏ ( μ e − 1 )!! D − μ e /2 ≤ ( j k )! D − ∣ S ∣ , since μ e ≥ 2 \mu_e\ge2 μ e ≥ 2 e ∈ S e\in S e ∈ S S S S M i M_i M i C 0 j C_0^j C 0 j S S S x x x supp M i \operatorname{supp} M_i supp M i x x x 2 ≤ v ≤ j k + 1 2\le v\le jk+1 2 ≤ v ≤ j k + 1 ∣ S ∣ ≥ v − 1 |S|\ge v-1 ∣ S ∣ ≥ v − 1 S S S ( k + 1 ) j ∣ S ∣ ≤ ( k + 1 ) j 2 k (k+1)^{j|S|}\le(k+1)^{j^2k} ( k + 1 ) j ∣ S ∣ ≤ ( k + 1 ) j 2 k k k k S S S Lemma 5.12 ,
∣ E [ Y 1 ⋯ Y j ] ∣ ≤ C 0 j ( 1 + ∑ v = 2 j k + 1 4 v v 2 v D v − 1 ( k + 1 ) j 2 k ( j k ) ! D − ( v − 1 ) ) , \left|\mathbb{E}[Y_1\cdots Y_j]\right|\le C_0^j\left(1+\sum_{v=2}^{jk+1}4^v v^{2v}D^{v-1}(k+1)^{j^2k}(jk)!\,D^{-(v-1)}\right), ∣ E [ Y 1 ⋯ Y j ] ∣ ≤ C 0 j ( 1 + v = 2 ∑ j k + 1 4 v v 2 v D v − 1 ( k + 1 ) j 2 k ( j k )! D − ( v − 1 ) ) , which depends only on j j j k k k C 0 C_0 C 0 Y 1 = ⋯ = Y 2 p ′ = Y Y_1=\cdots=Y_{2p'}=Y Y 1 = ⋯ = Y 2 p ′ = Y p ′ ≥ p / 2 p'\ge p/2 p ′ ≥ p /2
The next lemma replaces squared couplings by their variances in an Onsager coefficient. For dense matrices, the corresponding fourth-moment bound is proved in [22 Appendix A.2] . Hachem stated a version for sparse variance profiles in [80 Lemma 21] and omitted its proof. We give a complete proof for tame polynomials, along the lines of [22 ] .
Lemma 5.14. Let x ∈ V x\in V x ∈ V y y y x x x Y y Y_y Y y y y y ( k , C 0 ) (k,C_0) ( k , C 0 )
E ( ∑ y ∼ x ( W x y 2 − D − 1 ) Y y ) 4 ≤ C D 2 , \mathbb{E}\left(\sum_{y\sim x}(W_{xy}^2-D^{-1})Y_y\right)^4\le\frac{C}{D^2}, E ( y ∼ x ∑ ( W x y 2 − D − 1 ) Y y ) 4 ≤ D 2 C , where C C C k k k C 0 C_0 C 0
Proof. Expanding the fourth power and each Y y Y_y Y y ( y a , M a ) a ≤ 4 (y_a,M_a)_{a\le4} ( y a , M a ) a ≤ 4 y a ∼ x y_a\sim x y a ∼ x M a M_a M a Y y a Y_{y_a} Y y a
∏ a = 1 4 c M a ( y a ) ⋅ E ∏ a = 1 4 ( W x y a 2 − D − 1 ) M a . \prod_{a=1}^{4}c_{M_a}^{(y_a)}\cdot\mathbb{E}\prod_{a=1}^{4}(W_{xy_a}^2-D^{-1})M_a. a = 1 ∏ 4 c M a ( y a ) ⋅ E a = 1 ∏ 4 ( W x y a 2 − D − 1 ) M a . Fix a tuple. Let e a = x y a e_a=xy_a e a = x y a S = ⋃ a ( supp M a ∪ { e a } ) S=\bigcup_a(\operatorname{supp} M_a\cup\{e_a\}) S = ⋃ a ( supp M a ∪ { e a }) e ∈ S e\in S e ∈ S ν e = ∣ { a : e a = e } ∣ \nu_e=|\{a:e_a=e\}| ν e = ∣ { a : e a = e } ∣ μ e = ∑ a k e ( M a ) \mu_e=\sum_a k_e(M_a) μ e = ∑ a k e ( M a )
E ∏ a = 1 4 ( W x y a 2 − D − 1 ) M a = ∏ e ∈ S w e , w e = E [ W e μ e ( W e 2 − D − 1 ) ν e ] . \mathbb{E}\prod_{a=1}^{4}(W_{xy_a}^2-D^{-1})M_a=\prod_{e\in S}w_e,\qquad w_e=\mathbb{E}\left[W_e^{\mu_e}(W_e^2-D^{-1})^{\nu_e}\right]. E a = 1 ∏ 4 ( W x y a 2 − D − 1 ) M a = e ∈ S ∏ w e , w e = E [ W e μ e ( W e 2 − D − 1 ) ν e ] . Writing W e = D − 1 / 2 g e W_e=D^{-1/2}g_e W e = D − 1/2 g e w e = D − ( μ e + 2 ν e ) / 2 E [ g e μ e ( g e 2 − 1 ) ν e ] w_e=D^{-(\mu_e+2\nu_e)/2}\mathbb{E}[g_e^{\mu_e}(g_e^2-1)^{\nu_e}] w e = D − ( μ e + 2 ν e ) /2 E [ g e μ e ( g e 2 − 1 ) ν e ] w e w_e w e
∣ ∏ e ∈ S w e ∣ ≤ C D − ∣ S ∣ − 2 . (66) \left|\prod_{e\in S}w_e\right|\le C D^{-|S|-2}. \tag*{(66)} e ∈ S ∏ w e ≤ C D − ∣ S ∣ − 2 . (66) Suppose that no w e w_e w e μ e \mu_e μ e ν e = 0 \nu_e = 0 ν e = 0 μ e ≥ 2 \mu_e \ge2 μ e ≥ 2 e ∈ S e \in S e ∈ S ∣ w e ∣ ≤ C D − 1 |w_e| \le CD^{-1} ∣ w e ∣ ≤ C D − 1 ν e = 1 \nu_e = 1 ν e = 1 μ e ≥ 2 \mu_e \ge2 μ e ≥ 2 E [ g e 2 − 1 ] = 0 \mathbb{E}[g_e^2 - 1] = 0 E [ g e 2 − 1 ] = 0 ∣ w e ∣ ≤ C D − 2 |w_e| \le CD^{-2} ∣ w e ∣ ≤ C D − 2 μ e + 2 ν e ≥ 4 \mu_e + 2\nu_e \ge4 μ e + 2 ν e ≥ 4 ν e ≥ 2 \nu_e \ge2 ν e ≥ 2 ∣ w e ∣ ≤ C D − ν e |w_e| \le CD^{-\nu_e} ∣ w e ∣ ≤ C D − ν e ν e ≥ 1 \nu_e \ge1 ν e ≥ 1 ∣ w e ∣ ≤ C D − 1 − θ e |w_e| \le CD^{-1-\theta_e} ∣ w e ∣ ≤ C D − 1 − θ e θ e = max ( 1 , ν e − 1 ) \theta_e = \max(1,\nu_e - 1) θ e = max ( 1 , ν e − 1 ) ∑ e ν e = 4 \sum_e \nu_e = 4 ∑ e ν e = 4 ν e = 4 \nu_e = 4 ν e = 4 θ e = 3 \theta_e = 3 θ e = 3 ν e ≥ 1 \nu_e \ge1 ν e ≥ 1 ∑ e : ν e ≥ 1 θ e ≥ 2 \sum_{e:\nu_e\ge1}\theta_e \ge2 ∑ e : ν e ≥ 1 θ e ≥ 2
Each set supp M a ∪ { e a } \operatorname{supp} M_a \cup\{e_a\} supp M a ∪ { e a } e a e_a e a x x x M a M_a M a y a y_a y a e a = x y a e_a = xy_a e a = x y a S S S x x x 4 ( k + 1 ) 4(k+1) 4 ( k + 1 ) 2 ≤ v ≤ 4 k + 5 2 \le v \le4k+5 2 ≤ v ≤ 4 k + 5 ∣ S ∣ ≥ v − 1 |S| \ge v-1 ∣ S ∣ ≥ v − 1 S S S ∣ S ∣ 4 ( k + 1 ) 4 ∣ S ∣ ≤ C |S|^{4}(k+1)^{4|S|} \le C ∣ S ∣ 4 ( k + 1 ) 4∣ S ∣ ≤ C
C 0 4 ∑ v = 2 4 k + 5 4 v 2 v 2 D v − 1 ⋅ C ⋅ C D − ( v − 1 ) − 2 ≤ C D 2 . C_0^4 \sum_{v=2}^{4k+5} 4^v 2^{v^2} D^{v-1} \cdot C \cdot CD^{-(v-1)-2} \le\frac{C}{D^2}. C 0 4 v = 2 ∑ 4 k + 5 4 v 2 v 2 D v − 1 ⋅ C ⋅ C D − ( v − 1 ) − 2 ≤ D 2 C . Proof of the state evolution
We compare (5.3) with the recursion in which the Onsager coefficients are computed from the disorder: u ~ 1 = f 0 W 1 \widetilde{u}^1 = f_0W1 u 1 = f 0 W 1 1 ≤ t ≤ T − 1 1 \le t \le T-1 1 ≤ t ≤ T − 1
u ~ t + 1 = W f t ( u ‾ ~ t ) − ∑ s = 1 t b ~ t , s ⊙ f s − 1 ( u ‾ ~ s − 1 ) , ( b ~ t , s ) x = ∑ y W x y 2 ∂ s f t ( u ‾ ~ y t ) . (67) \widetilde{u}^{t+1} = Wf_t(\widetilde{\overline{u}}^t) - \sum_{s=1}^{t} \widetilde{b}_{t,s} \odot f_{s-1}(\widetilde{\overline{u}}^{s-1}), \qquad(\widetilde{b}_{t,s})_x = \sum_y W_{xy}^2 \partial_s f_t(\widetilde{\overline{u}}^t_y). \tag*{(67)} u t + 1 = W f t ( u t ) − s = 1 ∑ t b t , s ⊙ f s − 1 ( u s − 1 ) , ( b t , s ) x = y ∑ W x y 2 ∂ s f t ( u y t ) . (67) This recursion is used only in the proof; it is not a query history.
Lemma 5.15. Let G ∈ G D G \in G_D G ∈ G D x ∈ V x \in V x ∈ V 1 ≤ t ≤ T 1 \le t \le T 1 ≤ t ≤ T u ‾ x t \overline{u}^t_x u x t u ‾ ~ x t \widetilde{\overline{u}}^t_x u x t x x x t t t f 0 , … , f T − 1 f_0,\ldots,f_{T-1} f 0 , … , f T − 1 ψ : R 2 t → R \psi:\mathbb{R}^{2t} \to\mathbb{R} ψ : R 2 t → R p ≥ 1 p \ge1 p ≥ 1
sup D ≥ 1 sup G ∈ G D max x ∈ V E ∣ ψ ( u ‾ x t , u ‾ ~ x t ) ∣ p < ∞ . \sup_{D\ge1}\sup_{G\in G_D}\max_{x\in V}\mathbb{E}\left|\psi\left(\overline{u}^t_x,\widetilde{\overline{u}}^t_x\right)\right|^p < \infty. D ≥ 1 sup G ∈ G D sup x ∈ V max E ψ ( u x t , u x t ) p < ∞. In particular, Proposition 5.6(a) holds.
Proof. We induct on t t t u x 1 = u ~ x 1 = f 0 ∑ y ∼ x W x y u^1_x = \widetilde{u}^1_x = f_0 \sum_{y\sim x} W_{xy} u x 1 = u x 1 = f 0 ∑ y ∼ x W x y x x x Y y = f 0 Y_y = f_0 Y y = f 0 t t t f t ( u ‾ y t ) f_t(\overline{u}^t_y) f t ( u y t ) f t ( u ‾ ~ y t ) f_t(\widetilde{\overline{u}}^t_y) f t ( u y t ) ∂ s f t ( u ‾ ~ y t ) \partial_s f_t(\widetilde{\overline{u}}^t_y) ∂ s f t ( u y t ) y y y f s − 1 ( u ‾ x s − 1 ) f_{s-1}(\overline{u}^{s-1}_x) f s − 1 ( u x s − 1 ) f s − 1 ( u ‾ ~ x s − 1 ) f_{s-1}(\widetilde{\overline{u}}^{s-1}_x) f s − 1 ( u x s − 1 ) x x x ∑ y ∼ x W x y f t ( u ‾ y t ) \sum_{y\sim x} W_{xy}f_t(\overline{u}^t_y) ∑ y ∼ x W x y f t ( u y t ) ∑ y ∼ x W x y f t ( u ‾ ~ y t ) \sum_{y\sim x} W_{xy}f_t(\widetilde{\overline{u}}^t_y) ∑ y ∼ x W x y f t ( u y t ) ( b ~ t , s ) x = ∑ y ∼ x W x y 2 ∂ s f t ( u ‾ ~ y t ) (\widetilde{b}_{t,s})_x = \sum_{y\sim x} W_{xy}^2\partial_s f_t(\widetilde{\overline{u}}^t_y) ( b t , s ) x = ∑ y ∼ x W x y 2 ∂ s f t ( u y t ) x x x u x t + 1 u^{t+1}_x u x t + 1 u ~ x t + 1 \widetilde{u}^{t+1}_x u x t + 1 x x x t t t b t , s b_{t,s} b t , s
The next lemma identifies (5.6) as an instance of (5.4). The vector iterate stores the history of the scalar iterates; once a coordinate has appeared, the recursion recomputes the same value at every later step. The same reduction of a recursion with memory to a vector recursion appears in [103 Appendix A] , where it is combined with [91, Theorem 1.
Lemma 5.16. Let ( G d ) (G_d) ( G d ) T L d \mathbb{T}_L^d T L d L ≥ 3 L \ge3 L ≥ 3 Q d Q_d Q d 1 ≤ t ≤ T 1 \le t \le T 1 ≤ t ≤ T ψ : R t → R \psi:\mathbb{R}^t \to\mathbb{R} ψ : R t → R x = x d ∈ V x=x_d \in V x = x d ∈ V d → ∞ d \to\infty d → ∞
1 D ∑ y ∼ x ψ ( u ˉ ~ y t ) → E ψ ( U ˉ t ) and 1 n ∑ y ∈ V ψ ( u ˉ ~ y t ) → E ψ ( U ˉ t ) \frac{1}{D}\sum_{y\sim x}\psi\left(\widetilde{\bar{u}}_y^t\right) \to\mathbb{E}\psi\left(\bar{U}^t\right)
\qquad\text{and}\qquad
\frac{1}{n}\sum_{y\in V}\psi\left(\widetilde{\bar{u}}_y^t\right) \to\mathbb{E}\psi\left(\bar{U}^t\right) D 1 y ∼ x ∑ ψ ( u ˉ y t ) → E ψ ( U ˉ t ) and n 1 y ∈ V ∑ ψ ( u ˉ y t ) → E ψ ( U ˉ t ) in probability.
Proof. The graphs G d G_d G d k = d k=d k = d n k = n n_k=n n k = n D k = D D_k=D D k = D S ( k ) = S S^{(k)}=S S ( k ) = S S S S D D D D − 1 D^{-1} D − 1 D ≤ n D \le n D ≤ n W ( k ) W^{(k)} W ( k ) W W W g x y g_{xy} g x y x , y x,y x , y q = T q=T q = T 0 ≤ t ′ ≤ T − 1 0 \le t' \le T-1 0 ≤ t ′ ≤ T − 1 f t ′ : R T → R T f_{t'}:\mathbb{R}^T \to\mathbb{R}^T f t ′ : R T → R T
f t ′ , r ( z ) = f r − 1 ( z 1 , … , z r − 1 ) ( r ≤ t ′ + 1 ) , f t ′ , r ( z ) = 0 ( r > t ′ + 1 ) . f_{t',r}(z)=f_{r-1}(z_1,\ldots,z_{r-1})\quad(r\le t'+1),\qquad f_{t',r}(z)=0\quad(r>t'+1). f t ′ , r ( z ) = f r − 1 ( z 1 , … , z r − 1 ) ( r ≤ t ′ + 1 ) , f t ′ , r ( z ) = 0 ( r > t ′ + 1 ) . We claim that z x t ′ = ( u ~ x 1 , … , u ~ x t ′ , 0 , … , 0 ) z_x^{t'}=(\widetilde{u}_x^1,\ldots,\widetilde{u}_x^{t'},0,\ldots,0) z x t ′ = ( u x 1 , … , u x t ′ , 0 , … , 0 ) 0 ≤ t ′ ≤ T 0\le t'\le T 0 ≤ t ′ ≤ T x ∈ V x\in V x ∈ V t ′ = 0 t'=0 t ′ = 0 t ′ = 1 t'=1 t ′ = 1 f 0 ( 0 ) = ( f 0 , 0 , … , 0 ) f_0(0)=(f_0,0,\ldots,0) f 0 ( 0 ) = ( f 0 , 0 , … , 0 ) t ′ − 1 t'-1 t ′ − 1 t ′ t' t ′ 1 ≤ t ′ ≤ T − 1 1\le t'\le T-1 1 ≤ t ′ ≤ T − 1 r > t ′ + 1 r>t'+1 r > t ′ + 1 f t ′ , r = 0 f_{t',r}=0 f t ′ , r = 0 z x t ′ + 1 ( r ) = 0 z_x^{t'+1}(r)=0 z x t ′ + 1 ( r ) = 0 r ≤ t ′ + 1 r\le t'+1 r ≤ t ′ + 1 f t ′ , r ( z y t ′ ) = f r − 1 ( u ˉ ~ y r − 1 ) f_{t',r}(z_y^{t'})=f_{r-1}(\widetilde{\bar{u}}_y^{r-1}) f t ′ , r ( z y t ′ ) = f r − 1 ( u ˉ y r − 1 ) r − 1 ≤ t ′ r-1\le t' r − 1 ≤ t ′ ∂ s f t ′ , r \partial_s f_{t',r} ∂ s f t ′ , r ∂ s f r − 1 \partial_s f_{r-1} ∂ s f r − 1 s ≤ r − 1 s\le r-1 s ≤ r − 1 s ≥ r s\ge r s ≥ r s ≤ r − 1 ≤ t ′ s\le r-1\le t' s ≤ r − 1 ≤ t ′ f t ′ − 1 , s ( z x t ′ − 1 ) = f s − 1 ( u ˉ ~ x s − 1 ) f_{t'-1,s}(z_x^{t'-1})=f_{s-1}(\widetilde{\bar{u}}_x^{s-1}) f t ′ − 1 , s ( z x t ′ − 1 ) = f s − 1 ( u ˉ x s − 1 )
z x t ′ + 1 ( r ) = ∑ y W x y f r − 1 ( u ˉ ~ y r − 1 ) − ∑ s = 1 r − 1 f s − 1 ( u ˉ ~ x s − 1 ) ∑ y W x y 2 ∂ s f r − 1 ( u ˉ ~ y r − 1 ) = u ~ x r z_x^{t'+1}(r)=\sum_y W_{xy}f_{r-1}(\widetilde{\bar{u}}_y^{r-1})-\sum_{s=1}^{r-1}f_{s-1}(\widetilde{\bar{u}}_x^{s-1})\sum_y W_{xy}^2\partial_s f_{r-1}(\widetilde{\bar{u}}_y^{r-1})=\widetilde{u}_x^r z x t ′ + 1 ( r ) = y ∑ W x y f r − 1 ( u ˉ y r − 1 ) − s = 1 ∑ r − 1 f s − 1 ( u ˉ x s − 1 ) y ∑ W x y 2 ∂ s f r − 1 ( u ˉ y r − 1 ) = u x r by (5.6) at time r − 1 r-1 r − 1 r = 1 r=1 r = 1 u ~ 1 1 = f 0 W 1 \widetilde{u}_1^1=f_0W_1 u 1 1 = f 0 W 1
Next, we claim that Ξ x t ′ \Xi_x^{t'} Ξ x t ′ ( U 1 , … , U t ′ , 0 , … , 0 ) (U_1,\ldots,U_{t'},0,\ldots,0) ( U 1 , … , U t ′ , 0 , … , 0 ) x x x t ′ = 1 t'=1 t ′ = 1 ∑ y S x y = 1 \sum_y S_{xy}=1 ∑ y S x y = 1 E U 1 2 = f 0 2 \mathbb{E}U_1^2=f_0^2 E U 1 2 = f 0 2 t ′ t' t ′ ζ y t ′ \zeta_y^{t'} ζ y t ′ ( U ˉ t ′ , 0 , … , 0 ) (\bar{U}^{t'},0,\ldots,0) ( U ˉ t ′ , 0 , … , 0 ) y y y ∑ y S x y = 1 \sum_y S_{xy}=1 ∑ y S x y = 1 Ξ x t ′ + 1 \Xi_x^{t'+1} Ξ x t ′ + 1 E f r − 1 ( U ˉ r − 1 ) f r ′ − 1 ( U ˉ r ′ − 1 ) = E U r U r ′ \mathbb{E}f_{r-1}(\bar{U}^{r-1})f_{r'-1}(\bar{U}^{r'-1})=\mathbb{E}U_rU_{r'} E f r − 1 ( U ˉ r − 1 ) f r ′ − 1 ( U ˉ r ′ − 1 ) = E U r U r ′ r , r ′ ≤ t ′ + 1 r,r'\le t'+1 r , r ′ ≤ t ′ + 1
Finally, we apply the local conclusion of Theorem 5.7(ii) at time t t t z ↦ ψ ( z 1 , … , z t ) z\mapsto\psi(z_1,\ldots,z_t) z ↦ ψ ( z 1 , … , z t ) I k I_k I k x d x_d x d D D D ψ ( z ˉ y ) = ψ ( u ˉ ~ y t ) \psi(\bar{z}_y)=\psi(\widetilde{\bar{u}}_y^t) ψ ( z ˉ y ) = ψ ( u ˉ y t ) E ψ ( ζ y t ) = E ψ ( U ˉ t ) \mathbb{E}\psi(\zeta_y^t)=\mathbb{E}\psi(\bar{U}^t) E ψ ( ζ y t ) = E ψ ( U ˉ t ) y y y
We next use the symmetry of tori and hypercubes.
Lemma 5.17. Let G = G d G = G_d G = G d 1 ≤ t ≤ T 1 \le t \le T 1 ≤ t ≤ T ψ : R 2 t → R \psi: \mathbb{R}^{2t} \to\mathbb{R} ψ : R 2 t → R ξ y = ψ ( u ~ y t , u ~ y t ) \xi_y = \psi(\tilde{u}_y^t,\tilde{u}_y^t) ξ y = ψ ( u ~ y t , u ~ y t ) y ∈ V y \in V y ∈ V ξ x \xi_x ξ x D − 1 ∑ y ∼ x ξ y D^{-1}\sum_{y\sim x}\xi_y D − 1 ∑ y ∼ x ξ y ( W ξ ) x (W\xi)_x ( W ξ ) x x ∈ V x \in V x ∈ V
Proof. Let φ \varphi φ G G G W x y φ = W φ ( x ) φ ( y ) W^\varphi_{xy} = W_{\varphi(x)\varphi(y)} W x y φ = W φ ( x ) φ ( y ) φ \varphi φ W φ W^\varphi W φ W W W (64) and (67) use the same polynomials and the same real coefficients at every vertex. By induction on t t t W φ W^\varphi W φ W W W ( u φ ( x ) t ) x (u^t_{\varphi(x)})_x ( u φ ( x ) t ) x ( u ~ φ ( x ) t ) x (\tilde{u}^t_{\varphi(x)})_x ( u ~ φ ( x ) t ) x ∑ y W x y φ f t ( u ˉ φ ( y ) t ) = ∑ y ′ W φ ( x ) y ′ f t ( u ˉ y ′ t ) \sum_y W^\varphi_{xy}f_t(\bar{u}^t_{\varphi(y)}) = \sum_{y'} W_{\varphi(x)y'}f_t(\bar{u}^t_{y'}) ∑ y W x y φ f t ( u ˉ φ ( y ) t ) = ∑ y ′ W φ ( x ) y ′ f t ( u ˉ y ′ t ) ξ ( W ) \xi(W) ξ ( W ) ξ \xi ξ W W W ξ ( W φ ) x = ξ ( W ) φ ( x ) \xi(W^\varphi)_x = \xi(W)_{\varphi(x)} ξ ( W φ ) x = ξ ( W ) φ ( x )
1 D ∑ y ∼ x ξ ( W φ ) y = 1 D ∑ y ′ ∼ φ ( x ) ξ ( W ) y ′ , ( W φ ξ ( W φ ) ) x = ( W ξ ( W ) ) φ ( x ) . \frac{1}{D}\sum_{y\sim x}\xi(W^\varphi)_y = \frac{1}{D}\sum_{y'\sim\varphi(x)}\xi(W)_{y'}, \qquad(W^\varphi\xi(W^\varphi))_x = (W\xi(W))_{\varphi(x)}. D 1 y ∼ x ∑ ξ ( W φ ) y = D 1 y ′ ∼ φ ( x ) ∑ ξ ( W ) y ′ , ( W φ ξ ( W φ ) ) x = ( W ξ ( W ) ) φ ( x ) . Since G G G x , x ′ x,x' x , x ′ φ \varphi φ φ ( x ) = x ′ \varphi(x)=x' φ ( x ) = x ′ □ \square □
Lemma 5.18. Let ( G d ) (G_d) ( G d ) T L d \mathbb{T}_L^d T L d L ≥ 3 L \ge3 L ≥ 3 Q d Q_d Q d 1 ≤ s ≤ t ≤ T − 1 1 \le s \le t \le T-1 1 ≤ s ≤ t ≤ T − 1 x ∈ V x \in V x ∈ V lim d → ∞ E ∣ ( b ~ t , s ) x − b t , s ∣ 4 = 0 \lim_{d\to\infty}\mathbb{E}|(\tilde{b}_{t,s})_x-b_{t,s}|^4=0 lim d → ∞ E ∣ ( b ~ t , s ) x − b t , s ∣ 4 = 0
Proof. Since S x y = D − 1 S_{xy}=D^{-1} S x y = D − 1 y ∼ x y\sim x y ∼ x W x y = 0 W_{xy}=0 W x y = 0
( b ~ t , s ) x − b t , s = ∑ y ∼ x ( W x y 2 − D − 1 ) ∂ s f t ( u ˉ ~ y t ) + ( 1 D ∑ y ∼ x ∂ s f t ( u ˉ ~ y t ) − b t , s ) . (\tilde{b}_{t,s})_x-b_{t,s}=\sum_{y\sim x}(W_{xy}^2-D^{-1})\partial_s f_t(\tilde{\bar{u}}_y^t)+\left(\frac{1}{D}\sum_{y\sim x}\partial_s f_t(\tilde{\bar{u}}_y^t)-b_{t,s}\right). ( b ~ t , s ) x − b t , s = y ∼ x ∑ ( W x y 2 − D − 1 ) ∂ s f t ( u ˉ ~ y t ) + ( D 1 y ∼ x ∑ ∂ s f t ( u ˉ ~ y t ) − b t , s ) . By Lemmas 5.11 and 5.15, the polynomial ∂ s f t ( u ˉ ~ y t ) \partial_s f_t(\tilde{\bar{u}}_y^t) ∂ s f t ( u ˉ ~ y t ) y y y d d d y y y Lemma 5.14 implies that the fourth moment of the first term is at most C D − 2 CD^{-2} C D − 2 Lemma 5.16 with ψ = ∂ s f t \psi=\partial_s f_t ψ = ∂ s f t b t , s = E ∂ s f t ( U ˉ t ) b_{t,s}=\mathbb{E}\partial_s f_t(\bar{U}^t) b t , s = E ∂ s f t ( U ˉ t ) d d d □ \square □
Proof of Proposition 5.6. Part (a) is contained in Lemma 5.15. For the proof of (b), we use two elementary facts. First, if random variables ξ d \xi_d ξ d E ξ d 2 → 0 \mathbb{E}\xi_d^2\to0 E ξ d 2 → 0 sup d E ∣ ξ d ∣ 2 p < ∞ \sup_d\mathbb{E}|\xi_d|^{2p}<\infty sup d E ∣ ξ d ∣ 2 p < ∞ p ≥ 1 p\ge1 p ≥ 1 E ∣ ξ d ∣ p ≤ ( E ξ d 2 ) 1 / 2 ( E ∣ ξ d ∣ 2 p − 2 ) 1 / 2 → 0 \mathbb{E}|\xi_d|^p\le(\mathbb{E}\xi_d^2)^{1/2}(\mathbb{E}|\xi_d|^{2p-2})^{1/2}\to0 E ∣ ξ d ∣ p ≤ ( E ξ d 2 ) 1/2 ( E ∣ ξ d ∣ 2 p − 2 ) 1/2 → 0 p ≥ 1 p\ge1 p ≥ 1 P : R t → R P:\mathbb{R}^t\to\mathbb{R} P : R t → R C , k C,k C , k ∣ P ( z ) − P ( z ′ ) ∣ ≤ C ∣ z − z ′ ∣ ( 1 + ∣ z ∣ k + ∣ z ′ ∣ k ) |P(z)-P(z')|\le C|z-z'|(1+|z|^k+|z'|^k) ∣ P ( z ) − P ( z ′ ) ∣ ≤ C ∣ z − z ′ ∣ ( 1 + ∣ z ∣ k + ∣ z ′ ∣ k )
Step 1: comparison. Let w t = u t − u ~ t w^t=u^t-\tilde{u}^t w t = u t − u ~ t t ≤ T t\le T t ≤ T E ( w x t ) 2 → 0 \mathbb{E}(w_x^t)^2\to0 E ( w x t ) 2 → 0 x x x Lemma 5.17 , the law of w x t w_x^t w x t x x x w 1 = 0 w^1=0 w 1 = 0 E ( w x s ) 2 → 0 \mathbb{E}(w_x^s)^2\to0 E ( w x s ) 2 → 0 s ≤ t s\le t s ≤ t t ≤ T − 1 t\le T-1 t ≤ T − 1 P : R t → R P:\mathbb{R}^t\to\mathbb{R} P : R t → R p ≥ 1 p\ge1 p ≥ 1
E ∣ P ( u ˉ x t ) − P ( u ˉ ~ x t ) ∣ p ≤ C ( E ∣ u ˉ x t − u ˉ ~ x t ∣ 2 p ) 1 / 2 ( E ( 1 + ∣ u ˉ x t ∣ k + ∣ u ˉ ~ x t ∣ k ) 2 p ) 1 / 2 → 0. (68) \mathbb{E}|P(\bar{u}_x^t)-P(\tilde{\bar{u}}_x^t)|^p
\le C\left(\mathbb{E}|\bar{u}_x^t-\tilde{\bar{u}}_x^t|^{2p}\right)^{1/2}
\left(\mathbb{E}(1+|\bar{u}_x^t|^k+|\tilde{\bar{u}}_x^t|^k)^{2p}\right)^{1/2}\to0. \tag*{(68)} E ∣ P ( u ˉ x t ) − P ( u ˉ ~ x t ) ∣ p ≤ C ( E ∣ u ˉ x t − u ˉ ~ x t ∣ 2 p ) 1/2 ( E ( 1 + ∣ u ˉ x t ∣ k + ∣ u ˉ ~ x t ∣ k ) 2 p ) 1/2 → 0. (68) By (5.3) and (5.6), w x t + 1 = W ξ − ∑ s = 1 t ρ x s w_x^{t+1}=W\xi-\sum_{s=1}^{t}\rho_x^s w x t + 1 = W ξ − ∑ s = 1 t ρ x s ξ = f t ( u ˉ t ) − f t ( u ~ ˉ t ) \xi=f_t(\bar{u}^t)-f_t(\bar{\tilde{u}}^t) ξ = f t ( u ˉ t ) − f t ( u ~ ˉ t )
ρ x s = b t , s ( f s − 1 ( u ˉ x s − 1 ) − f s − 1 ( u ~ ˉ x s − 1 ) ) + ( b t , s 1 − b ~ t , s ) ⊙ f s − 1 ( u ~ ˉ x s − 1 ) . \rho_x^s=b_{t,s}\left(f_{s-1}(\bar{u}_x^{s-1})-f_{s-1}(\bar{\tilde{u}}_x^{s-1})\right)+(b_{t,s1}-\tilde{b}_{t,s})\odot f_{s-1}(\bar{\tilde{u}}_x^{s-1}). ρ x s = b t , s ( f s − 1 ( u ˉ x s − 1 ) − f s − 1 ( u ~ ˉ x s − 1 ) ) + ( b t , s 1 − b ~ t , s ) ⊙ f s − 1 ( u ~ ˉ x s − 1 ) . By Lemma 5.17 and the Cauchy–Schwarz inequality,
E ( W ξ ) x 2 = 1 n E ∥ W ξ ∥ 2 ≤ ( E ∥ W ∥ 4 ) 1 / 2 ( E ( 1 n ∑ y ξ y 2 ) 2 ) 1 / 2 ≤ ( E ∥ W ∥ 4 ) 1 / 2 ( E ξ x 4 ) 1 / 2 . \mathbb{E}(W\xi)_x^2=\frac{1}{n}\mathbb{E}\lVert W\xi\rVert^2\leq\left(\mathbb{E}\lVert W\rVert^4\right)^{1/2}\left(\mathbb{E}\left(\frac{1}{n}\sum_y \xi_y^2\right)^2\right)^{1/2}\leq\left(\mathbb{E}\lVert W\rVert^4\right)^{1/2}\left(\mathbb{E}\xi_x^4\right)^{1/2}. E ( W ξ ) x 2 = n 1 E ∥ W ξ ∥ 2 ≤ ( E ∥ W ∥ 4 ) 1/2 E ( n 1 y ∑ ξ y 2 ) 2 1/2 ≤ ( E ∥ W ∥ 4 ) 1/2 ( E ξ x 4 ) 1/2 . where in the last inequality we used Jensen’s inequality and that the law of ξ y \xi_y ξ y y y y ρ x s \rho_x^s ρ x s L 2 L^2 L 2
E [ ( b t , s − ( b ~ t , s ) x ) 2 f s − 1 ( u ~ ˉ x s − 1 ) 2 ] ≤ ( E ∣ b t , s − ( b ~ t , s ) x ∣ 4 ) 1 / 2 ( E f s − 1 ( u ~ ˉ x s − 1 ) 4 ) 1 / 2 → 0 , \mathbb{E}\left[(b_{t,s}-(\tilde{b}_{t,s})_x)^2f_{s-1}(\bar{\tilde{u}}_x^{s-1})^2\right]\leq\left(\mathbb{E}|b_{t,s}-(\tilde{b}_{t,s})_x|^4\right)^{1/2}\left(\mathbb{E}f_{s-1}(\bar{\tilde{u}}_x^{s-1})^4\right)^{1/2}\to0, E [ ( b t , s − ( b ~ t , s ) x ) 2 f s − 1 ( u ~ ˉ x s − 1 ) 2 ] ≤ ( E ∣ b t , s − ( b ~ t , s ) x ∣ 4 ) 1/2 ( E f s − 1 ( u ~ ˉ x s − 1 ) 4 ) 1/2 → 0 , by Lemmas 5.15 and 5.18. We conclude that E ( w x t + 1 ) 2 → 0 \mathbb{E}(w_x^{t+1})^2\to0 E ( w x t + 1 ) 2 → 0 t ≤ T t\leq T t ≤ T
Step 2: polynomial test functions. Let ψ : R t → R \psi:\mathbb{R}^t\to\mathbb{R} ψ : R t → R
E ∣ 1 n ∑ y ψ ( u ˉ y t ) − E ψ ( U ˉ t ) ∣ ≤ E ∣ ψ ( u ˉ x t ) − ψ ( u ~ ˉ x t ) ∣ + E ∣ 1 n ∑ y ψ ( u ~ ˉ y t ) − E ψ ( U ˉ t ) ∣ . \mathbb{E}\left|\frac{1}{n}\sum_y\psi(\bar{u}_y^t)-\mathbb{E}\psi(\bar{U}^t)\right|\leq\mathbb{E}|\psi(\bar{u}_x^t)-\psi(\bar{\tilde{u}}_x^t)|+\mathbb{E}\left|\frac{1}{n}\sum_y\psi(\bar{\tilde{u}}_y^t)-\mathbb{E}\psi(\bar{U}^t)\right|. E n 1 y ∑ ψ ( u ˉ y t ) − E ψ ( U ˉ t ) ≤ E ∣ ψ ( u ˉ x t ) − ψ ( u ~ ˉ x t ) ∣ + E n 1 y ∑ ψ ( u ~ ˉ y t ) − E ψ ( U ˉ t ) . The first term tends to zero by (5.7). The random variable in the second term tends to zero in probability by Lemma 5.16, and it tends to zero in L 1 L^1 L 1 d d d D − 1 ∑ y ∼ x ψ ( u ˉ y t ) D^{-1}\sum_{y\sim x}\psi(\bar{u}_y^t) D − 1 ∑ y ∼ x ψ ( u ˉ y t ) E ψ ( u ˉ x t ) = E [ n − 1 ∑ y ψ ( u ˉ y t ) ] → E ψ ( U ˉ t ) \mathbb{E}\psi(\bar{u}_x^t)=\mathbb{E}[n^{-1}\sum_y\psi(\bar{u}_y^t)]\to\mathbb{E}\psi(\bar{U}^t) E ψ ( u ˉ x t ) = E [ n − 1 ∑ y ψ ( u ˉ y t )] → E ψ ( U ˉ t )
Step 3: continuous test functions. By Step 2, all moments of u ˉ x t \bar{u}_x^t u ˉ x t U ˉ t \bar{U}^t U ˉ t u ˉ x t → U ˉ t \bar{u}_x^t\to\bar{U}^t u ˉ x t → U ˉ t φ \varphi φ sup d E φ ( u ˉ x t ) 2 < ∞ \sup_d\mathbb{E}\varphi(\bar{u}_x^t)^2<\infty sup d E φ ( u ˉ x t ) 2 < ∞ φ ( u ˉ x t ) \varphi(\bar{u}_x^t) φ ( u ˉ x t ) E φ ( u ˉ x t ) → E φ ( U ˉ t ) \mathbb{E}\varphi(\bar{u}_x^t)\to\mathbb{E}\varphi(\bar{U}^t) E φ ( u ˉ x t ) → E φ ( U ˉ t ) ψ \psi ψ η > 0 \eta>0 η > 0 P P P E ∣ ψ ( U ˉ t ) − P ( U ˉ t ) ∣ ≤ η \mathbb{E}|\psi(\bar{U}^t)-P(\bar{U}^t)|\leq\eta E ∣ ψ ( U ˉ t ) − P ( U ˉ t ) ∣ ≤ η
E ∣ 1 n ∑ y ψ ( u ˉ y t ) − E ψ ( U ˉ t ) ∣ ≤ E ∣ ( ψ − P ) ( u ˉ x t ) ∣ + E ∣ 1 n ∑ y P ( u ˉ y t ) − E P ( U ˉ t ) ∣ + E ∣ ( P − ψ ) ( U ˉ t ) ∣ . \mathbb{E}\left|\frac{1}{n}\sum_y\psi(\bar{u}_y^t)-\mathbb{E}\psi(\bar{U}^t)\right|\leq\mathbb{E}|(\psi-P)(\bar{u}_x^t)|+\mathbb{E}\left|\frac{1}{n}\sum_yP(\bar{u}_y^t)-\mathbb{E}P(\bar{U}^t)\right|+\mathbb{E}|(P-\psi)(\bar{U}^t)|. E n 1 y ∑ ψ ( u ˉ y t ) − E ψ ( U ˉ t ) ≤ E ∣ ( ψ − P ) ( u ˉ x t ) ∣ + E n 1 y ∑ P ( u ˉ y t ) − E P ( U ˉ t ) + E ∣ ( P − ψ ) ( U ˉ t ) ∣. As d → ∞ d\to\infty d → ∞ E ∣ ( ψ − P ) ( U ˉ t ) ∣ ≤ η \mathbb{E}|(\psi-P)(\bar{U}^t)|\leq\eta E ∣ ( ψ − P ) ( U ˉ t ) ∣ ≤ η φ = ∣ ψ − P ∣ \varphi=|\psi-P| φ = ∣ ψ − P ∣ lim sup \limsup lim sup 2 η 2\eta 2 η η \eta η E ∣ D − 1 ∑ y ∼ x ( ψ − P ) ( u ˉ y t ) ∣ ≤ E ∣ ( ψ − P ) ( u ˉ x t ) ∣ \mathbb{E}|D^{-1}\sum_{y\sim x}(\psi-P)(\bar{u}_y^t)|\leq\mathbb{E}|(\psi-P)(\bar{u}_x^t)| E ∣ D − 1 ∑ y ∼ x ( ψ − P ) ( u ˉ y t ) ∣ ≤ E ∣ ( ψ − P ) ( u ˉ x t ) ∣
Energy of the final message
The energy of the final polynomial message is determined by the next iterate and the Onsager term. A similar computation appears in the proof of [103 Lemma 2.3] .
Lemma 5.19. In the setting of Proposition 5.6 , let 1 ≤ K ≤ T − 1 1 \le K \le T-1 1 ≤ K ≤ T − 1 F = f K ( u ˉ K ) ∈ R V F=f_K(\bar{u}^{K})\in\mathbb{R}^{V} F = f K ( u ˉ K ) ∈ R V F = f K ( U ˉ K ) \mathbf{F}=f_K(\bar{U}^{K}) F = f K ( U ˉ K )
lim d → ∞ 1 n E H G d ( F ) = lim d → ∞ 1 2 n E ⟨ F , W F ⟩ = E [ F U K + 1 ] . \lim_{d\to\infty}\frac{1}{n}\mathbb{E}H_{G_d}(F)=\lim_{d\to\infty}\frac{1}{2n}\mathbb{E}\langle F,WF\rangle=\mathbb{E}[FU_{K+1}]. d → ∞ lim n 1 E H G d ( F ) = d → ∞ lim 2 n 1 E ⟨ F , W F ⟩ = E [ F U K + 1 ] . Proof. Let O = ∑ s = 1 K b K , s f s − 1 ( u ˉ s − 1 ) O=\sum_{s=1}^{K}b_{K,s}f_{s-1}(\bar{u}^{s-1}) O = ∑ s = 1 K b K , s f s − 1 ( u ˉ s − 1 ) W F = u K + 1 + O WF=u^{K+1}+O W F = u K + 1 + O
1 n ⟨ F , W F ⟩ = 1 n ∑ x f K ( u ˉ x K ) ( u x K + 1 + ∑ s = 1 K b K , s f s − 1 ( u ˉ x s − 1 ) ) , \frac{1}{n}\langle F,WF\rangle=\frac{1}{n}\sum_x f_K(\bar{u}^{K}_x)\left(u^{K+1}_x+\sum_{s=1}^{K}b_{K,s}f_{s-1}(\bar{u}^{s-1}_x)\right), n 1 ⟨ F , W F ⟩ = n 1 x ∑ f K ( u ˉ x K ) ( u x K + 1 + s = 1 ∑ K b K , s f s − 1 ( u ˉ x s − 1 ) ) , which is the empirical average of a polynomial in u ˉ x K + 1 \bar{u}^{K+1}_x u ˉ x K + 1
E [ F U K + 1 ] + ∑ s = 1 K b K , s E [ F f s − 1 ( U ˉ s − 1 ) ] . \mathbb{E}[FU_{K+1}]+\sum_{s=1}^{K}b_{K,s}\mathbb{E}[Ff_{s-1}(\bar{U}^{s-1})]. E [ F U K + 1 ] + s = 1 ∑ K b K , s E [ F f s − 1 ( U ˉ s − 1 )] . By Gaussian integration by parts (F1), (5.1), and (5.2), we have
E [ U K + 1 f K ( U ˉ K ) ] = ∑ s = 1 K Cov ( U K + 1 , U s ) E ∂ s f K ( U ˉ K ) = ∑ s = 1 K E [ f K ( U ˉ K ) f s − 1 ( U ˉ s − 1 ) ] b K , s . \mathbb{E}[U_{K+1}f_K(\bar{U}^{K})]=\sum_{s=1}^{K}\operatorname{Cov}(U_{K+1},U_s)\mathbb{E}\partial_s f_K(\bar{U}^{K})=\sum_{s=1}^{K}\mathbb{E}[f_K(\bar{U}^{K})f_{s-1}(\bar{U}^{s-1})]b_{K,s}. E [ U K + 1 f K ( U ˉ K )] = s = 1 ∑ K Cov ( U K + 1 , U s ) E ∂ s f K ( U ˉ K ) = s = 1 ∑ K E [ f K ( U ˉ K ) f s − 1 ( U ˉ s − 1 )] b K , s . We conclude that the limit equals 2 E [ F U K + 1 ] 2\mathbb{E}[FU_{K+1}] 2 E [ F U K + 1 ] 2 H G d ( F ) = ⟨ F , W F ⟩ 2H_{G_d}(F)=\langle F,WF\rangle 2 H G d ( F ) = ⟨ F , W F ⟩
The Onsager term contributes exactly half of the energy. The center is a bounded function of the history close to F F F W W W
Lemma 5.20. In the setting of Lemma 5.19, let θ : R K → [ − 1 , 1 ] \theta:\mathbb{R}^{K}\to[-1,1] θ : R K → [ − 1 , 1 ] m = θ ( u ˉ K ) m=\theta(\bar{u}^{K}) m = θ ( u ˉ K ) Θ = θ ( U ˉ K ) \Theta=\theta(\bar{U}^{K}) Θ = θ ( U ˉ K ) M W = sup d ( E ∥ W ∥ 2 ) 1 / 2 M_W=\sup_d(\mathbb{E}\lVert W\rVert^2)^{1/2} M W = sup d ( E ∥ W ∥ 2 ) 1/2
lim sup d → ∞ ∣ 1 n E H G d ( m ) − E [ F U K + 1 ] ∣ ≤ M W 2 ∥ Θ − F ∥ 2 ( ∥ Θ ∥ 2 + ∥ F ∥ 2 ) . \limsup_{d\to\infty}\left|\frac{1}{n}\mathbb{E}H_{G_d}(m)-\mathbb{E}[FU_{K+1}]\right|\le\frac{M_W}{2}\lVert\Theta-F\rVert_2(\lVert\Theta\rVert_2+\lVert F\rVert_2). d → ∞ lim sup n 1 E H G d ( m ) − E [ F U K + 1 ] ≤ 2 M W ∥ Θ − F ∥ 2 (∥ Θ ∥ 2 + ∥ F ∥ 2 ) . Proof. Since 2 ( H G d ( m ) − H G d ( F ) ) = ⟨ m − F , W ( m + F ) ⟩ 2(H_{G_d}(m)-H_{G_d}(F))=\langle m-F,W(m+F)\rangle 2 ( H G d ( m ) − H G d ( F )) = ⟨ m − F , W ( m + F )⟩ 2 n − 1 ∣ H G d ( m ) − H G d ( F ) ∣ ≤ ∥ W ∥ Ξ 2n^{-1}|H_{G_d}(m)-H_{G_d}(F)|\le\lVert W\rVert\Xi 2 n − 1 ∣ H G d ( m ) − H G d ( F ) ∣ ≤ ∥ W ∥ Ξ Ξ = ∥ m − F ∥ n ( ∥ m ∥ n + ∥ F ∥ n ) \Xi=\lVert m-F\rVert_n(\lVert m\rVert_n+\lVert F\rVert_n) Ξ = ∥ m − F ∥ n (∥ m ∥ n + ∥ F ∥ n ) E [ ∥ W ∥ Ξ ] ≤ M W ( E Ξ 2 ) 1 / 2 \mathbb{E}[\lVert W\rVert\Xi]\le M_W(\mathbb{E}\Xi^2)^{1/2} E [∥ W ∥ Ξ ] ≤ M W ( E Ξ 2 ) 1/2 E Ξ 2 → ∥ Θ − F ∥ 2 2 ( ∥ Θ ∥ 2 + ∥ F ∥ 2 ) 2 \mathbb{E}\Xi^2\to\lVert\Theta-F\rVert_2^2(\lVert\Theta\rVert_2+\lVert F\rVert_2)^2 E Ξ 2 → ∥ Θ − F ∥ 2 2 (∥ Θ ∥ 2 + ∥ F ∥ 2 ) 2
Let A 1 = ∥ m − F ∥ n 2 A_1=\|m-F\|_n^2 A 1 = ∥ m − F ∥ n 2 A 2 = ∥ m ∥ n 2 A_2=\|m\|_n^2 A 2 = ∥ m ∥ n 2 A 3 = ∥ F ∥ n 2 A_3=\|F\|_n^2 A 3 = ∥ F ∥ n 2 A i A_i A i n − 1 ∑ y ψ i ( u ˉ K y ) n^{-1}\sum_y\psi_i(\bar{u}_K^y) n − 1 ∑ y ψ i ( u ˉ K y ) ψ 1 = ( θ − f K ) 2 \psi_1=(\theta-f_K)^2 ψ 1 = ( θ − f K ) 2 ψ 2 = θ 2 \psi_2=\theta^2 ψ 2 = θ 2 ψ 3 = f K 2 \psi_3=f_K^2 ψ 3 = f K 2 Proposition 5.6 (b), A 1 A_1 A 1 A 2 A_2 A 2 A 3 A_3 A 3 L 1 L^1 L 1 ∥ Θ − F ∥ 2 2 \|\Theta-F\|_2^2 ∥Θ − F ∥ 2 2 ∥ Θ ∥ 2 2 \|\Theta\|_2^2 ∥Θ ∥ 2 2 ∥ F ∥ 2 2 \|F\|_2^2 ∥ F ∥ 2 2 Ξ 2 = A 1 ( A 2 1 / 2 + A 3 1 / 2 ) 2 \Xi^2=A_1(A_2^{1/2}+A_3^{1/2})^2 Ξ 2 = A 1 ( A 2 1/2 + A 3 1/2 ) 2 ∥ Θ − F ∥ 2 2 ( ∥ Θ ∥ 2 + ∥ F ∥ 2 ) 2 \|\Theta-F\|_2^2(\|\Theta\|_2+\|F\|_2)^2 ∥Θ − F ∥ 2 2 ( ∥Θ ∥ 2 + ∥ F ∥ 2 ) 2 p ≥ 1 p\geq1 p ≥ 1 A i p ≤ n − 1 ∑ y ψ i ( u ˉ K y ) p A_i^p\leq n^{-1}\sum_y\psi_i(\bar{u}_K^y)^p A i p ≤ n − 1 ∑ y ψ i ( u ˉ K y ) p ψ i p \psi_i^p ψ i p 2 p ( 1 + f K 2 ) ⌈ p ⌉ 2^p(1+f_K^2)^{\lceil p\rceil} 2 p ( 1 + f K 2 ) ⌈ p ⌉ ∣ θ ∣ ≤ 1 |\theta|\leq1 ∣ θ ∣ ≤ 1 Proposition 5.6 (a) implies that sup d E A i p < ∞ \sup_d\mathbb{E}A_i^p<\infty sup d E A i p < ∞ sup d E Ξ 4 < ∞ \sup_d\mathbb{E}\Xi^4<\infty sup d E Ξ 4 < ∞ Ξ 2 \Xi^2 Ξ 2 E Ξ 2 \mathbb{E}\Xi^2 E Ξ 2 □ \square □
The two-phase scheme and controls
We now choose the polynomials f t f_t f t [125 ] , with the incremental nonlinearities of Montanari [103 ] : a fixed-point iteration first produces a field of variance q 0 > 0 q_0>0 q 0 > 0 (62) involves only the polynomials f t f_t f t
For q 0 ∈ ( 0 , 1 ) q_0\in(0,1) q 0 ∈ ( 0 , 1 ) τ ∈ ( 0 , 1 ] \tau\in(0,1] τ ∈ ( 0 , 1 ] N = N q 0 , τ \mathcal{N}=\mathcal{N}_{q_0,\tau} N = N q 0 , τ ( Y , Δ 1 , Δ 2 , … ) (Y,\Delta_1,\Delta_2,\ldots) ( Y , Δ 1 , Δ 2 , … ) Y ∼ N ( 0 , q 0 ) Y\sim N(0,q_0) Y ∼ N ( 0 , q 0 ) Δ j \Delta_j Δ j N ( 0 , τ ) N(0,\tau) N ( 0 , τ ) Y Y Y p p p E p ( q 0 Z ) 2 < ∞ \mathbb{E}p(\sqrt{q_0}Z)^2<\infty E p ( q 0 Z ) 2 < ∞ c ∈ [ 0 , q 0 ] c\in[0,q_0] c ∈ [ 0 , q 0 ] φ p ( c ) = E [ p ( A ) p ( A ′ ) ] \varphi_p(c)=\mathbb{E}[p(A)p(A')] φ p ( c ) = E [ p ( A ) p ( A ′ )] ( A , A ′ ) (A,A') ( A , A ′ ) Var A = Var A ′ = q 0 \operatorname{Var}A=\operatorname{Var}A'=q_0 Var A = Var A ′ = q 0 Cov ( A , A ′ ) = c \operatorname{Cov}(A,A')=c Cov ( A , A ′ ) = c A = c Z 0 + q 0 − c Z 1 A=\sqrt{c}Z_0+\sqrt{q_0-c}Z_1 A = c Z 0 + q 0 − c Z 1 A ′ = c Z 0 + q 0 − c Z 2 A'=\sqrt{c}Z_0+\sqrt{q_0-c}Z_2 A ′ = c Z 0 + q 0 − c Z 2 Z 0 , Z 1 , Z 2 Z_0,Z_1,Z_2 Z 0 , Z 1 , Z 2 φ p ( c ) = E [ ( E [ p ( A ) ∣ Z 0 ] ) 2 ] ≥ 0 \varphi_p(c)=\mathbb{E}[(\mathbb{E}[p(A)\mid Z_0])^2]\geq0 φ p ( c ) = E [( E [ p ( A ) ∣ Z 0 ] ) 2 ] ≥ 0
Definition 5.21. Let q 0 ∈ ( 0 , 1 ) q_0\in(0,1) q 0 ∈ ( 0 , 1 ) τ ∈ ( 0 , 1 ] \tau\in(0,1] τ ∈ ( 0 , 1 ] K 2 ≥ 0 K_2\geq0 K 2 ≥ 0 q 0 + K 2 τ ≤ 1 q_0+K_2\tau\leq1 q 0 + K 2 τ ≤ 1 K 1 ≥ 1 K_1\geq1 K 1 ≥ 1 p p p E p ( q 0 Z ) 2 = q 0 \mathbb{E}p(\sqrt{q_0}Z)^2=q_0 E p ( q 0 Z ) 2 = q 0 P j ( y , z 1 , … , z j ) P_j(y,z_1,\ldots,z_j) P j ( y , z 1 , … , z j ) 0 ≤ j < K 2 0\leq j<K_2 0 ≤ j < K 2
E N P j ( Y , Δ 1 , … , Δ j ) 2 = 1 ( 0 ≤ j < K 2 ) . (69) \mathbb{E}_{\mathcal{N}}P_j(Y,\Delta_1,\ldots,\Delta_j)^2=1 \qquad(0\leq j<K_2). \tag*{(69)} E N P j ( Y , Δ 1 , … , Δ j ) 2 = 1 ( 0 ≤ j < K 2 ) . (69) The two-phase scheme is the recursion (64) with T = K 1 + K 2 + 1 T=K_1+K_2+1 T = K 1 + K 2 + 1 f 0 , … , f T − 1 f_0,\ldots,f_{T-1} f 0 , … , f T − 1 K = K 1 + K 2 K=K_1+K_2 K = K 1 + K 2
Root phase. Let f 0 = q 0 f_0=\sqrt{q_0} f 0 = q 0 E p ( q 0 Z ) ≥ 0 \mathbb{E}p(\sqrt{q_0}Z)\geq0 E p ( q 0 Z ) ≥ 0 f 0 = − q 0 f_0=-\sqrt{q_0} f 0 = − q 0 f k ( u ˉ k ) = p ( u k ) f_k(\bar{u}^k)=p(u^k) f k ( u ˉ k ) = p ( u k ) 1 ≤ k ≤ K 1 1\leq k\leq K_1 1 ≤ k ≤ K 1 y = u K 1 y=u^{K_1} y = u K 1 F ^ 0 = p ( y ) \widehat{F}_0=p(y) F 0 = p ( y )
Innovations. Let ∑ j ≤ K 1 γ j U j \sum_{j\leq K_1}\gamma_jU_j ∑ j ≤ K 1 γ j U j U K 1 + 1 U_{K_1+1} U K 1 + 1 U K 1 \mathcal{U}_{K_1} U K 1 τ 1 = E ( U K 1 + 1 − ∑ j ≤ K 1 γ j U j ) 2 \tau_1=\mathbb{E}(U_{K_1+1}-\sum_{j\leq K_1}\gamma_jU_j)^2 τ 1 = E ( U K 1 + 1 − ∑ j ≤ K 1 γ j U j ) 2 Δ 1 = ( τ / τ 1 ) 1 / 2 ( u K 1 + 1 − ∑ j ≤ K 1 γ j u j ) \Delta^1=(\tau/\tau_1)^{1/2}(u^{K_1+1}-\sum_{j\leq K_1}\gamma_ju^j) Δ 1 = ( τ / τ 1 ) 1/2 ( u K 1 + 1 − ∑ j ≤ K 1 γ j u j ) r ≥ 2 r\geq2 r ≥ 2 Δ r = u K 1 + r − u K 1 + r − 1 \Delta^r=u^{K_1+r}-u^{K_1+r-1} Δ r = u K 1 + r − u K 1 + r − 1
Incremental phase. For 1 ≤ r ≤ K 2 1\leq r\leq K_2 1 ≤ r ≤ K 2
f K 1 + r ( u ˉ K 1 + r ) = F ^ 0 + ∑ i = 1 r P i − 1 ( y , Δ 1 , … , Δ i − 1 ) Δ i . (70) f_{K_1+r}(\bar{u}^{K_1+r})=\widehat{F}_0+\sum_{i=1}^{r}P_{i-1}(y,\Delta^1,\ldots,\Delta^{i-1})\Delta^i. \tag*{(70)} f K 1 + r ( u ˉ K 1 + r ) = F 0 + i = 1 ∑ r P i − 1 ( y , Δ 1 , … , Δ i − 1 ) Δ i . (70) :::_
On the state-evolution side, we write Y = U K 1 \mathsf{Y}=U_{K_1} Y = U K 1 F ^ 0 = p ( Y ) \widehat{F}_0=p(Y) F 0 = p ( Y ) Δ 1 = ( τ / τ 1 ) 1 / 2 ( U K 1 + 1 − ∑ j ≤ K 1 γ j U j ) \Delta_1=(\tau/\tau_1)^{1/2}(U_{K_1+1}-\sum_{j\le K_1}\gamma_j U_j) Δ 1 = ( τ / τ 1 ) 1/2 ( U K 1 + 1 − ∑ j ≤ K 1 γ j U j ) Δ r = U K 1 + r − U K 1 + r − 1 \Delta_r=U_{K_1+r}-U_{K_1+r-1} Δ r = U K 1 + r − U K 1 + r − 1 r ≥ 2 r\ge2 r ≥ 2
The numbers γ j \gamma_j γ j τ 1 \tau_1 τ 1 f 0 , … , f K 1 f_0,\ldots,f_{K_1} f 0 , … , f K 1 τ 1 > 0 \tau_1>0 τ 1 > 0 Δ i \Delta^i Δ i u 1 , … , u K 1 + i u^1,\ldots,u^{K_1+i} u 1 , … , u K 1 + i f t f_t f t u ˉ t \bar{u}^t u ˉ t ( Y , Δ 1 , Δ 2 , … ) (Y,\Delta_1,\Delta_2,\ldots) ( Y , Δ 1 , Δ 2 , … ) N \mathsf{N} N
Lemma 5.22. In the setting of Definition 5.21, the following hold.
(a) The variables U 1 , … , U K 1 + 1 U_1,\ldots,U_{K_1+1} U 1 , … , U K 1 + 1 N ( 0 , q 0 ) N(0,q_0) N ( 0 , q 0 ) τ 1 > 0 \tau_1>0 τ 1 > 0 ϱ k = E U k U k + 1 \varrho_k=\mathbb{E}U_kU_{k+1} ϱ k = E U k U k + 1 ϱ 1 = q 0 ∣ E p ( q 0 Z ) ∣ < q 0 \varrho_1=\sqrt{q_0}\lvert\mathbb{E}p(\sqrt{q_0}Z)\rvert<q_0 ϱ 1 = q 0 ∣ E p ( q 0 Z )∣ < q 0 ϱ k + 1 = ϕ p ( ϱ k ) \varrho_{k+1}=\phi_p(\varrho_k) ϱ k + 1 = ϕ p ( ϱ k ) 1 ≤ k < K 1 1\le k<K_1 1 ≤ k < K 1 0 ≤ ϱ k ≤ q 0 0\le\varrho_k\le q_0 0 ≤ ϱ k ≤ q 0 k ≤ K 1 k\le K_1 k ≤ K 1 p p p q 0 q_0 q 0 K 1 K_1 K 1
(b) The variables Δ 1 , … , Δ K 2 + 1 \Delta_1,\ldots,\Delta_{K_2+1} Δ 1 , … , Δ K 2 + 1 N ( 0 , τ ) N(0,\tau) N ( 0 , τ ) U 1 , … , U K 1 U_1,\ldots,U_{K_1} U 1 , … , U K 1 ( Y , Δ 1 , … , Δ K 2 ) (Y,\Delta_1,\ldots,\Delta_{K_2}) ( Y , Δ 1 , … , Δ K 2 ) N \mathsf{N} N
(c) The final message F = f K ( U ˉ K ) = F ^ 0 + ∑ i = 1 K 2 P i − 1 Δ i \mathbf{F}=f_K(\bar{U}^K)=\widehat{F}_0+\sum_{i=1}^{K_2}P_{i-1}\Delta_i F = f K ( U ˉ K ) = F 0 + ∑ i = 1 K 2 P i − 1 Δ i E F 2 ≤ 1 \mathbb{E}F^2\le1 E F 2 ≤ 1
E [ F U K + 1 ] = ϱ K 1 E p ′ ( Y ) + ( τ τ 1 ) 1 / 2 E P 0 + τ ∑ j = 1 K 2 − 1 E P j , \mathbb{E}[FU_{K+1}]=\varrho_{K_1}\mathbb{E}p'(Y)+(\tau\tau_1)^{1/2}\mathbb{E}P_0+\tau\sum_{j=1}^{K_2-1}\mathbb{E}P_j, E [ F U K + 1 ] = ϱ K 1 E p ′ ( Y ) + ( τ τ 1 ) 1/2 E P 0 + τ j = 1 ∑ K 2 − 1 E P j , where we abbreviate P j = P j ( Y , Δ 1 , … , Δ j ) P_j=P_j(Y,\Delta_1,\ldots,\Delta_j) P j = P j ( Y , Δ 1 , … , Δ j ) P j P_j P j j ≥ K 2 j\ge K_2 j ≥ K 2
Proof. (a) We have Var U 1 = f 0 2 = q 0 \operatorname{Var}U_1=f_0^2=q_0 Var U 1 = f 0 2 = q 0 Var U k + 1 = E p ( U k ) 2 = q 0 \operatorname{Var}U_{k+1}=\mathbb{E}p(U_k)^2=q_0 Var U k + 1 = E p ( U k ) 2 = q 0 k ≤ K 1 k\le K_1 k ≤ K 1 ϱ 1 = E [ f 0 p ( U 1 ) ] = q 0 ∣ E p ( q 0 Z ) ∣ \varrho_1=\mathbb{E}[f_0p(U_1)]=\sqrt{q_0}\lvert\mathbb{E}p(\sqrt{q_0}Z)\rvert ϱ 1 = E [ f 0 p ( U 1 )] = q 0 ∣ E p ( q 0 Z )∣ f 0 f_0 f 0 q 0 q_0 q 0 p ( q 0 Z ) p(\sqrt{q_0}Z) p ( q 0 Z ) E p ( q 0 Z ) 2 = q 0 \mathbb{E}p(\sqrt{q_0}Z)^2=q_0 E p ( q 0 Z ) 2 = q 0 ϱ k + 1 = E [ p ( U k ) p ( U k + 1 ) ] = ϕ p ( ϱ k ) \varrho_{k+1}=\mathbb{E}[p(U_k)p(U_{k+1})]=\phi_p(\varrho_k) ϱ k + 1 = E [ p ( U k ) p ( U k + 1 )] = ϕ p ( ϱ k ) 0 ≤ ϱ k ≤ q 0 0\le\varrho_k\le q_0 0 ≤ ϱ k ≤ q 0 ϕ p ≥ 0 \phi_p\ge0 ϕ p ≥ 0 k ≤ K 1 k\le K_1 k ≤ K 1 U 1 U_1 U 1 ( U 1 , … , U k ) (U_1,\ldots,U_k) ( U 1 , … , U k ) U k = Λ + ς ξ U_k=\Lambda+\varsigma\xi U k = Λ + ς ξ Λ \Lambda Λ U 1 , … , U k − 1 U_1,\ldots,U_{k-1} U 1 , … , U k − 1 ς > 0 \varsigma>0 ς > 0 ξ ∼ N ( 0 , 1 ) \xi\sim N(0,1) ξ ∼ N ( 0 , 1 ) U 1 , … , U k − 1 U_1,\ldots,U_{k-1} U 1 , … , U k − 1 U 1 , … , U k − 1 U_1,\ldots,U_{k-1} U 1 , … , U k − 1 p ( U k ) p(U_k) p ( U k ) U 1 , … , U k − 1 U_1,\ldots,U_{k-1} U 1 , … , U k − 1 F k \mathcal{F}_k F k p ( U k ) ∉ F k p(U_k)\notin\mathcal{F}_k p ( U k ) ∈ / F k U k + 1 U_{k+1} U k + 1 F k \mathcal{F}_k F k ( U 1 , … , U k + 1 ) (U_1,\ldots,U_{k+1}) ( U 1 , … , U k + 1 ) k = K 1 k=K_1 k = K 1 τ 1 \tau_1 τ 1
(b) Let G \mathcal{G} G σ \sigma σ U 1 , … , U K 1 U_1,\ldots,U_{K_1} U 1 , … , U K 1 Δ 1 ∼ N ( 0 , τ ) \Delta_1\sim N(0,\tau) Δ 1 ∼ N ( 0 , τ ) G \mathcal{G} G U K 1 + 1 − ∑ j γ j U j U_{K_1+1}-\sum_j\gamma_jU_j U K 1 + 1 − ∑ j γ j U j U 1 , … , U K 1 U_1,\ldots,U_{K_1} U 1 , … , U K 1 τ 1 \tau_1 τ 1 1 ≤ r ≤ K 2 1\le r\le K_2 1 ≤ r ≤ K 2 Δ 1 , … , Δ r \Delta_1,\ldots,\Delta_r Δ 1 , … , Δ r N ( 0 , τ ) N(0,\tau) N ( 0 , τ ) G \mathcal{G} G G r \mathcal{G}_r G r σ \sigma σ G \mathcal{G} G Δ 1 , … , Δ r − 1 \Delta_1,\ldots,\Delta_{r-1} Δ 1 , … , Δ r − 1 f K 1 + r − f K 1 + r − 1 = P r − 1 Δ r f_{K_1+r}-f_{K_1+r-1}=P_{r-1}\Delta_r f K 1 + r − f K 1 + r − 1 = P r − 1 Δ r f K 1 = F ^ 0 f_{K_1}=\widehat{F}_0 f K 1 = F 0 f j f_j f j j < K 1 + r j<K_1+r j < K 1 + r G r \mathcal{G}_r G r P r − 1 P_{r-1} P r − 1 Δ r \Delta_r Δ r G r \mathcal{G}_r G r P r − 1 Δ r P_{r-1}\Delta_r P r − 1 Δ r F K 1 + r \mathcal{F}_{K_1+r} F K 1 + r Δ r + 1 = U K 1 + r + 1 − U K 1 + r \Delta_{r+1}=U_{K_1+r+1}-U_{K_1+r} Δ r + 1 = U K 1 + r + 1 − U K 1 + r U K 1 + r \mathcal{U}_{K_1+r} U K 1 + r U 1 , … , U K 1 + r U_1,\ldots,U_{K_1+r} U 1 , … , U K 1 + r G \mathcal{G} G Δ 1 , … , Δ r \Delta_1,\ldots,\Delta_r Δ 1 , … , Δ r Δ r + 1 \Delta_{r+1} Δ r + 1 E [ P r − 1 2 Δ r 2 ] = τ E P r − 1 2 = τ \mathbb{E}[P_{r-1}^2\Delta_r^2]=\tau\mathbb{E}P_{r-1}^2=\tau E [ P r − 1 2 Δ r 2 ] = τ E P r − 1 2 = τ (69) . This completes the induction.
(c) By (b), a product P i − 1 Δ i P_{i-1}\Delta_i P i − 1 Δ i G i \mathcal{G}_i G i F ^ 0 \widehat{F}_0 F 0 G \mathcal{G} G Δ 1 , … , Δ i − 1 \Delta_1,\ldots,\Delta_{i-1} Δ 1 , … , Δ i − 1 P j − 1 Δ j P_{j-1}\Delta_j P j − 1 Δ j j < i j<i j < i Δ r \Delta_r Δ r r > i r>i r > i E [ P i − 1 Δ i 2 ] = τ E P i − 1 2 \mathbb{E}[P_{i-1}\Delta_i^2]=\tau\mathbb{E}P_{i-1}^2 E [ P i − 1 Δ i 2 ] = τ E P i − 1 2 E F 2 = E F ^ 0 2 + K 2 τ ≤ q 0 + ( 1 − q 0 ) = 1 \mathbb{E}F^2=\mathbb{E}\widehat{F}_0^2+K_2\tau\le q_0+(1-q_0)=1 E F 2 = E F 0 2 + K 2 τ ≤ q 0 + ( 1 − q 0 ) = 1 U K + 1 = U K 1 + 1 + ∑ r = 2 K 2 + 1 Δ r U_{K+1}=U_{K_1+1}+\sum_{r=2}^{K_2+1}\Delta_r U K + 1 = U K 1 + 1 + ∑ r = 2 K 2 + 1 Δ r U K 1 + 1 = ∑ j γ j U j + ( τ 1 / τ ) 1 / 2 Δ 1 U_{K_1+1}=\sum_j\gamma_jU_j+(\tau_1/\tau)^{1/2}\Delta_1 U K 1 + 1 = ∑ j γ j U j + ( τ 1 / τ ) 1/2 Δ 1 F ^ 0 \widehat{F}_0 F 0 U K 1 + 1 U_{K_1+1} U K 1 + 1 P 0 Δ 1 P_0\Delta_1 P 0 Δ 1 ( τ 1 / τ ) 1 / 2 Δ 1 (\tau_1/\tau)^{1/2}\Delta_1 ( τ 1 / τ ) 1/2 Δ 1 ( τ τ 1 ) 1 / 2 E P 0 (\tau\tau_1)^{1/2}\mathbb{E}P_0 ( τ τ 1 ) 1/2 E P 0 i ≥ 2 i\ge2 i ≥ 2 P i − 1 Δ i P_{i-1}\Delta_i P i − 1 Δ i Δ i \Delta_i Δ i τ E P i − 1 \tau\mathbb{E}P_{i-1} τ E P i − 1 ( U K 1 , U K 1 + 1 ) (U_{K_1},U_{K_1+1}) ( U K 1 , U K 1 + 1 ) E [ F ^ 0 U K 1 + 1 ] = E [ p ( U K 1 ) U K 1 + 1 ] = ϱ K 1 E p ′ ( U K 1 ) \mathbb{E}[\widehat{F}_0U_{K_1+1}]=\mathbb{E}[p(U_{K_1})U_{K_1+1}]=\varrho_{K_1}\mathbb{E}p'(U_{K_1}) E [ F 0 U K 1 + 1 ] = E [ p ( U K 1 ) U K 1 + 1 ] = ϱ K 1 E p ′ ( U K 1 )
A control specifies the ideal root message, the ideal coefficients, of which the P j P_j P j
Definition 5.23. Let q 0 ∈ ( 0 , 1 ) q_0\in(0,1) q 0 ∈ ( 0 , 1 ) τ ∈ ( 0 , 1 ] \tau\in(0,1] τ ∈ ( 0 , 1 ] K 2 ≥ 0 K_2\ge0 K 2 ≥ 0 control of step τ \tau τ K 2 K_2 K 2 q 0 q_0 q 0
(a) A root map : a nonconstant C 1 C^1 C 1 f : R → [ − 1 , 1 ] f:\mathbb{R}\to[-1,1] f : R → [ − 1 , 1 ] E f ( q 0 Z ) 2 = q 0 \mathbb{E}f(\sqrt{q_0}Z)^2=q_0 E f ( q 0 Z ) 2 = q 0 ϕ f ( c ) > c \phi_f(c)>c ϕ f ( c ) > c c ∈ [ 0 , q 0 ) c\in[0,q_0) c ∈ [ 0 , q 0 )
(b) Random variables G j = g j ( Y , Δ 1 , … , Δ j ) G_j=g_j(Y,\Delta_1,\ldots,\Delta_j) G j = g j ( Y , Δ 1 , … , Δ j ) 0 ≤ j < K 2 0\le j<K_2 0 ≤ j < K 2 g j g_j g j E N G j 2 = 1 \mathbb{E}_{\mathsf{N}}G_j^2=1 E N G j 2 = 1
(c) A continuous output function Θ : R 1 + K 2 → ( − 1 , 1 ) \Theta:\mathbb{R}^{1+K_2}\to(-1,1) Θ : R 1 + K 2 → ( − 1 , 1 )
Let F 0 = f ( Y ) F_0=f(Y) F 0 = f ( Y ) stochastic integral, value, and defect of the control are
F = F 0 + ∑ j = 0 K 2 − 1 G j Δ j + 1 , V = E [ Y F 0 ] + τ ∑ j = 0 K 2 − 1 E G j , ϑ = ∥ F − Θ ∥ 2 , (71) F=F_0+\sum_{j=0}^{K_2-1}G_j\Delta_{j+1},\qquad V=\mathbb{E}[YF_0]+\tau\sum_{j=0}^{K_2-1}\mathbb{E}G_j,\qquad\vartheta=\lVert F-\Theta\rVert_2, \tag*{(71)} F = F 0 + j = 0 ∑ K 2 − 1 G j Δ j + 1 , V = E [ Y F 0 ] + τ j = 0 ∑ K 2 − 1 E G j , ϑ = ∥ F − Θ ∥ 2 , (71) where Θ \Theta Θ Θ ( Y , Δ 1 , … , Δ K 2 ) \Theta(Y,\Delta_1,\ldots,\Delta_{K_2}) Θ ( Y , Δ 1 , … , Δ K 2 ) N \mathsf{N} N μ Θ \mu_\Theta μ Θ Θ \Theta Θ N \mathsf{N} N
If K 2 = 0 K_2=0 K 2 = 0 F = F 0 F=F_0 F = F 0 V = E [ Y F 0 ] V=\mathbb{E}[YF_0] V = E [ Y F 0 ]
The algorithm. Fix a control and accuracy parameters: an integer ℓ ≥ 2 \ell\ge2 ℓ ≥ 2 η ∈ ( 0 , q 0 ) \eta\in(0,q_0) η ∈ ( 0 , q 0 )
(i) Root polynomial. By dominated convergence, the function ϕ f \phi_f ϕ f [ 0 , q 0 ] [0,q_0] [ 0 , q 0 ] c η = min c ∈ [ 0 , q 0 − η ] ( ϕ f ( c ) − c ) > 0 c_\eta=\min_{c\in[0,q_0-\eta]}(\phi_f(c)-c)>0 c η = min c ∈ [ 0 , q 0 − η ] ( ϕ f ( c ) − c ) > 0 f f f N ( 0 , q 0 ) \mathsf{N}(0,q_0) N ( 0 , q 0 ) ∥ f − E f ∥ 2 > 0 \lVert f-\mathbb{E}f\rVert_2>0 ∥ f − E f ∥ 2 > 0 Q ⋆ Q_\star Q ⋆
∥ Q ⋆ − f ∥ 2 ≤ min { 1 ℓ , c η 8 q 0 , ∥ f − E f ∥ 2 3 } \lVert Q_\star-f\rVert_2\le\min\left\{\frac{1}{\ell},\frac{c_\eta}{8\sqrt{q_0}},\frac{\lVert f-\mathbb{E}f\rVert_2}{3}\right\} ∥ Q ⋆ − f ∥ 2 ≤ min { ℓ 1 , 8 q 0 c η , 3 ∥ f − E f ∥ 2 } in L 2 ( N ( 0 , q 0 ) ) L^2(\mathcal{N}(0,q_0)) L 2 ( N ( 0 , q 0 )) ∥ Q ⋆ ∥ 2 ≥ ∥ f ∥ 2 − ∥ Q ⋆ − f ∥ 2 ≥ 2 q 0 / 3 > 0 \lVert Q_\star\rVert_2 \ge\lVert f\rVert_2-\lVert Q_\star-f\rVert_2 \ge2\sqrt{q_0}/3>0 ∥ Q ⋆ ∥ 2 ≥ ∥ f ∥ 2 − ∥ Q ⋆ − f ∥ 2 ≥ 2 q 0 /3 > 0 p = q 0 Q ⋆ / ∥ Q ⋆ ∥ 2 p=\sqrt{q_0}\,Q_\star/\lVert Q_\star\rVert_2 p = q 0 Q ⋆ / ∥ Q ⋆ ∥ 2 K 1 K_1 K 1 k ≥ 1 k\ge1 k ≥ 1 ϱ k > q 0 − η \varrho_k>q_0-\eta ϱ k > q 0 − η ϱ k \varrho_k ϱ k Lemma 5.22 (a).
(ii) Coefficient polynomials. By (F3), there exist polynomials Q j Q_j Q j ( y , z 1 , … , z j ) (y,z_1,\ldots,z_j) ( y , z 1 , … , z j ) ∥ Q j − G j ∥ 2 ≤ 1 / ℓ \lVert Q_j-G_j\rVert_2\le1/\ell ∥ Q j − G j ∥ 2 ≤ 1/ ℓ N \mathcal{N} N ∥ Q j ∥ 2 ≥ 1 − 1 / ℓ ≥ 1 / 2 \lVert Q_j\rVert_2\ge1-1/\ell\ge1/2 ∥ Q j ∥ 2 ≥ 1 − 1/ ℓ ≥ 1/2 P j = Q j / ∥ Q j ∥ 2 P_j=Q_j/\lVert Q_j\rVert_2 P j = Q j / ∥ Q j ∥ 2
(iii) Center. We run the two-phase scheme of Definition 5.21 with p p p K 1 K_1 K 1 ( P j ) (P_j) ( P j ) G d G_d G d m = Θ ( y , Δ 1 , … , Δ K 2 ) ∈ ( − 1 , 1 ) V m=\Theta(y,\Delta^1,\ldots,\Delta^{K_2})\in(-1,1)^V m = Θ ( y , Δ 1 , … , Δ K 2 ) ∈ ( − 1 , 1 ) V Lemma 5.4 , applied with θ \theta θ Θ \Theta Θ u ˉ K ↦ ( y , Δ 1 , … , Δ K 2 ) \bar{u}^{K}\mapsto(y,\Delta^1,\ldots,\Delta^{K_2}) u ˉ K ↦ ( y , Δ 1 , … , Δ K 2 ) x t = f t − 1 ( u ˉ t − 1 ) x^t=f_{t-1}(\bar{u}^{t-1}) x t = f t − 1 ( u ˉ t − 1 ) 1 ≤ t ≤ K 1\le t\le K 1 ≤ t ≤ K x K + 1 = m x^{K+1}=m x K + 1 = m K + 1 K+1 K + 1 m m m
The update rule depends only on the control and the accuracy parameters, not on d d d
Lemma 5.24. In the algorithm, p p p E p ( q 0 Z ) 2 = q 0 \mathbb{E}p(\sqrt{q_0}Z)^2=q_0 E p ( q 0 Z ) 2 = q 0 ∥ p − f ∥ 2 ≤ 2 / ℓ \lVert p-f\rVert_2\le2/\ell ∥ p − f ∥ 2 ≤ 2/ ℓ K 1 K_1 K 1 K 1 ≤ 1 + 2 q 0 / c η K_1\le1+2q_0/c_\eta K 1 ≤ 1 + 2 q 0 / c η q 0 − η < ϱ K 1 ≤ q 0 q_0-\eta<\varrho_{K_1}\le q_0 q 0 − η < ϱ K 1 ≤ q 0 j < K 2 j<K_2 j < K 2 P j P_j P j ∥ P j − G j ∥ 2 ≤ 2 / ℓ \lVert P_j-G_j\rVert_2\le2/\ell ∥ P j − G j ∥ 2 ≤ 2/ ℓ
Proof. Since ∣ ∥ Q ⋆ ∥ 2 − q 0 ∣ ≤ ∥ Q ⋆ − f ∥ 2 \left|\lVert Q_\star\rVert_2-\sqrt{q_0}\right|\le\lVert Q_\star-f\rVert_2 ∥ Q ⋆ ∥ 2 − q 0 ≤ ∥ Q ⋆ − f ∥ 2 ∥ p − Q ⋆ ∥ 2 = ∣ q 0 − ∥ Q ⋆ ∥ 2 ∣ ≤ ∥ Q ⋆ − f ∥ 2 \lVert p-Q_\star\rVert_2=\left|\sqrt{q_0}-\lVert Q_\star\rVert_2\right|\le\lVert Q_\star-f\rVert_2 ∥ p − Q ⋆ ∥ 2 = q 0 − ∥ Q ⋆ ∥ 2 ≤ ∥ Q ⋆ − f ∥ 2 ∥ p − f ∥ 2 ≤ 2 ∥ Q ⋆ − f ∥ 2 ≤ 2 / ℓ \lVert p-f\rVert_2\le2\lVert Q_\star-f\rVert_2\le2/\ell ∥ p − f ∥ 2 ≤ 2 ∥ Q ⋆ − f ∥ 2 ≤ 2/ ℓ p p p ∥ ( p − E p ) − ( f − E f ) ∥ 2 ≤ ∥ p − f ∥ 2 ≤ ( 2 / 3 ) ∥ f − E f ∥ 2 \lVert(p-\mathbb{E}p)-(f-\mathbb{E}f)\rVert_2\le\lVert p-f\rVert_2\le(2/3)\lVert f-\mathbb{E}f\rVert_2 ∥( p − E p ) − ( f − E f ) ∥ 2 ≤ ∥ p − f ∥ 2 ≤ ( 2/3 ) ∥ f − E f ∥ 2 E p ( q 0 Z ) 2 = q 0 \mathbb{E}p(\sqrt{q_0}Z)^2=q_0 E p ( q 0 Z ) 2 = q 0 c ∈ [ 0 , q 0 ] c\in[0,q_0] c ∈ [ 0 , q 0 ]
∣ φ p ( c ) − φ f ( c ) ∣ ≤ ∥ p − f ∥ 2 ( ∥ p ∥ 2 + ∥ f ∥ 2 ) = 2 q 0 ∥ p − f ∥ 2 ≤ 4 q 0 ∥ Q ⋆ − f ∥ 2 ≤ c η 2 . |\varphi_p(c)-\varphi_f(c)|\le\lVert p-f\rVert_2(\lVert p\rVert_2+\lVert f\rVert_2)=2\sqrt{q_0}\lVert p-f\rVert_2\le4\sqrt{q_0}\lVert Q_\star-f\rVert_2\le\frac{c_\eta}{2}. ∣ φ p ( c ) − φ f ( c ) ∣ ≤ ∥ p − f ∥ 2 (∥ p ∥ 2 + ∥ f ∥ 2 ) = 2 q 0 ∥ p − f ∥ 2 ≤ 4 q 0 ∥ Q ⋆ − f ∥ 2 ≤ 2 c η . which implies that φ p ( c ) ≥ c + c η / 2 \varphi_p(c)\ge c+c_\eta/2 φ p ( c ) ≥ c + c η /2 [ 0 , q 0 − η ] [0,q_0-\eta] [ 0 , q 0 − η ] Lemma 5.22 (a), ϱ 1 ∈ [ 0 , q 0 ) \varrho_1\in[0,q_0) ϱ 1 ∈ [ 0 , q 0 ) ϱ k + 1 = φ p ( ϱ k ) \varrho_{k+1}=\varphi_p(\varrho_k) ϱ k + 1 = φ p ( ϱ k ) K 1 K_1 K 1 ϱ k \varrho_k ϱ k c η / 2 c_\eta/2 c η /2 ϱ k ≤ q 0 − η \varrho_k\le q_0-\eta ϱ k ≤ q 0 − η k k k ϱ k > q 0 − η \varrho_k>q_0-\eta ϱ k > q 0 − η 1 + 2 q 0 / c η 1+2q_0/c_\eta 1 + 2 q 0 / c η ϱ K 1 ≤ q 0 \varrho_{K_1}\le q_0 ϱ K 1 ≤ q 0 Lemma 5.22 (a). For the coefficients, ∥ Q j ∥ 2 ≥ 1 − 1 / ℓ > 0 \lVert Q_j\rVert_2\ge1-1/\ell>0 ∥ Q j ∥ 2 ≥ 1 − 1/ ℓ > 0 ∥ P j − G j ∥ 2 ≤ ∣ 1 − ∥ Q j ∥ 2 ∣ + ∥ Q j − G j ∥ 2 ≤ 2 / ℓ \lVert P_j-G_j\rVert_2\le|1-\lVert Q_j\rVert_2|+\lVert Q_j-G_j\rVert_2\le2/\ell ∥ P j − G j ∥ 2 ≤ ∣1 − ∥ Q j ∥ 2 ∣ + ∥ Q j − G j ∥ 2 ≤ 2/ ℓ
Proposition 5.25. Let ( G d ) (G_d) ( G d ) T L d T_L^d T L d L ≥ 3 L\ge3 L ≥ 3 Q d Q_d Q d M W M_W M W Lemma 5.20 . Fix a control with root variance q 0 q_0 q 0 τ \tau τ K 2 K_2 K 2 V V V ϑ \vartheta ϑ ℓ \ell ℓ η \eta η m m m G d G_d G d κ x = 1 − m x 2 \kappa_x=1-m_x^2 κ x = 1 − m x 2
(a) As d → ∞ d\to\infty d → ∞ n − 1 E ∑ x m x → E Θ n^{-1}\mathbb{E}\sum_xm_x\to\mathbb{E}\Theta n − 1 E ∑ x m x → E Θ n − 1 E ∑ x ent ( m x ) → E ent ( Θ ) n^{-1}\mathbb{E}\sum_x\operatorname{ent}(m_x)\to\mathbb{E}\operatorname{ent}(\Theta) n − 1 E ∑ x ent ( m x ) → E ent ( Θ ) n − 1 E ⟨ κ , S κ ⟩ → ( 1 − E Θ 2 ) 2 n^{-1}\mathbb{E}\langle\kappa,S\kappa\rangle\to(1-\mathbb{E}\Theta^2)^2 n − 1 E ⟨ κ , S κ ⟩ → ( 1 − E Θ 2 ) 2 Θ \Theta Θ μ Θ \mu_\Theta μ Θ
(b) We have
lim sup d → ∞ ∣ 1 n E H G d ( m ) − V ∣ ≤ 2 τ 1 { K 2 ≥ 1 } + η + 4 ℓ + M W ( ϑ + 4 ℓ ) . \limsup_{d\to\infty}\left|\frac{1}{n}\mathbb{E}H_{G_d}(m)-V\right|\le2\sqrt{\tau}\,\mathbf{1}_{\{K_2\ge1\}}+\eta+\frac{4}{\ell}+M_W\left(\vartheta+\frac{4}{\ell}\right). d → ∞ lim sup n 1 E H G d ( m ) − V ≤ 2 τ 1 { K 2 ≥ 1 } + η + ℓ 4 + M W ( ϑ + ℓ 4 ) . (c) For every x ∈ V x\in V x ∈ V m x m_x m x x x x μ Θ \mu_\Theta μ Θ d → ∞ d\to\infty d → ∞
Proof. Let θ \theta θ m = θ ( U ˉ K ) m=\theta(\bar{U}^{K}) m = θ ( U ˉ K ) θ ( U ˉ K ) = Θ ( Y , Δ 1 , … , Δ K 2 ) \theta(\bar{U}^{K})=\Theta(Y,\Delta_1,\ldots,\Delta_{K_2}) θ ( U ˉ K ) = Θ ( Y , Δ 1 , … , Δ K 2 ) μ Θ \mu_\Theta μ Θ Θ \Theta Θ
(a) and (c). Since the functions θ \theta θ ent ∘ θ \operatorname{ent}\circ\theta ent ∘ θ φ ∘ θ \varphi\circ\theta φ ∘ θ φ \varphi φ m x m_x m x x x x κ ˉ = E ( 1 − Θ 2 ) \bar{\kappa}=\mathbb{E}(1-\Theta^2) κ ˉ = E ( 1 − Θ 2 ) ( S κ ) x = D − 1 ∑ y ∼ x ( 1 − θ ( u ˉ y K ) 2 ) (S\kappa)_x=D^{-1}\sum_{y\sim x}(1-\theta(\bar{u}^{K}_y)^2) ( S κ ) x = D − 1 ∑ y ∼ x ( 1 − θ ( u ˉ y K ) 2 ) E ∣ ( S κ ) x − κ ˉ ∣ → 0 \mathbb{E}|(S\kappa)_x-\bar{\kappa}|\to0 E ∣ ( S κ ) x − κ ˉ ∣ → 0 0 ≤ κ x ≤ 1 0\leq\kappa_x\leq1 0 ≤ κ x ≤ 1
∣ 1 n E ⟨ κ , S κ ⟩ − κ ˉ 1 n E ∑ x κ x ∣ ≤ 1 n ∑ x E ∣ ( S κ ) x − κ ˉ ∣ → 0 , \left|\frac{1}{n}\mathbb{E}\langle\kappa,S\kappa\rangle-\bar{\kappa}\frac{1}{n}\mathbb{E}\sum_x\kappa_x\right|\leq\frac{1}{n}\sum_x\mathbb{E}|(S\kappa)_x-\bar{\kappa}|\to0, n 1 E ⟨ κ , S κ ⟩ − κ ˉ n 1 E x ∑ κ x ≤ n 1 x ∑ E ∣ ( S κ ) x − κ ˉ ∣ → 0 , and n − 1 E ∑ x κ x → κ ˉ n^{-1}\mathbb{E}\sum_x\kappa_x\to\bar{\kappa} n − 1 E ∑ x κ x → κ ˉ
(b) Let F = f K ( U ˉ K ) = F ^ 0 + ∑ j < K 2 P j Δ j + 1 F=f_K(\bar{U}^{K})=\hat{F}_0+\sum_{j<K_2}P_j\Delta_{j+1} F = f K ( U ˉ K ) = F ^ 0 + ∑ j < K 2 P j Δ j + 1 ( Y , Δ 1 , … , Δ K 2 ) (Y,\Delta_1,\ldots,\Delta_{K_2}) ( Y , Δ 1 , … , Δ K 2 ) N \mathcal{N} N E [ F U K + 1 ] \mathbb{E}[FU_{K+1}] E [ F U K + 1 ] V V V
Root message. By Stein’s identity, we have q 0 E p ′ ( Y ) = E [ Y p ( Y ) ] q_0\mathbb{E}p'(Y)=\mathbb{E}[Yp(Y)] q 0 E p ′ ( Y ) = E [ Y p ( Y )] ∣ E p ′ ( Y ) ∣ ≤ ∥ Y ∥ 2 ∥ p ( Y ) ∥ 2 / q 0 = 1 |\mathbb{E}p'(Y)|\leq\|Y\|_2\|p(Y)\|_2/q_0=1 ∣ E p ′ ( Y ) ∣ ≤ ∥ Y ∥ 2 ∥ p ( Y ) ∥ 2 / q 0 = 1
∣ q K 1 E p ′ ( Y ) − E [ Y f ( Y ) ] ∣ ≤ ( q 0 − q K 1 ) ∣ E p ′ ( Y ) ∣ + ∣ E [ Y ( p − f ) ( Y ) ] ∣ ≤ η + 2 ℓ . |q_{K_1}\mathbb{E}p'(Y)-\mathbb{E}[Yf(Y)]|\leq(q_0-q_{K_1})|\mathbb{E}p'(Y)|+|\mathbb{E}[Y(p-f)(Y)]|\leq\eta+\frac{2}{\ell}. ∣ q K 1 E p ′ ( Y ) − E [ Y f ( Y )] ∣ ≤ ( q 0 − q K 1 ) ∣ E p ′ ( Y ) ∣ + ∣ E [ Y ( p − f ) ( Y )] ∣ ≤ η + ℓ 2 . First innovation (K 2 ≥ 1 K_2\geq1 K 2 ≥ 1 E P 0 \mathbb{E}P_0 E P 0 ( τ τ 1 ) 1 / 2 (\tau\tau_1)^{1/2} ( τ τ 1 ) 1/2 τ \tau τ τ 1 ≤ E U K 1 + 1 2 = q 0 ≤ 1 \tau_1\leq\mathbb{E}U_{K_1+1}^2=q_0\leq1 τ 1 ≤ E U K 1 + 1 2 = q 0 ≤ 1 ∣ E P 0 ∣ , ∣ E G 0 ∣ ≤ 1 |\mathbb{E}P_0|,|\mathbb{E}G_0|\leq1 ∣ E P 0 ∣ , ∣ E G 0 ∣ ≤ 1 ∣ ( τ τ 1 ) 1 / 2 E P 0 − τ E G 0 ∣ ≤ τ + τ ≤ 2 τ |(\tau\tau_1)^{1/2}\mathbb{E}P_0-\tau\mathbb{E}G_0|\leq\sqrt{\tau}+\tau\leq2\sqrt{\tau} ∣ ( τ τ 1 ) 1/2 E P 0 − τ E G 0 ∣ ≤ τ + τ ≤ 2 τ
Combining the three bounds, we obtain ∣ E [ F U K + 1 ] − V ∣ ≤ 2 τ 1 { K 2 ≥ 1 } + η + 4 / ℓ |\mathbb{E}[FU_{K+1}]-V|\leq2\sqrt{\tau}\mathbf{1}_{\{K_2\geq1\}}+\eta+4/\ell ∣ E [ F U K + 1 ] − V ∣ ≤ 2 τ 1 { K 2 ≥ 1 } + η + 4/ ℓ F − F ^ = ( F ^ 0 − F 0 ) + ∑ j ( P j − G j ) Δ j + 1 F-\hat{F}=(\hat{F}_0-F_0)+\sum_j(P_j-G_j)\Delta_{j+1} F − F ^ = ( F ^ 0 − F 0 ) + ∑ j ( P j − G j ) Δ j + 1 τ ∥ P j − G j ∥ 2 \sqrt{\tau}\|P_j-G_j\|_2 τ ∥ P j − G j ∥ 2 K 2 τ ≤ 1 K_2\tau\leq1 K 2 τ ≤ 1 ∥ F − F ^ ∥ 2 ≤ 4 / ℓ \|F-\hat{F}\|_2\leq4/\ell ∥ F − F ^ ∥ 2 ≤ 4/ ℓ ∥ Θ − F ∥ 2 ≤ ϑ + 4 / ℓ \|\Theta-F\|_2\leq\vartheta+4/\ell ∥Θ − F ∥ 2 ≤ ϑ + 4/ ℓ ∥ Θ ∥ 2 ≤ 1 \|\Theta\|_2\leq1 ∥Θ ∥ 2 ≤ 1 ∥ F ∥ 2 ≤ 1 \|F\|_2\leq1 ∥ F ∥ 2 ≤ 1
lim sup d → ∞ ∣ 1 n E H G d ( m ) − E [ F U K + 1 ] ∣ ≤ M W ( ϑ + 4 ℓ ) . \limsup_{d\to\infty}\left|\frac{1}{n}\mathbb{E}H_{G_d}(m)-\mathbb{E}[FU_{K+1}]\right|\leq M_W\left(\vartheta+\frac{4}{\ell}\right). d → ∞ lim sup n 1 E H G d ( m ) − E [ F U K + 1 ] ≤ M W ( ϑ + ℓ 4 ) . Combining the two bounds, we obtain (b). □ \square □
The Parisi control
In this subsection, β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 supp ν = [ q − , q + ] \operatorname{supp}\nu=[q_-,q_+] supp ν = [ q − , q + ] 0 < q − ≤ q + < 1 0<q_-\leq q_+<1 0 < q − ≤ q + < 1 ν ( t ) = ν ( [ 0 , t ] ) \nu(t)=\nu([0,t]) ν ( t ) = ν ([ 0 , t ])
∣ ∂ x Φ ∣ ≤ 1 |\partial_x\Phi| \le1 ∣ ∂ x Φ∣ ≤ 1 0 < ∂ x 2 Φ ≤ 1 0 < \partial_x^2\Phi\le1 0 < ∂ x 2 Φ ≤ 1 ∣ ∂ x 3 Φ ∣ ≤ 4 |\partial_x^3\Phi| \le4 ∣ ∂ x 3 Φ∣ ≤ 4 ∣ v ∣ ≤ β 2 |v| \le\beta^2 ∣ v ∣ ≤ β 2 0 ≤ ∂ x v ≤ β 2 0 \le\partial_xv \le\beta^2 0 ≤ ∂ x v ≤ β 2 0 < a ≤ β 0 < a \le\beta 0 < a ≤ β ∣ ∂ x a ∣ ≤ 4 β |\partial_xa| \le4\beta ∣ ∂ x a ∣ ≤ 4 β
For s , t ∈ [ 0 , 1 ] s,t \in[0,1] s , t ∈ [ 0 , 1 ] x ∈ R x \in\mathbb{R} x ∈ R ∣ ∂ x Φ ( t , x ) − ∂ x Φ ( s , x ) ∣ ≤ 3 β 2 ∣ t − s ∣ |\partial_x\Phi(t,x)-\partial_x\Phi(s,x)| \le3\beta^2|t-s| ∣ ∂ x Φ ( t , x ) − ∂ x Φ ( s , x ) ∣ ≤ 3 β 2 ∣ t − s ∣ ∣ a ( t , x ) − a ( s , x ) ∣ ≤ 4 3 β 2 ∣ t − s ∣ 1 / 2 |a(t,x)-a(s,x)| \le4\sqrt{3}\beta^2|t-s|^{1/2} ∣ a ( t , x ) − a ( s , x ) ∣ ≤ 4 3 β 2 ∣ t − s ∣ 1/2
For 0 ≤ s ≤ t ≤ 1 0 \le s \le t \le1 0 ≤ s ≤ t ≤ 1 M t − M s = ∫ s t a ( r , X r ) d B r M_t-M_s=\int_s^t a(r,X_r)\,\mathrm{d}B_r M t − M s = ∫ s t a ( r , X r ) d B r
∂ x 2 Φ ( t , X t ) − ∂ x 2 Φ ( s , X s ) = − ∫ s t β 2 ν ( r ) ( ∂ x 2 Φ ( r , X r ) ) 2 d r + ∫ s t β ∂ x 3 Φ ( r , X r ) d B r . \partial_x^2\Phi(t,X_t)-\partial_x^2\Phi(s,X_s)=-\int_s^t \beta^2\nu(r)\bigl(\partial_x^2\Phi(r,X_r)\bigr)^2\,\mathrm{d}r+\int_s^t \beta\partial_x^3\Phi(r,X_r)\,\mathrm{d}B_r. ∂ x 2 Φ ( t , X t ) − ∂ x 2 Φ ( s , X s ) = − ∫ s t β 2 ν ( r ) ( ∂ x 2 Φ ( r , X r ) ) 2 d r + ∫ s t β ∂ x 3 Φ ( r , X r ) d B r . E M t 2 = t \mathbb{E}M_t^2=t E M t 2 = t t ∈ [ q − , q + ] t\in[q_-,q_+] t ∈ [ q − , q + ] q − < q + q_-<q_+ q − < q + E a ( t , X t ) 2 = 1 \mathbb{E}a(t,X_t)^2=1 E a ( t , X t ) 2 = 1 t ∈ [ q − , q + ] t\in[q_-,q_+] t ∈ [ q − , q + ] M q + = tanh X q + M_{q_+}=\tanh X_{q_+} M q + = tanh X q +
The function f ( y ) = ∂ x Φ ( q − , h + β y ) f(y)=\partial_x\Phi(q_-,h+\beta y) f ( y ) = ∂ x Φ ( q − , h + β y ) q − q_- q − f ( B q − ) = M q − f(B_{q_-})=M_{q_-} f ( B q − ) = M q − ϕ f \phi_f ϕ f ϕ p \phi_p ϕ p p = f p=f p = f
Since ν ( t ) = 0 \nu(t)=0 ν ( t ) = 0 t < q − t<q_- t < q − v v v [ 0 , q − ) × R [0,q_-)\times\mathbb{R} [ 0 , q − ) × R X t = h + β B t X_t=h+\beta B_t X t = h + β B t t ≤ q − t\le q_- t ≤ q − ∂ x 2 Φ ( t , X t ) \partial_x^2\Phi(t,X_t) ∂ x 2 Φ ( t , X t ) [ 0 , q − ] [0,q_-] [ 0 , q − ] E a ( t , X t ) \mathbb{E}a(t,X_t) E a ( t , X t ) [ 0 , q − ] [0,q_-] [ 0 , q − ] E ( β , h ) = ∫ 0 q + E a ( t , X t ) d t \mathcal{E}(\beta,h)=\int_0^{q_+}\mathbb{E}a(t,X_t)\,\mathrm{d}t E ( β , h ) = ∫ 0 q + E a ( t , X t ) d t
The control. If q − < q + q_-<q_+ q − < q + K 2 ≥ 1 K_2\ge1 K 2 ≥ 1 τ = ( q + − q − ) / K 2 \tau=(q_+-q_-)/K_2 τ = ( q + − q − ) / K 2 t j = q − + j τ t_j=q_-+j\tau t j = q − + j τ 0 ≤ j ≤ K 2 0\le j\le K_2 0 ≤ j ≤ K 2 t K 2 = q + t_{K_2}=q_+ t K 2 = q + N \mathcal{N} N q 0 = q − q_0=q_- q 0 = q − Y = B q − Y=B_{q_-} Y = B q − Δ j = B t j − B t j − 1 \Delta_j=B_{t_j}-B_{t_{j-1}} Δ j = B t j − B t j − 1 X X X
X ^ 0 = h + β Y , X ^ j + 1 = X ^ j + v ( t j , X ^ j ) τ + β Δ j + 1 , λ j = ∥ a ( t j , X ^ j ) ∥ 2 , G j = a ( t j , X ^ j ) λ j , Θ = tanh X ^ K 2 . (72) \begin{aligned}
\widehat{X}_0&=h+\beta Y,\qquad\widehat{X}_{j+1}=\widehat{X}_j+v(t_j,\widehat{X}_j)\tau+\beta\Delta_{j+1},\\
\lambda_j&=\|a(t_j,\widehat{X}_j)\|_2,\qquad G_j=\frac{a(t_j,\widehat{X}_j)}{\lambda_j},\qquad\Theta=\tanh\widehat{X}_{K_2}. \tag*{(72)}
\end{aligned} X 0 λ j = h + β Y , X j + 1 = X j + v ( t j , X j ) τ + β Δ j + 1 , = ∥ a ( t j , X j ) ∥ 2 , G j = λ j a ( t j , X j ) , Θ = tanh X K 2 . (72) with the root map f ( y ) = ∂ x Φ ( q − , h + β y ) f(y)=\partial_x\Phi(q_-,h+\beta y) f ( y ) = ∂ x Φ ( q − , h + β y ) [103 (2.37)] , the coefficients are normalized to have unit second moment under the law of the Euler scheme. If q − = q + q_-=q_+ q − = q + K 2 = 0 K_2=0 K 2 = 0 f f f Θ ( y ) = f ( y ) \Theta(y)=f(y) Θ ( y ) = f ( y )
Proposition 5.26. Let β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 C < ∞ C<\infty C < ∞ β \beta β h h h
(a) If q − = q + q_-=q_+ q − = q + q − q_- q − 0 0 0 V = E ( β , h ) V=\mathcal{E}(\beta,h) V = E ( β , h ) ϑ = 0 \vartheta=0 ϑ = 0 μ Θ = μ + \mu_\Theta=\mu_+ μ Θ = μ +
(b) If q − < q + q_-<q_+ q − < q + K 2 ≥ 1 K_2\ge1 K 2 ≥ 1 q − q_- q − τ = ( q + − q − ) / K 2 \tau=(q_+-q_-)/K_2 τ = ( q + − q − ) / K 2 K 2 K_2 K 2
∣ V − E ( β , h ) ∣ ≤ C τ , ϑ ≤ C τ , ∥ Θ − M q + ∥ 2 ≤ C τ . |V-\mathcal{E}(\beta,h)|\le C\sqrt{\tau},\qquad\vartheta\le C\sqrt{\tau},\qquad\|\Theta-M_{q_+}\|_2\le C\sqrt{\tau}. ∣ V − E ( β , h ) ∣ ≤ C τ , ϑ ≤ C τ , ∥Θ − M q + ∥ 2 ≤ C τ . In particular, W 1 ( μ Θ , μ + ) ≤ C τ W_1(\mu_\Theta,\mu_+)\le C\sqrt{\tau} W 1 ( μ Θ , μ + ) ≤ C τ
Proof. (a) Since q − = q + q_{-}=q_{+} q − = q + ∂ x Φ ( q − , x ) = tanh x \partial_x\Phi(q_{-},x)=\tanh x ∂ x Φ ( q − , x ) = tanh x f ( y ) = tanh ( h + β y ) f(y)=\tanh(h+\beta y) f ( y ) = tanh ( h + β y ) Θ = f \Theta=f Θ = f ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) F = F 0 = f ( Y ) = Θ F=F_0=f(Y)=\Theta F = F 0 = f ( Y ) = Θ ϑ = 0 \vartheta=0 ϑ = 0 Θ = f ( B q − ) = M q − = M q + \Theta=f(B_{q_{-}})=M_{q_{-}}=M_{q_{+}} Θ = f ( B q − ) = M q − = M q + μ Θ = μ + \mu_{\Theta}=\mu_{+} μ Θ = μ + V = E [ Y f ( Y ) ] = q − E f ′ ( Y ) = q − E a ( q − , X q − ) V=\mathbb{E}[Yf(Y)]=q_{-}\mathbb{E}f'(Y)=q_{-}\mathbb{E}a(q_{-},X_{q_{-}}) V = E [ Y f ( Y )] = q − E f ′ ( Y ) = q − E a ( q − , X q − ) E a ( t , X t ) \mathbb{E}a(t,X_t) E a ( t , X t ) [ 0 , q − ] [0,q_{-}] [ 0 , q − ] E ( β , h ) \mathcal{E}(\beta,h) E ( β , h )
(b) Let C C C β \beta β h h h e ^ j = X ^ j − X t j \widehat{e}_j=\widehat{X}_j-X_{t_j} e j = X j − X t j r ∈ [ t j , t j + 1 ] r\in[t_j,t_{j+1}] r ∈ [ t j , t j + 1 ] x ∈ R x\in\mathbb{R} x ∈ R
∣ a ( t j , x ) − a ( r , x ) ∣ ≤ 4 3 β 2 τ , ∣ v ( t j , x ) − v ( r , x ) ∣ ≤ 3 β 4 τ + β 2 ( ν ( t j + 1 ) − ν ( t j ) ) , \left|a(t_j,x)-a(r,x)\right|\le4\sqrt{3}\,\beta^2\sqrt{\tau},\qquad\left|v(t_j,x)-v(r,x)\right|\le3\beta^4\tau+\beta^2\bigl(\nu(t_{j+1})-\nu(t_j)\bigr), ∣ a ( t j , x ) − a ( r , x ) ∣ ≤ 4 3 β 2 τ , ∣ v ( t j , x ) − v ( r , x ) ∣ ≤ 3 β 4 τ + β 2 ( ν ( t j + 1 ) − ν ( t j ) ) , the second because v = β 2 ν ∂ x Φ v=\beta^2\nu\,\partial_x\Phi v = β 2 ν ∂ x Φ ν \nu ν ∣ ∂ x Φ ∣ ≤ 1 |\partial_x\Phi|\le1 ∣ ∂ x Φ∣ ≤ 1 ∥ X r − X t j ∥ 2 ≤ β 2 τ + β τ \|X_r-X_{t_j}\|_2\le\beta^2\tau+\beta\sqrt{\tau} ∥ X r − X t j ∥ 2 ≤ β 2 τ + β τ ∣ v ∣ ≤ β 2 |v|\le\beta^2 ∣ v ∣ ≤ β 2
Euler error. We have e ^ 0 = 0 \widehat{e}_0=0 e 0 = 0 X q − = h + β B q − X_{q_{-}}=h+\beta B_{q_{-}} X q − = h + β B q − X ^ \widehat{X} X X X X e ^ j + 1 = e ^ j + ∫ t j t j + 1 ( v ( t j , X ^ j ) − v ( r , X r ) ) d r \widehat{e}_{j+1}=\widehat{e}_j+\int_{t_j}^{t_{j+1}}\bigl(v(t_j,\widehat{X}_j)-v(r,X_r)\bigr)\,\mathrm{d}r e j + 1 = e j + ∫ t j t j + 1 ( v ( t j , X j ) − v ( r , X r ) ) d r
[ v ( t j , X ^ j ) − v ( t j , X t j ) ] + [ v ( t j , X t j ) − v ( t j , X r ) ] + [ v ( t j , X r ) − v ( r , X r ) ] . \left[v(t_j,\widehat{X}_j)-v(t_j,X_{t_j})\right]+\left[v(t_j,X_{t_j})-v(t_j,X_r)\right]+\left[v(t_j,X_r)-v(r,X_r)\right]. [ v ( t j , X j ) − v ( t j , X t j ) ] + [ v ( t j , X t j ) − v ( t j , X r ) ] + [ v ( t j , X r ) − v ( r , X r ) ] . The first two brackets are at most β 2 ∣ e ^ j ∣ \beta^2|\widehat{e}_j| β 2 ∣ e j ∣ β 2 ∣ X t j − X r ∣ \beta^2|X_{t_j}-X_r| β 2 ∣ X t j − X r ∣ 3 β 4 τ + β 2 ( ν ( t j + 1 ) − ν ( t j ) ) 3\beta^4\tau+\beta^2\bigl(\nu(t_{j+1})-\nu(t_j)\bigr) 3 β 4 τ + β 2 ( ν ( t j + 1 ) − ν ( t j ) ) j j j ν \nu ν ∑ j τ ( ν ( t j + 1 ) − ν ( t j ) ) ≤ τ \sum_j\tau\bigl(\nu(t_{j+1})-\nu(t_j)\bigr)\le\tau ∑ j τ ( ν ( t j + 1 ) − ν ( t j ) ) ≤ τ ν \nu ν ν \nu ν L 2 L^2 L 2 j < k j<k j < k K 2 τ ≤ 1 K_2\tau\le1 K 2 τ ≤ 1 τ ≤ 1 \tau\le1 τ ≤ 1 ∥ e ^ k ∥ 2 ≤ β 2 τ ∑ j < k ∥ e ^ j ∥ 2 + C τ \|\widehat{e}_k\|_2\le\beta^2\tau\sum_{j<k}\|\widehat{e}_j\|_2+C\sqrt{\tau} ∥ e k ∥ 2 ≤ β 2 τ ∑ j < k ∥ e j ∥ 2 + C τ max k ≤ K 2 ∥ e ^ k ∥ 2 ≤ C τ ( 1 + β 2 τ ) K 2 ≤ C τ \max_{k\le K_2}\|\widehat{e}_k\|_2\le C\sqrt{\tau}(1+\beta^2\tau)^{K_2}\le C\sqrt{\tau} max k ≤ K 2 ∥ e k ∥ 2 ≤ C τ ( 1 + β 2 τ ) K 2 ≤ C τ
Normalization. For j < K 2 j<K_2 j < K 2 t j ∈ [ q − , q + ) t_j\in[q_{-},q_{+}) t j ∈ [ q − , q + ) E a ( t j , X t j ) 2 = 1 \mathbb{E}a(t_j,X_{t_j})^2=1 E a ( t j , X t j ) 2 = 1
∣ λ j 2 − 1 ∣ ≤ E ∣ a ( t j , X ^ j ) 2 − a ( t j , X t j ) 2 ∣ ≤ 2 β ⋅ 4 β ∥ e ^ j ∥ 1 ≤ C τ , |\lambda_j^2-1|\le\mathbb{E}\left|a(t_j,\widehat{X}_j)^2-a(t_j,X_{t_j})^2\right|\le2\beta\cdot4\beta\|\widehat{e}_j\|_1\le C\sqrt{\tau}, ∣ λ j 2 − 1∣ ≤ E a ( t j , X j ) 2 − a ( t j , X t j ) 2 ≤ 2 β ⋅ 4 β ∥ e j ∥ 1 ≤ C τ , and ∣ λ j − 1 ∣ = ∣ λ j 2 − 1 ∣ / ( λ j + 1 ) ≤ C τ |\lambda_j-1|=|\lambda_j^2-1|/(\lambda_j+1)\le C\sqrt{\tau} ∣ λ j − 1∣ = ∣ λ j 2 − 1∣/ ( λ j + 1 ) ≤ C τ a > 0 a>0 a > 0 λ j > 0 \lambda_j>0 λ j > 0 G j G_j G j E G j 2 = 1 \mathbb{E}G_j^2=1 E G j 2 = 1
Admissibility. By the last of the listed facts, f f f G j G_j G j ( Y , Δ 1 , … , Δ j ) (Y,\Delta_1,\ldots,\Delta_j) ( Y , Δ 1 , … , Δ j ) x ↦ x + v ( t j , x ) τ x\mapsto x+v(t_j,x)\tau x ↦ x + v ( t j , x ) τ Θ \Theta Θ ( Y , Δ 1 , … , Δ K 2 ) (Y,\Delta_1,\ldots,\Delta_{K_2}) ( Y , Δ 1 , … , Δ K 2 ) ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) q − + K 2 τ = q + ≤ 1 q_{-}+K_2\tau=q_{+}\le1 q − + K 2 τ = q + ≤ 1
Coefficient error. For r ∈ [ t j , t j + 1 ] r\in[t_j,t_{j+1}] r ∈ [ t j , t j + 1 ] ∥ G j − a ( t j , X ^ j ) ∥ 2 = ∣ 1 − λ j ∣ \|G_j-a(t_j,\widehat{X}_j)\|_2=|1-\lambda_j| ∥ G j − a ( t j , X j ) ∥ 2 = ∣1 − λ j ∣
∥ G j − a ( r , X r ) ∥ 2 ≤ ∣ 1 − λ j ∣ + 4 β ∥ e ^ j ∥ 2 + 4 β ∥ X t j − X r ∥ 2 + 4 3 β 2 τ ≤ C τ . \|G_j-a(r,X_r)\|_2\le|1-\lambda_j|+4\beta\|\widehat{e}_j\|_2+4\beta\|X_{t_j}-X_r\|_2+4\sqrt{3}\,\beta^2\sqrt{\tau}\le C\sqrt{\tau}. ∥ G j − a ( r , X r ) ∥ 2 ≤ ∣1 − λ j ∣ + 4 β ∥ e j ∥ 2 + 4 β ∥ X t j − X r ∥ 2 + 4 3 β 2 τ ≤ C τ . Defect, output, and value. By the martingale identity, M q + = M q − + ∑ j < K 2 ∫ t j t j + 1 a ( r , X r ) d B r M_{q_{+}}=M_{q_{-}}+\sum_{j<K_2}\int_{t_j}^{t_{j+1}}a(r,X_r)\,\mathrm{d}B_r M q + = M q − + ∑ j < K 2 ∫ t j t j + 1 a ( r , X r ) d B r G j G_j G j ( B s ) s ≤ t j (B_s)_{s\le t_j} ( B s ) s ≤ t j
G j Δ j + 1 = ∫ t j t j + 1 G j d B r . G_j\Delta_{j+1}=\int_{t_j}^{t_{j+1}}G_j\,\mathrm{d}B_r. G j Δ j + 1 = ∫ t j t j + 1 G j d B r . Since the first term of F F F F 0 = f ( Y ) = M q − F_0=f(Y)=M_{q_-} F 0 = f ( Y ) = M q −
F − M q + = ∑ j < K 2 ∫ t j t j + 1 ( G j − a ( r , X r ) ) d B r . F-M_{q_+}=\sum_{j<K_2}\int_{t_j}^{t_{j+1}}\left(G_j-a(r,X_r)\right)\,\mathrm{d}B_r. F − M q + = j < K 2 ∑ ∫ t j t j + 1 ( G j − a ( r , X r ) ) d B r . and by the Itô isometry, the coefficient error, and K 2 τ ≤ 1 K_2\tau\leq1 K 2 τ ≤ 1 ∥ F − M q + ∥ 2 ≤ C τ \lVert F-M_{q_+}\rVert_2\leq C\sqrt{\tau} ∥ F − M q + ∥ 2 ≤ C τ M q + = tanh X q + M_{q_+}=\tanh X_{q_+} M q + = tanh X q + tanh \tanh tanh ∥ Θ − M q + ∥ 2 ≤ ∥ e ^ K 2 ∥ 2 ≤ C τ \lVert\Theta-M_{q_+}\rVert_2\leq\lVert\widehat{e}_{K_2}\rVert_2\leq C\sqrt{\tau} ∥ Θ − M q + ∥ 2 ≤ ∥ e K 2 ∥ 2 ≤ C τ ϑ = ∥ F − Θ ∥ 2 ≤ C τ \vartheta=\lVert F-\Theta\rVert_2\leq C\sqrt{\tau} ϑ = ∥ F − Θ ∥ 2 ≤ C τ ( Θ , M q + ) (\Theta,M_{q_+}) ( Θ , M q + ) μ Θ \mu_\Theta μ Θ μ + \mu_+ μ + W 1 ( μ Θ , μ + ) ≤ ∥ Θ − M q + ∥ 2 W_1(\mu_\Theta,\mu_+)\leq\lVert\Theta-M_{q_+}\rVert_2 W 1 ( μ Θ , μ + ) ≤ ∥ Θ − M q + ∥ 2 E [ Y F 0 ] = E [ Y f ( Y ) ] = q − E f ′ ( Y ) = q − E a ( q − , X q − ) = ∫ 0 q − E a ( t , X t ) d t \mathbb{E}[YF_0]=\mathbb{E}[Yf(Y)]=q_-\mathbb{E}f'(Y)=q_-\mathbb{E}a(q_-,X_{q_-})=\int_0^{q_-}\mathbb{E}a(t,X_t)\,\mathrm{d}t E [ Y F 0 ] = E [ Y f ( Y )] = q − E f ′ ( Y ) = q − E a ( q − , X q − ) = ∫ 0 q − E a ( t , X t ) d t V = E [ Y F 0 ] + τ ∑ j < K 2 E G j V=\mathbb{E}[YF_0]+\tau\sum_{j<K_2}\mathbb{E}G_j V = E [ Y F 0 ] + τ ∑ j < K 2 E G j t K 2 = q + t_{K_2}=q_+ t K 2 = q +
∣ V − E ( β , h ) ∣ ≤ ∑ j < K 2 ∫ t j t j + 1 ∥ G j − a ( r , X r ) ∥ 1 d r ≤ C τ . \left|V-\mathcal{E}(\beta,h)\right|\leq\sum_{j<K_2}\int_{t_j}^{t_{j+1}}\lVert G_j-a(r,X_r)\rVert_1\,\mathrm{d}r\leq C\sqrt{\tau}. ∣ V − E ( β , h ) ∣ ≤ j < K 2 ∑ ∫ t j t j + 1 ∥ G j − a ( r , X r ) ∥ 1 d r ≤ C τ . Homogenization
The law of the center is homogeneous only after averaging over the disorder. This subsection shows that its empirical behavior is nevertheless close to the limiting law, uniformly over the fields in a 1 + c S [ 0 , B ] V a1+cS[0,B]^V a 1 + c S [ 0 , B ] V m x m_x m x S S S
Lemma 5.27. Let G ∈ G D G\in\mathcal{G}_D G ∈ G D (64) . For 1 ≤ t ≤ T 1\leq t\leq T 1 ≤ t ≤ T x ∈ V x\in V x ∈ V u ˉ x t \bar{u}_x^t u ˉ x t J e J_e J e e e e t − 1 t-1 t − 1 x x x 1 ≤ K ≤ T 1\leq K\leq T 1 ≤ K ≤ T θ : R K → R \theta:\mathbb{R}^K\to\mathbb{R} θ : R K → R m = θ ( u ˉ K ) m=\theta(\bar{u}^K) m = θ ( u ˉ K ) V V V ( D + 1 ) 2 K − 1 (D+1)^{2K-1} ( D + 1 ) 2 K − 1 m x m_x m x
Proof. The first claim holds for t = 1 t=1 t = 1 u x 1 = f 0 ∑ y ∼ x W x y u_x^1=f_0\sum_{y\sim x}W_{xy} u x 1 = f 0 ∑ y ∼ x W x y t t t (64) , u x t + 1 = ∑ y ∼ x W x y f t ( u ˉ y t ) − ∑ s b t , s f s − 1 ( u ˉ x s − 1 ) u_x^{t+1}=\sum_{y\sim x}W_{xy}f_t(\bar{u}_y^t)-\sum_s b_{t,s}f_{s-1}(\bar{u}_x^{s-1}) u x t + 1 = ∑ y ∼ x W x y f t ( u ˉ y t ) − ∑ s b t , s f s − 1 ( u ˉ x s − 1 ) x y xy x y t − 1 t-1 t − 1 x x x t − 2 t-2 t − 2 x x x t t t x x x b t , s b_{t,s} b t , s E x E_x E x K − 1 K-1 K − 1 x x x E x ∩ E x ′ E_x\cap E_{x'} E x ∩ E x ′ dist ( x , x ′ ) ≤ 2 K − 1 \operatorname{dist}(x,x')\leq2K-1 dist ( x , x ′ ) ≤ 2 K − 1 V V V x ≠ x ′ x\ne x' x = x ′ dist ( x , x ′ ) ≤ 2 K − 1 \operatorname{dist}(x,x')\leq2K-1 dist ( x , x ′ ) ≤ 2 K − 1 ∑ j = 1 2 K − 1 D j ≤ ( D + 1 ) 2 K − 1 − 1 \sum_{j=1}^{2K-1}D^j\leq(D+1)^{2K-1}-1 ∑ j = 1 2 K − 1 D j ≤ ( D + 1 ) 2 K − 1 − 1 ( D + 1 ) 2 K − 1 (D+1)^{2K-1} ( D + 1 ) 2 K − 1 m x m_x m x E x E_x E x
If the Onsager coefficients were computed as empirical averages over all vertices, the recursion would generally introduce global dependence, and the preceding locality argument would no longer apply. The deterministic coefficients avoid this.
Lemma 5.28. Let G d G_d G d T L d \mathbb{T}_L^d T L d L ≥ 3 L \ge3 L ≥ 3 Q d Q_d Q d 0 < α ≤ 1 0 < \alpha\le1 0 < α ≤ 1 Π α \Pi_\alpha Π α S S S α \alpha α k α = rank Π α k_\alpha= \operatorname{rank} \Pi_\alpha k α = rank Π α k α ≤ 2 n e − d α 2 / 2 k_\alpha\le2n e^{-d\alpha^2/2} k α ≤ 2 n e − d α 2 /2
Proof. For the torus, the characters χ ω ( x ) = e 2 π i ⟨ ω , x ⟩ / L \chi_\omega(x) = e^{2\pi i\langle\omega,x\rangle/L} χ ω ( x ) = e 2 π i ⟨ ω , x ⟩ / L ω ∈ ( Z / L Z ) d \omega\in(\mathbb{Z}/L\mathbb{Z})^d ω ∈ ( Z / L Z ) d C V \mathbb{C}^V C V x x x x ± e j x \pm e_j x ± e j 1 ≤ j ≤ d 1 \le j \le d 1 ≤ j ≤ d S χ ω = λ ( ω ) χ ω S\chi_\omega= \lambda(\omega)\chi_\omega S χ ω = λ ( ω ) χ ω λ ( ω ) = d − 1 ∑ j = 1 d cos ( 2 π ω j / L ) \lambda(\omega) = d^{-1}\sum_{j=1}^d \cos(2\pi\omega_j/L) λ ( ω ) = d − 1 ∑ j = 1 d cos ( 2 π ω j / L ) S S S ω \omega ω λ ( ω ) \lambda(\omega) λ ( ω ) k α = ∣ { ω : ∣ λ ( ω ) ∣ ≥ α } ∣ k_\alpha= |\{\omega: |\lambda(\omega)| \ge\alpha\}| k α = ∣ { ω : ∣ λ ( ω ) ∣ ≥ α } ∣ ω \omega ω ( Z / L Z ) d (\mathbb{Z}/L\mathbb{Z})^d ( Z / L Z ) d cos ( 2 π ω j / L ) \cos(2\pi\omega_j/L) cos ( 2 π ω j / L ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ∑ k = 0 L − 1 e 2 π i k / L = 0 \sum_{k=0}^{L-1}e^{2\pi i k/L}=0 ∑ k = 0 L − 1 e 2 π ik / L = 0 [84 ] gives k α / n = P ( ∣ ∑ j cos ( 2 π ω j / L ) ∣ ≥ d α ) ≤ 2 e − d α 2 / 2 k_\alpha/n = \mathbb{P}(|\sum_j \cos(2\pi\omega_j/L)| \ge d\alpha) \le2e^{-d\alpha^2/2} k α / n = P ( ∣ ∑ j cos ( 2 π ω j / L ) ∣ ≥ d α ) ≤ 2 e − d α 2 /2 χ ω ( x ) = ( − 1 ) ⟨ ω , x ⟩ \chi_\omega(x) = (-1)^{\langle\omega,x\rangle} χ ω ( x ) = ( − 1 ) ⟨ ω , x ⟩ ω ∈ { 0 , 1 } d \omega\in\{0,1\}^d ω ∈ { 0 , 1 } d S S S d − 1 ∑ j ( − 1 ) ω j d^{-1}\sum_j(-1)^{\omega_j} d − 1 ∑ j ( − 1 ) ω j □ \square □
The next lemma is a Hoeffding inequality for sums with a partition into independent classes; the argument is that of Janson [90 ] .
Lemma 5.29. Let Λ > 0 \Lambda> 0 Λ > 0 ( ξ x ) x ∈ V (\xi_x)_{x\in V} ( ξ x ) x ∈ V [ − Λ , Λ ] [-\Lambda,\Lambda] [ − Λ , Λ ] V V V χ \chi χ ξ x \xi_x ξ x b > 0 b > 0 b > 0
P ( ∣ ∑ x ∈ V ( ξ x − E ξ x ) ∣ ≥ n b ) ≤ 2 exp ( − n b 2 2 χ Λ 2 ) . \mathbb{P}\left(\left|\sum_{x\in V}(\xi_x-\mathbb{E}\xi_x)\right|\ge nb\right)\le2\exp\left(-\frac{nb^2}{2\chi\Lambda^2}\right). P ( x ∈ V ∑ ( ξ x − E ξ x ) ≥ nb ) ≤ 2 exp ( − 2 χ Λ 2 n b 2 ) . Proof. Let V 1 , … , V χ V_1,\ldots,V_\chi V 1 , … , V χ λ > 0 \lambda> 0 λ > 0 [84 ] , E e s ( ξ x − E ξ x ) ≤ e s 2 Λ 2 / 2 \mathbb{E}e^{s(\xi_x-\mathbb{E}\xi_x)}\le e^{s^2\Lambda^2/2} E e s ( ξ x − E ξ x ) ≤ e s 2 Λ 2 /2 s ∈ R s\in\mathbb{R} s ∈ R χ \chi χ
E exp ( λ ∑ x ( ξ x − E ξ x ) ) ≤ ∏ i = 1 χ ( ∏ x ∈ V i E e χ λ ( ξ x − E ξ x ) ) 1 / χ ≤ exp ( χ λ 2 Λ 2 n 2 ) . \mathbb{E}\exp\left(\lambda\sum_x(\xi_x-\mathbb{E}\xi_x)\right)\le\prod_{i=1}^{\chi}\left(\prod_{x\in V_i}\mathbb{E}e^{\chi\lambda(\xi_x-\mathbb{E}\xi_x)}\right)^{1/\chi}\le\exp\left(\frac{\chi\lambda^2\Lambda^2 n}{2}\right). E exp ( λ x ∑ ( ξ x − E ξ x ) ) ≤ i = 1 ∏ χ ( x ∈ V i ∏ E e χ λ ( ξ x − E ξ x ) ) 1/ χ ≤ exp ( 2 χ λ 2 Λ 2 n ) . Using Markov’s inequality with λ = b / ( χ Λ 2 ) \lambda=b/(\chi\Lambda^2) λ = b / ( χ Λ 2 ) e − n b 2 / ( 2 χ Λ 2 ) e^{-nb^2/(2\chi\Lambda^2)} e − n b 2 / ( 2 χ Λ 2 ) □ \square □
Proposition 5.30. Let ( G d ) (G_d) ( G d ) T L d \mathbb{T}_L^d T L d L ≥ 3 L \ge3 L ≥ 3 Q d Q_d Q d μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] k ≥ 1 k \ge1 k ≥ 1 d d d m = m ( d ) m=m^{(d)} m = m ( d ) [ − 1 , 1 ] V [-1,1]^V [ − 1 , 1 ] V
(i) The set V V V ( D + 1 ) k (D+1)^k ( D + 1 ) k m x m_x m x
(ii) The law of m x m_x m x x ∈ V x\in V x ∈ V μ \mu μ d → ∞ d\to\infty d → ∞
Then the conclusion of Proposition 5.2 (c) holds with μ \mu μ μ ϵ \mu_\epsilon μ ϵ
Proof. Fix Υ \Upsilon Υ L Υ L_\Upsilon L Υ a a a c c c B B B Proposition 5.2 (c), and let Z = a 1 + c S [ 0 , B ] V \mathcal{Z}=a\mathbf{1}+cS[0,B]^V Z = a 1 + c S [ 0 , B ] V Υ ‾ ( s ) = ∫ Υ ( m ′ , s ) μ ( d m ′ ) \overline{\Upsilon}(s)=\int\Upsilon(m',s)\,\mu(\mathrm{d}m') Υ ( s ) = ∫ Υ ( m ′ , s ) μ ( d m ′ ) s ≥ 0 s\geq0 s ≥ 0 r ∈ [ 0 , ∞ ) V r\in[0,\infty)^V r ∈ [ 0 , ∞ ) V
A ( r ) = 1 n ∑ x ( Υ ( m x , r x ) − Υ ‾ ( r x ) ) . \mathcal{A}(r)=\frac{1}{n}\sum_x\left(\Upsilon(m_x,r_x)-\overline{\Upsilon}(r_x)\right). A ( r ) = n 1 x ∑ ( Υ ( m x , r x ) − Υ ( r x ) ) . Both Υ ( m ′ , ⋅ ) \Upsilon(m',\mathord\cdot) Υ ( m ′ , ⋅ ) Υ ‾ \overline{\Upsilon} Υ L Υ L_\Upsilon L Υ r , r ′ ∈ [ 0 , ∞ ) V r,r'\in[0,\infty)^V r , r ′ ∈ [ 0 , ∞ ) V
∣ A ( r ) − A ( r ′ ) ∣ ≤ 2 L Υ n ∑ x ∣ r x − r x ′ ∣ ≤ 2 L Υ ∥ r − r ′ ∥ n . (73) \left|\mathcal{A}(r)-\mathcal{A}(r')\right|
\leq\frac{2L_\Upsilon}{n}\sum_x|r_x-r'_x|
\leq2L_\Upsilon\lVert r-r'\rVert_n. \tag*{(73)} ∣ A ( r ) − A ( r ′ ) ∣ ≤ n 2 L Υ x ∑ ∣ r x − r x ′ ∣ ≤ 2 L Υ ∥ r − r ′ ∥ n . (73) Let M = a + c B M=a+cB M = a + c B Λ = 1 + max [ − 1 , 1 ] × [ 0 , M ] ∣ Υ ∣ \Lambda=1+\max_{[-1,1]\times[0,M]}|\Upsilon| Λ = 1 + max [ − 1 , 1 ] × [ 0 , M ] ∣Υ∣ ∣ A ( r ) ∣ ≤ 2 Λ |\mathcal{A}(r)|\leq2\Lambda ∣ A ( r ) ∣ ≤ 2Λ r ∈ [ 0 , M ] V r\in[0,M]^V r ∈ [ 0 , M ] V α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] ζ > 0 \zeta>0 ζ > 0 ξ > 0 \xi>0 ξ > 0 clip : R V → [ 0 , M ] V \operatorname{clip}:\mathbb{R}^V\to[0,M]^V clip : R V → [ 0 , M ] V [ 0 , M ] [0,M] [ 0 , M ]
Step 1: reduction to a ball in the range of Π α \Pi_\alpha Π α Let r ∈ Z r\in\mathcal{Z} r ∈ Z r = a 1 + c S u r=a\mathbf{1}+cSu r = a 1 + c S u u ∈ [ 0 , B ] V u\in[0,B]^V u ∈ [ 0 , B ] V S S S r ∈ [ 0 , M ] V r\in[0,M]^V r ∈ [ 0 , M ] V Π α \Pi_\alpha Π α k α k_\alpha k α Lemma 5.28 . Since S 1 = 1 S\mathbf{1}=\mathbf{1} S 1 = 1 1 \mathbf{1} 1 Π α \Pi_\alpha Π α r H = a 1 + c Π α S u r_H=a\mathbf{1}+c\Pi_\alpha Su r H = a 1 + c Π α S u r H r_H r H
K α = { w ∈ range Π α : ∥ w ∥ n ≤ M } , \mathcal{K}_\alpha=\left\{w\in\operatorname{range}\Pi_\alpha:\lVert w\rVert_n\leq M\right\}, K α = { w ∈ range Π α : ∥ w ∥ n ≤ M } , because ∥ Π α S u ∥ n ≤ ∥ u ∥ n ≤ B \lVert\Pi_\alpha Su\rVert_n\leq\lVert u\rVert_n\leq B ∥ Π α S u ∥ n ≤ ∥ u ∥ n ≤ B ∥ r − r H ∥ n = c ∥ ( I − Π α ) S u ∥ n ≤ c α B \lVert r-r_H\rVert_n=c\lVert(I-\Pi_\alpha)Su\rVert_n\leq c\alpha B ∥ r − r H ∥ n = c ∥( I − Π α ) S u ∥ n ≤ c α B S S S I − Π α I-\Pi_\alpha I − Π α α \alpha α clip \operatorname{clip} clip r r r ∥ r − clip ( r H ) ∥ n ≤ c α B \lVert r-\operatorname{clip}(r_H)\rVert_n\leq c\alpha B ∥ r − clip ( r H ) ∥ n ≤ c α B
sup r ∈ Z ∣ A ( r ) ∣ ≤ 2 L Υ c α B + sup w ∈ K α ∣ A ( clip ( w ) ) ∣ . (74) \sup_{r\in\mathcal{Z}}|\mathcal{A}(r)|
\leq2L_\Upsilon c\alpha B+\sup_{w\in\mathcal{K}_\alpha}|\mathcal{A}(\operatorname{clip}(w))|. \tag*{(74)} r ∈ Z sup ∣ A ( r ) ∣ ≤ 2 L Υ c α B + w ∈ K α sup ∣ A ( clip ( w )) ∣. (74) Step 2: a net. The set K α \mathcal{K}_\alpha K α M M M k α k_\alpha k α range Π α \operatorname{range}\Pi_\alpha range Π α ∥ ⋅ ∥ n \lVert\mathord\cdot\rVert_n ∥ ⋅ ∥ n ζ \zeta ζ N ⊂ K α \mathcal{N}\subset\mathcal{K}_\alpha N ⊂ K α ζ \zeta ζ ζ / 2 \zeta/2 ζ /2 M + ζ / 2 M+\zeta/2 M + ζ /2 ∣ N ∣ ≤ ( 1 + 2 M / ζ ) k α |\mathcal{N}|\leq(1+2M/\zeta)^{k_\alpha} ∣ N ∣ ≤ ( 1 + 2 M / ζ ) k α w ∈ K α w\in\mathcal{K}_\alpha w ∈ K α w ′ ∈ N w'\in\mathcal{N} w ′ ∈ N ∥ w − w ′ ∥ n ≤ ζ \lVert w-w'\rVert_n\leq\zeta ∥ w − w ′ ∥ n ≤ ζ ∥ clip ( w ) − clip ( w ′ ) ∥ n ≤ ζ \lVert\operatorname{clip}(w)-\operatorname{clip}(w')\rVert_n\leq\zeta ∥ clip ( w ) − clip ( w ′ ) ∥ n ≤ ζ
sup w ∈ K α ∣ A ( clip ( w ) ) ∣ ≤ 2 L Υ ζ + max w ∈ N ∣ A ( clip ( w ) ) ∣ . (75) \sup_{w\in\mathcal{K}_\alpha}|\mathcal{A}(\operatorname{clip}(w))|
\leq2L_\Upsilon\zeta+\max_{w\in\mathcal{N}}|\mathcal{A}(\operatorname{clip}(w))|. \tag*{(75)} w ∈ K α sup ∣ A ( clip ( w )) ∣ ≤ 2 L Υ ζ + w ∈ N max ∣ A ( clip ( w )) ∣. (75) Step 3: a fixed field. Let r ∈ [ 0 , M ] V r\in[0,M]^V r ∈ [ 0 , M ] V λ d \lambda_d λ d m x m_x m x x x x
∣ E A ( r ) ∣ ≤ ϵ d , ϵ d = sup s ∈ [ 0 , M ] ∣ ∫ Υ ( m ′ , s ) λ d ( d m ′ ) − ∫ Υ ( m ′ , s ) μ ( d m ′ ) ∣ . |\mathbb{E}\mathcal{A}(r)|\leq\epsilon_d,\qquad
\epsilon_d=\sup_{s\in[0,M]}\left|\int\Upsilon(m',s)\,\lambda_d(\mathrm{d}m')-\int\Upsilon(m',s)\,\mu(\mathrm{d}m')\right|. ∣ E A ( r ) ∣ ≤ ϵ d , ϵ d = s ∈ [ 0 , M ] sup ∫ Υ ( m ′ , s ) λ d ( d m ′ ) − ∫ Υ ( m ′ , s ) μ ( d m ′ ) . For each s s s Υ ( ⋅ , s ) \Upsilon(\cdot,s) Υ ( ⋅ , s ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] L Υ L_{\Upsilon} L Υ s s s [ 0 , M ] [0,M] [ 0 , M ] ε d → 0 \varepsilon_d \to0 ε d → 0 Υ ( m x , r x ) ∈ [ − Λ , Λ ] \Upsilon(m_x,r_x) \in[-\Lambda,\Lambda] Υ ( m x , r x ) ∈ [ − Λ , Λ ] Lemma 5.29 with χ ≤ ( D + 1 ) k \chi\le(D+1)^k χ ≤ ( D + 1 ) k
P ( ∣ A ( r ) − E A ( r ) ∣ ≥ ξ ) ≤ 2 exp ( − n ξ 2 2 ( D + 1 ) k Λ 2 ) . \mathbb{P}\left(\left|\mathcal{A}(r)-\mathbb{E}\mathcal{A}(r)\right|\ge\xi\right)\le2\exp\left(-\frac{n\xi^2}{2(D+1)^k\Lambda^2}\right). P ( ∣ A ( r ) − E A ( r ) ∣ ≥ ξ ) ≤ 2 exp ( − 2 ( D + 1 ) k Λ 2 n ξ 2 ) . Step 4: union bound. By Step 3, the bound on ∣ N ∣ |\mathcal{N}| ∣ N ∣ Lemma 5.28 ,
P ( max w ∈ N ∣ A ( clip ( w ) ) ∣ ≥ ε d + ξ ) ≤ 2 exp ( n [ 2 e − d α 2 / 2 log ( 1 + 2 M ζ ) − ξ 2 2 ( D + 1 ) k Λ 2 ] ) . \mathbb{P}\left(\max_{w\in\mathcal{N}}|\mathcal{A}(\operatorname{clip}(w))|\ge\varepsilon_d+\xi\right)\le2\exp\left(n\left[2e^{-d\alpha^2/2}\log\left(1+\frac{2M}{\zeta}\right)-\frac{\xi^2}{2(D+1)^k\Lambda^2}\right]\right). P ( w ∈ N max ∣ A ( clip ( w )) ∣ ≥ ε d + ξ ) ≤ 2 exp ( n [ 2 e − d α 2 /2 log ( 1 + ζ 2 M ) − 2 ( D + 1 ) k Λ 2 ξ 2 ] ) . Since D ≤ 2 d D\le2d D ≤ 2 d e − d α 2 / 2 ( D + 1 ) k → 0 e^{-d\alpha^2/2}(D+1)^k\to0 e − d α 2 /2 ( D + 1 ) k → 0 − ξ 2 / ( 4 ( D + 1 ) k Λ 2 ) -\xi^2/(4(D+1)^k\Lambda^2) − ξ 2 / ( 4 ( D + 1 ) k Λ 2 ) d d d n ≥ 2 d n\ge2^d n ≥ 2 d n / ( D + 1 ) k → ∞ n/(D+1)^k\to\infty n / ( D + 1 ) k → ∞ ∣ A ( clip ( w ) ) ∣ ≤ 2 Λ |\mathcal{A}(\operatorname{clip}(w))|\le2\Lambda ∣ A ( clip ( w )) ∣ ≤ 2Λ
lim sup d → ∞ E max w ∈ N ∣ A ( clip ( w ) ) ∣ ≤ lim sup d → ∞ ( ε d + ξ + 2 Λ P ( max w ∈ N ∣ A ( clip ( w ) ) ∣ ≥ ε d + ξ ) ) = ξ . \limsup_{d\to\infty}\mathbb{E}\max_{w\in\mathcal{N}}|\mathcal{A}(\operatorname{clip}(w))|
\le\limsup_{d\to\infty}\left(\varepsilon_d+\xi+2\Lambda\mathbb{P}\left(\max_{w\in\mathcal{N}}|\mathcal{A}(\operatorname{clip}(w))|\ge\varepsilon_d+\xi\right)\right)=\xi. d → ∞ lim sup E w ∈ N max ∣ A ( clip ( w )) ∣ ≤ d → ∞ lim sup ( ε d + ξ + 2Λ P ( w ∈ N max ∣ A ( clip ( w )) ∣ ≥ ε d + ξ ) ) = ξ . Combining this with (74) and (75) , we find
lim sup d → ∞ E sup r ∈ Z ∣ A ( r ) ∣ ≤ 2 L Υ c α B + 2 L Υ ζ + ξ . \limsup_{d\to\infty}\mathbb{E}\sup_{r\in\mathbb{Z}}|\mathcal{A}(r)|\le2L_{\Upsilon}c\alpha B+2L_{\Upsilon}\zeta+\xi. d → ∞ lim sup E r ∈ Z sup ∣ A ( r ) ∣ ≤ 2 L Υ c α B + 2 L Υ ζ + ξ . The left side does not depend on α \alpha α ζ \zeta ζ ξ \xi ξ
Proof of Proposition 5.2
Proof of Proposition 5.2. Fix β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 ϵ > 0 \epsilon>0 ϵ > 0 M W M_W M W Lemma 5.20 . By Lemma 2.6(a), (b), and (c), we have E M q + 2 = q + \mathbb{E}M_{q_+}^2=q_+ E M q + 2 = q + M q + = tanh X q + M_{q_+}=\tanh X_{q_+} M q + = tanh X q +
P ( β , h ) = β E ( β , h ) + h E M q + + E ent ( M q + ) + β 2 4 ( 1 − q + ) 2 . (76) \mathcal{P}(\beta,h)=\beta\mathcal{E}(\beta,h)+h\mathbb{E}M_{q_+}+\mathbb{E}\operatorname{ent}(M_{q_+})+\frac{\beta^2}{4}(1-q_+)^2. \tag*{(76)} P ( β , h ) = β E ( β , h ) + h E M q + + E ent ( M q + ) + 4 β 2 ( 1 − q + ) 2 . (76) We choose the control of Proposition 5.26. If q − = q + q_-=q_+ q − = q + τ = 1 \tau=1 τ = 1 q − < q + q_-<q_+ q − < q + K 2 ≥ 1 K_2\ge1 K 2 ≥ 1 Θ \Theta Θ μ Θ \mu_{\Theta} μ Θ
e 1 ( τ ) = ( β + β M W + h + β 2 ) C τ + 2 β τ + ∣ E ent ( Θ ) − E ent ( M q + ) ∣ , e_1(\tau)=(\beta+\beta M_W+h+\beta^2)C\sqrt{\tau}+2\beta\sqrt{\tau}+\left|\mathbb{E}\operatorname{ent}(\Theta)-\mathbb{E}\operatorname{ent}(M_{q_+})\right|, e 1 ( τ ) = ( β + β M W + h + β 2 ) C τ + 2 β τ + ∣ E ent ( Θ ) − E ent ( M q + ) ∣ , where Θ \Theta Θ M q + M_{q_+} M q + (72) , and we set e 1 = 0 e_1=0 e 1 = 0 ent \operatorname{ent} ent [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ∥ Θ − M q + ∥ 2 ≤ C τ \|\Theta-M_{q_+}\|_2\le C\sqrt{\tau} ∥Θ − M q + ∥ 2 ≤ C τ e 1 ( τ ) → 0 e_1(\tau) \to0 e 1 ( τ ) → 0 K 2 → ∞ K_2 \to\infty K 2 → ∞ K 2 K_2 K 2 e 1 ( τ ) ≤ ϵ / 2 e_1(\tau) \leq\epsilon/2 e 1 ( τ ) ≤ ϵ /2 C τ ≤ ϵ C\sqrt{\tau} \leq\epsilon C τ ≤ ϵ μ ϵ = μ Θ \mu_\epsilon= \mu_\Theta μ ϵ = μ Θ η ∈ ( 0 , q − ) \eta\in(0,q_-) η ∈ ( 0 , q − ) ℓ ≥ 2 \ell\geq2 ℓ ≥ 2
β ( η + 4 ℓ + 4 M W ℓ ) ≤ ϵ 2 . \beta\left(\eta+\frac{4}{\ell}+\frac{4M_W}{\ell}\right)\leq\frac{\epsilon}{2}. β ( η + ℓ 4 + ℓ 4 M W ) ≤ 2 ϵ . This fixes p p p K 1 K_1 K 1 P j P_j P j K = K 1 + K 2 K=K_1+K_2 K = K 1 + K 2 T = K + 1 T=K+1 T = K + 1 m m m G d G_d G d T T T
(b) By Proposition 5.26, W 1 ( μ ϵ , μ + ) = 0 W_1(\mu_\epsilon,\mu_+)=0 W 1 ( μ ϵ , μ + ) = 0 W 1 ( μ ϵ , μ + ) ≤ C τ ≤ ϵ W_1(\mu_\epsilon,\mu_+)\leq C\sqrt{\tau}\leq\epsilon W 1 ( μ ϵ , μ + ) ≤ C τ ≤ ϵ
(a) By Proposition 5.25(a) and (b), the lim inf in (a) is at least
β V − β ( 2 τ 1 { K 2 ≥ 1 } + η + 4 ℓ + M W ϑ + 4 M W ℓ ) + h E Θ + E ent ( Θ ) + β 2 4 ( 1 − E Θ 2 ) 2 . \beta V-\beta\left(2\sqrt{\tau}\mathbb{1}_{\{K_2\geq1\}}+\eta+\frac{4}{\ell}+M_W\vartheta+\frac{4M_W}{\ell}\right)+h\mathbb{E}\Theta+\mathbb{E}\operatorname{ent}(\Theta)+\frac{\beta^2}{4}(1-\mathbb{E}\Theta^2)^2. β V − β ( 2 τ 1 { K 2 ≥ 1 } + η + ℓ 4 + M W ϑ + ℓ 4 M W ) + h E Θ + E ent ( Θ ) + 4 β 2 ( 1 − E Θ 2 ) 2 . In case (a) of Proposition 5.26, we have V = E ( β , h ) V=\mathcal{E}(\beta,h) V = E ( β , h ) ϑ = 0 \vartheta=0 ϑ = 0 Θ = M q + \Theta=M_{q_+} Θ = M q + E M q + 2 = q + \mathbb{E}M_{q_+}^2=q_+ E M q + 2 = q + P ( β , h ) − ϵ / 2 \mathcal{P}(\beta,h)-\epsilon/2 P ( β , h ) − ϵ /2 V ≥ E ( β , h ) − C τ V\geq\mathcal{E}(\beta,h)-C\sqrt{\tau} V ≥ E ( β , h ) − C τ ϑ ≤ C τ \vartheta\leq C\sqrt{\tau} ϑ ≤ C τ ∣ h E Θ − h E M q + ∣ ≤ h C τ \lvert h\mathbb{E}\Theta-h\mathbb{E}M_{q_+}\rvert\leq hC\sqrt{\tau} ∣ h E Θ − h E M q + ∣ ≤ h C τ ∣ s 2 − s ′ 2 ∣ ≤ 2 ∣ s − s ′ ∣ \lvert s^2-s'^2\rvert\leq2\lvert s-s'\rvert ∣ s 2 − s ′2 ∣ ≤ 2 ∣ s − s ′ ∣ s , s ′ ∈ [ − 1 , 1 ] s,s'\in[-1,1] s , s ′ ∈ [ − 1 , 1 ]
∣ ( 1 − E Θ 2 ) 2 − ( 1 − E M q + 2 ) 2 ∣ ≤ 2 ∣ E Θ 2 − M q + 2 ∣ ≤ 4 ∥ Θ − M q + ∥ 2 ≤ 4 C τ . \left|(1-\mathbb{E}\Theta^2)^2-(1-\mathbb{E}M_{q_+}^2)^2\right|\leq2\left|\mathbb{E}\Theta^2-M_{q_+}^2\right|\leq4\|\Theta-M_{q_+}\|_2\leq4C\sqrt{\tau}. ( 1 − E Θ 2 ) 2 − ( 1 − E M q + 2 ) 2 ≤ 2 E Θ 2 − M q + 2 ≤ 4∥Θ − M q + ∥ 2 ≤ 4 C τ . By (5.15) and E M q + 2 = q + \mathbb{E}M_{q_+}^2=q_+ E M q + 2 = q + P ( β , h ) − e 1 ( τ ) − ϵ / 2 ≥ P ( β , h ) − ϵ \mathcal{P}(\beta,h)-e_1(\tau)-\epsilon/2\geq\mathcal{P}(\beta,h)-\epsilon P ( β , h ) − e 1 ( τ ) − ϵ /2 ≥ P ( β , h ) − ϵ
(c) By Lemma 5.27 , applied with θ \theta θ V V V ( D + 1 ) 2 K − 1 (D+1)^{2K-1} ( D + 1 ) 2 K − 1 m x m_x m x m x m_x m x x x x μ Θ = μ ϵ \mu_\Theta=\mu_\epsilon μ Θ = μ ϵ Proposition 5.30 with k = 2 K − 1 k=2K-1 k = 2 K − 1
Gaussian conditioning and the Onsager term
The goal of this section is to prove Theorem 6.2, which states that on every regular graph of large degree, the pressure is at least the expected value of
Θ G ( m ) = β H G ( m ) + h ∑ x ∈ V m x + ∑ x ∈ V ent ( m x ) + β 2 4 ⟨ κ , S κ ⟩ (77) \Theta_G(m)=\beta H_G(m)+h\sum_{x\in V}m_x+\sum_{x\in V}\operatorname{ent}(m_x)+\frac{\beta^2}{4}\langle\kappa,S\kappa\rangle \tag*{(77)} Θ G ( m ) = β H G ( m ) + h x ∈ V ∑ m x + x ∈ V ∑ ent ( m x ) + 4 β 2 ⟨ κ , S κ ⟩ (77) per vertex, up to an error that tends to zero with the degree, provided that two conditions hold. The functional Θ G \Theta_G Θ G G G G [135 ] . First, the center m m m m m m o ( n ) o(n) o ( n ) κ x = 1 − m x 2 \kappa_x=1-m_x^2 κ x = 1 − m x 2 Θ G \Theta_G Θ G β \beta β h h h
We first explain the structure of the bound. Let
Z G = ∑ σ ∈ { − 1 , 1 } V exp ( β H G ( σ ) + h ∑ x ∈ V σ x ) . Z_G=\sum_{\sigma\in\{-1,1\}^{V}}\exp\left(\beta H_G(\sigma)+h\sum_{x\in V}\sigma_x\right). Z G = σ ∈ { − 1 , 1 } V ∑ exp ( β H G ( σ ) + h x ∈ V ∑ σ x ) . so that p G ( β , h ) = n − 1 E log Z G p_G(\beta,h)=n^{-1}\mathbb{E}\log Z_G p G ( β , h ) = n − 1 E log Z G m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V log Z G ≥ Θ G ( m ) − ( β 2 / 4 ) ⟨ κ , S κ ⟩ \log Z_G\geq\Theta_G(m)-(\beta^2/4)\langle\kappa,S\kappa\rangle log Z G ≥ Θ G ( m ) − ( β 2 /4 ) ⟨ κ , S κ ⟩ π m \pi_m π m H G H_G H G σ = m + v \sigma=m+v σ = m + v z z z β ⟨ z , Π a ( v ) ⟩ \beta\langle z,\Pi a(v)\rangle β ⟨ z , Π a ( v )⟩ Π \Pi Π a ( v ) a(v) a ( v ) z z z exp ( β 2 ∥ Π a ( v ) ∥ 2 / 2 ) \exp(\beta^2\|\Pi a(v)\|^2/2) exp ( β 2 ∥Π a ( v ) ∥ 2 /2 ) v v v exp ( ( β 2 / 4 ) ⟨ κ , S κ ⟩ ) \exp((\beta^2/4)\langle\kappa,S\kappa\rangle) exp (( β 2 /4 ) ⟨ κ , S κ ⟩) KL ( P 0 ∥ Q ) \operatorname{KL}(P_0\mathbin\Vert Q) KL ( P 0 ∥ Q ) Q Q Q [132 ] , we bound it by 4 β ( n D ( m ) ) 1 / 2 4\beta(nD(m))^{1/2} 4 β ( n D ( m ) ) 1/2 m m m v v v π m \pi_m π m
Conditioning on the history of an iterative algorithm goes back to Bolthausen’s construction of solutions of the TAP equations [25 ] , and Bayati and Montanari used it to prove the state evolution of message passing [21 ] . Bolthausen [26 ] used this conditioning to compute the free energy of the SK model at high temperature. He showed that the quenched free energy equals the annealed free energy conditional on the iterates. Brennecke and Yau [29 ] extended his argument, which required β \beta β ( β , h ) (\beta,h) ( β , h ) h > 0 h>0 h > 0 β 2 E sech 2 ( β q Z + h ) ≤ 1 \beta^2\mathbb{E}\,\operatorname{sech}^2(\beta\sqrt{q}Z+h)\leq1 β 2 E sech 2 ( β q Z + h ) ≤ 1 q = E tanh 2 ( β q Z + h ) q=\mathbb{E}\,\tanh^2(\beta\sqrt{q}Z+h) q = E tanh 2 ( β q Z + h ) Z Z Z [64 ] . These works use a conditional second-moment method. Here the gap between the conditional quenched and annealed free energies is the relative entropy above, and we bound it through the planted model. For the SK model in zero field, the normalized partition function is likewise the likelihood ratio of a planted model with respect to the null model, and El Alaoui, Montanari, and Sellke used this together with contiguity at high temperature [61 Section 4.1] . Planting that preserves the typical properties of the null ensemble is called quiet [96 ] .
Restricting the free energy to a thin band around the output of an algorithm was used by Fan, Li, and Sen [66 ] for spin glasses with orthogonally invariant couplings. Bands around a magnetization also appear in Subag’s TAP representation of the free energy of spherical models [131 ] , and they define the generalized TAP free energy of Chen, Panchenko, and Subag [41 ] . Huang and Sellke [85 ] proved the lower bound in the Parisi formula for spherical models by constructing an ultrametric tree of states, partly with an optimization algorithm, and bounding the free energy of bands around its leaves. For mixed p p p [39 Theorem 1] . For the SK model, let q + q_+ q + § 3.3 . Then their theorem states that P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) N → ∞ N\to\infty N → ∞ ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 q + − ε q_+-\varepsilon q + − ε q + q_+ q + P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) G G G m m m
Section 6.1 states Definition 6.1 and Theorem 6.2. Section 6.2 describes the law of the disorder given a query history. Section 6.3 expands the partition function around the center, and Section 6.4 restricts the expansion to a set of configurations of large product probability on which the error terms are small. Section 6.5 bounds the remaining Gaussian likelihood in terms of D ( m ) \mathcal{D}(m) D ( m )
The planted model and the main result
We use the query histories of Definition 5.1. We recall that a query history of depth T T T G G G U U U ( g e ) e ∈ E (g_e)_{e\in E} ( g e ) e ∈ E x 1 , … , x T ∈ R V x^1,\ldots,x^T\in\mathbb{R}^V x 1 , … , x T ∈ R V x t x^t x t σ \sigma σ U U U W x 1 , … , W x t − 1 Wx^1,\ldots,Wx^{t-1} W x 1 , … , W x t − 1 σ \sigma σ H \mathcal{H} H U U U W x 1 , … , W x T Wx^1,\ldots,Wx^T W x 1 , … , W x T H \mathcal{H} H m ∈ ( − 1 , 1 ) V m\in(-1,1)^V m ∈ ( − 1 , 1 ) V m = x t m=x^t m = x t t ≤ T t\le T t ≤ T W m Wm W m
The following definition is also used in the proof of Theorem 7.5.
Definition 6.1. Let S S S n × n n\times n n × n E = { { x , y } : S x y > 0 } E=\{\{x,y\}:S_{xy}>0\} E = {{ x , y } : S x y > 0 } β ≥ 0 \beta\ge0 β ≥ 0 m ∈ [ − 1 , 1 ] n m\in[-1,1]^n m ∈ [ − 1 , 1 ] n σ ∗ \sigma^* σ ∗ π m \pi_m π m v ∗ = σ ∗ − m v^*=\sigma^*-m v ∗ = σ ∗ − m Z Z Z R E \mathbb{R}^E R E σ ∗ \sigma^* σ ∗ Q m Q_m Q m Y = β a ( v ∗ ) + Z Y=\beta a(v^*)+Z Y = β a ( v ∗ ) + Z a ( v ) x y = S x y v x v y a(v)_{xy}=\sqrt{S_{xy}}v_xv_y a ( v ) x y = S x y v x v y P 0 P_0 P 0 R E \mathbb{R}^E R E D ( m ) = KL ( Q m ∥ P 0 ) D(m)=\operatorname{KL}(Q_m\mathbin\Vert P_0) D ( m ) = KL ( Q m ∥ P 0 )
For G ∈ G D G\in\mathcal{G}_D G ∈ G D S = A G / D S=A_G/D S = A G / D { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } V V V E = E E=E E = E R E \mathbb{R}^E R E a ( v ) x y = D − 1 / 2 v x v y a(v)_{xy}=D^{-1/2}v_xv_y a ( v ) x y = D − 1/2 v x v y x y ∈ E xy\in E x y ∈ E E π m \mathbb{E}_{\pi_m} E π m σ \sigma σ π m \pi_m π m v = σ − m v=\sigma-m v = σ − m R E \mathbb{R}^E R E b b b exp ( ⟨ y , b ⟩ − ∥ b ∥ 2 / 2 ) \exp(\langle y,b\rangle-\|b\|^2/2) exp (⟨ y , b ⟩ − ∥ b ∥ 2 /2 ) P 0 P_0 P 0 Q m Q_m Q m
L m ( y ) = E π m exp ( β ⟨ y , a ( v ) ⟩ − β 2 2 ∥ a ( v ) ∥ 2 ) ( y ∈ R E ) (78) L_m(y)=\mathbb{E}_{\pi_m}\exp\left(\beta\langle y,a(v)\rangle-\frac{\beta^2}{2}\|a(v)\|^2\right)\qquad(y\in\mathbb{R}^E) \tag*{(78)} L m ( y ) = E π m exp ( β ⟨ y , a ( v )⟩ − 2 β 2 ∥ a ( v ) ∥ 2 ) ( y ∈ R E ) (78) with respect to P 0 P_0 P 0 m ↦ D ( m ) m\mapsto D(m) m ↦ D ( m ) [ − 1 , 1 ] V [-1,1]^V [ − 1 , 1 ] V D ( m ) D(m) D ( m ) m m m [ − 1 , 1 ] V [-1,1]^V [ − 1 , 1 ] V m m m D ( m ) D(m) D ( m ) m m m D D D g g g
Theorem 6.2. Let β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R δ ≥ 0 \delta\ge0 δ ≥ 0 T ≥ 1 T\ge1 T ≥ 1 k ≥ 1 k\ge1 k ≥ 1 D k ≥ 1 D_k\ge1 D k ≥ 1 G k ∈ G D k G_k\in\mathcal{G}_{D_k} G k ∈ G D k n k n_k n k S k = A G k / D k S_k=A_{G_k}/D_k S k = A G k / D k T T T G k G_k G k m = m ( k ) m=m^{(k)} m = m ( k ) D D D S = S k S=S_k S = S k D k → ∞ D_k\to\infty D k → ∞
lim sup k → ∞ 1 n k E D ( m ( k ) ) ≤ δ . (79) \limsup_{k\to\infty}\frac{1}{n_k}\mathbb{E}\,D\left(m^{(k)}\right)\le\delta. \tag*{(79)} k → ∞ lim sup n k 1 E D ( m ( k ) ) ≤ δ . (79) Then
lim inf k → ∞ p G k ( β , h ) ≥ lim inf k → ∞ 1 n k E [ β H G k ( m ) + h ∑ x m x + ∑ x ent ( m x ) + β 2 4 ⟨ κ , S k κ ⟩ ] − 4 β δ . (80) \liminf_{k\to\infty}p_{G_k}(\beta,h)\ge\liminf_{k\to\infty}\frac{1}{n_k}\mathbb{E}\left[\beta H_{G_k}(m)+h\sum_xm_x+\sum_x\operatorname{ent}(m_x)+\frac{\beta^2}{4}\langle\kappa,S_k\kappa\rangle\right]-4\beta\sqrt{\delta}. \tag*{(80)} k → ∞ lim inf p G k ( β , h ) ≥ k → ∞ lim inf n k 1 E [ β H G k ( m ) + h x ∑ m x + x ∑ ent ( m x ) + 4 β 2 ⟨ κ , S k κ ⟩ ] − 4 β δ . (80) where m = m ( k ) m = m^{(k)} m = m ( k ) κ x = 1 − m x 2 \kappa_x = 1 - m_x^2 κ x = 1 − m x 2
Remark 6.3. Theorem 6.2 follows from the nonasymptotic bound of Proposition 6.12, which holds for every G ∈ G D G \in\mathcal{G}_D G ∈ G D T T T G G G ( G k ) (G_k) ( G k ) D k → ∞ D_k \to\infty D k → ∞ W W W n k n_k n k D k D_k D k
Gaussian conditioning
In this subsection, we fix G ∈ G D G \in\mathcal{G}_D G ∈ G D T T T G G G ( Ω , A , P ) (\Omega,\mathcal{A},\mathbb{P}) ( Ω , A , P ) g = ( g e ) e ∈ E g = (g_e)_{e \in E} g = ( g e ) e ∈ E B ( R E ) \mathcal{B}(\mathbb{R}^E) B ( R E ) σ \sigma σ R E \mathbb{R}^E R E 0 ≤ t ≤ T 0 \le t \le T 0 ≤ t ≤ T F t \mathcal{F}_t F t σ \sigma σ U U U W x 1 , … , W x t Wx^1,\ldots,Wx^t W x 1 , … , W x t F T = H \mathcal{F}_T = \mathcal{H} F T = H
For b ∈ R V b \in\mathbb{R}^V b ∈ R V Λ b : R E → R V \Lambda_b:\mathbb{R}^E \to\mathbb{R}^V Λ b : R E → R V
( Λ b w ) x = D − 1 / 2 ∑ y : x y ∈ E w x y b y ( x ∈ V ) . (\Lambda_b w)_x = D^{-1/2} \sum_{y:xy\in E} w_{xy}b_y \qquad(x \in V). ( Λ b w ) x = D − 1/2 y : x y ∈ E ∑ w x y b y ( x ∈ V ) . Then W b = Λ b g Wb = \Lambda_b g W b = Λ b g n n n g g g 0 ≤ t ≤ T 0 \le t \le T 0 ≤ t ≤ T Λ ( t ) : R E → ( R V ) t \Lambda^{(t)}:\mathbb{R}^E \to(\mathbb{R}^V)^t Λ ( t ) : R E → ( R V ) t w ↦ ( Λ x 1 w , … , Λ x t w ) w \mapsto(\Lambda_{x^1}w,\ldots,\Lambda_{x^t}w) w ↦ ( Λ x 1 w , … , Λ x t w ) Π t \Pi_t Π t R E \mathbb{R}^E R E ker Λ ( t ) \ker\Lambda^{(t)} ker Λ ( t ) Π 0 = I \Pi_0 = I Π 0 = I
Π = Π T , Π ⊥ = I − Π , g ˉ = Π ⊥ g . \Pi= \Pi_T,\qquad\Pi^\perp= I-\Pi,\qquad\bar{g} = \Pi^\perp g. Π = Π T , Π ⊥ = I − Π , g ˉ = Π ⊥ g . and we let W ˉ \bar{W} W ˉ V × V V \times V V × V W ˉ x y = D − 1 / 2 g ˉ x y \bar{W}_{xy} = D^{-1/2}\bar{g}_{xy} W ˉ x y = D − 1/2 g ˉ x y x y ∈ E xy \in E x y ∈ E W ˉ x y = 0 \bar{W}_{xy}=0 W ˉ x y = 0 K K K ∣ K ∣ = ( K 2 ) 1 / 2 |K| = (K^2)^{1/2} ∣ K ∣ = ( K 2 ) 1/2 ∥ K ∥ t r = tr ∣ K ∣ \|K\|_{\mathrm{tr}} = \operatorname{tr}|K| ∥ K ∥ tr = tr ∣ K ∣
The next lemma is a version of the conditioning lemmas of Bolthausen [25 ] and Bayati and Montanari [21 Lemmas 11 and 12] , for linear functionals of the edge disorder that are chosen adaptively and may depend on the auxiliary randomness U U U
Lemma 6.4. Let G ∈ G D G \in\mathcal{G}_D G ∈ G D T T T G G G
(a) The projection Π \Pi Π g ˉ \bar{g} g ˉ H \mathcal{H} H rank Π ⊥ ≤ n T \operatorname{rank}\Pi^\perp\le nT rank Π ⊥ ≤ n T m m m W m Wm W m H \mathcal{H} H
(b) For every H ⊗ B ( R E ) \mathcal{H} \otimes\mathcal{B}(\mathbb{R}^E) H ⊗ B ( R E ) F : Ω × R E → [ 0 , ∞ ] F:\Omega\times\mathbb{R}^E \to[0,\infty] F : Ω × R E → [ 0 , ∞ ]
E [ F ( ⋅ , g ) ∣ H ] = ∫ F ( ⋅ , g ˉ + Π w ) P 0 ( d w ) almost surely . (81) \mathbb{E}\left[F(\cdot,g)\mid\mathcal{H}\right] = \int F(\cdot,\bar{g}+\Pi w)\,P_0(dw)\quad\text{almost surely}. \tag*{(81)} E [ F ( ⋅ , g ) ∣ H ] = ∫ F ( ⋅ , g ˉ + Π w ) P 0 ( d w ) almost surely . (81) (c) We have E ∥ g ˉ ∥ 2 = E rank Π ⊥ ≤ n T \mathbb{E}\|\bar{g}\|^2 = \mathbb{E}\operatorname{rank}\Pi^\perp\le nT E ∥ g ˉ ∥ 2 = E rank Π ⊥ ≤ n T E ∥ W ˉ ∥ t r ≤ n ( 2 T / D ) 1 / 2 \mathbb{E}\|\bar{W}\|_{\mathrm{tr}} \le n(2T/D)^{1/2} E ∥ W ˉ ∥ tr ≤ n ( 2 T / D ) 1/2
Part (b) states that, conditional on H \mathcal{H} H g ˉ + Π z \bar{g}+\Pi z g ˉ + Π z z z z R E \mathbb{R}^{E} R E H \mathcal{H} H F F F H \mathcal{H} H Π \Pi Π E [ g ∣ H ] = g ˉ \mathbb{E}[g\mid\mathcal{H}]=\bar{g} E [ g ∣ H ] = g ˉ W ˉ = E [ W ∣ H ] \bar{W}=\mathbb{E}[W\mid\mathcal{H}] W ˉ = E [ W ∣ H ]
Proof. For a linear map M M M M + M^{+} M + M + M M^{+}M M + M ( ker M ) ⊥ (\ker M)^{\perp} ( ker M ) ⊥ M ↦ M + M\mapsto M^{+} M ↦ M + ( M T M + ϵ I ) − 1 M T (M^{\mathsf{T}}M+\epsilon I)^{-1}M^{\mathsf{T}} ( M T M + ϵ I ) − 1 M T ϵ ↓ 0 \epsilon\downarrow0 ϵ ↓ 0
Π t ⊥ = I − Π t = ( Λ ( t ) ) + Λ ( t ) . \Pi_{t}^{\perp}=I-\Pi_{t}=(\Lambda^{(t)})^{+}\Lambda^{(t)}. Π t ⊥ = I − Π t = ( Λ ( t ) ) + Λ ( t ) . (a) Fix 1 ≤ t ≤ T 1\le t\le T 1 ≤ t ≤ T x s x^{s} x s F s − 1 \mathcal{F}_{s-1} F s − 1 b ↦ Λ b b\mapsto\Lambda_{b} b ↦ Λ b Λ ( t ) \Lambda^{(t)} Λ ( t ) F t − 1 \mathcal{F}_{t-1} F t − 1 Π t \Pi_{t} Π t Λ ( t ) g = ( W x 1 , … , W x t ) \Lambda^{(t)}g=(Wx^{1},\ldots,Wx^{t}) Λ ( t ) g = ( W x 1 , … , W x t ) F t \mathcal{F}_{t} F t Π t ⊥ g = ( Λ ( t ) ) + Λ ( t ) g \Pi_{t}^{\perp}g=(\Lambda^{(t)})^{+}\Lambda^{(t)}g Π t ⊥ g = ( Λ ( t ) ) + Λ ( t ) g t = T t=T t = T Π \Pi Π g ˉ \bar{g} g ˉ H \mathcal{H} H Π ⊥ \Pi^{\perp} Π ⊥ Λ ( T ) \Lambda^{(T)} Λ ( T ) n T nT n T Λ ( T ) \Lambda^{(T)} Λ ( T ) n T nT n T m m m τ \tau τ t t t m = x t m=x^{t} m = x t { τ = t } \{\tau=t\} { τ = t } H \mathcal{H} H m m m x 1 , … , x T x^{1},\ldots,x^{T} x 1 , … , x T H \mathcal{H} H W m = ∑ t ≤ T 1 { τ = t } W x t Wm=\sum_{t\le T}1_{\{\tau=t\}}Wx^{t} W m = ∑ t ≤ T 1 { τ = t } W x t H \mathcal{H} H
(b) For 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T ( b t ) (b_{t}) ( b t ) F t \mathcal{F}_{t} F t Π t \Pi_{t} Π t H \mathcal{H} H Π \Pi Π ( b t ) (b_{t}) ( b t ) t t t ( b t ) (b_{t}) ( b t ) F t \mathcal{F}_{t} F t Π t ⊥ g \Pi_{t}^{\perp}g Π t ⊥ g Π t \Pi_{t} Π t F t \mathcal{F}_{t} F t F t ⊗ B ( R E ) \mathcal{F}_{t}\otimes\mathcal{B}(\mathbb{R}^{E}) F t ⊗ B ( R E ) F F F F F F ( b t ) (b_{t}) ( b t ) F ( ω , w ) = 1 A ( ω ) f ( w ) F(\omega,w)=1_{A}(\omega)f(w) F ( ω , w ) = 1 A ( ω ) f ( w ) A ∈ F t A\in\mathcal{F}_{t} A ∈ F t f : R E → [ 0 , ∞ ) f:\mathbb{R}^{E}\to[0,\infty) f : R E → [ 0 , ∞ ) 1 A 1_{A} 1 A F t \mathcal{F}_{t} F t
E [ f ( g ) ∣ F t ] = ∫ f ( Π t ⊥ g + Π t w ) P 0 ( d w ) . (82) \mathbb{E}\left[f(g)\mid\mathcal{F}_{t}\right]
=
\int f\left(\Pi_{t}^{\perp}g+\Pi_{t}w\right)P_{0}(dw). \tag*{(82)} E [ f ( g ) ∣ F t ] = ∫ f ( Π t ⊥ g + Π t w ) P 0 ( d w ) . (82) For t = 0 t=0 t = 0 Π 0 = I \Pi_{0}=I Π 0 = I g g g F 0 \mathcal{F}_{0} F 0 U U U
Let 1 ≤ t ≤ T 1\le t\le T 1 ≤ t ≤ T ( b t − 1 ) (b_{t-1}) ( b t − 1 ) Π ′ = Π t − 1 \Pi'=\Pi_{t-1} Π ′ = Π t − 1 Λ = Λ x t \Lambda=\Lambda_{x^{t}} Λ = Λ x t ker Λ ( t ) ⊂ ker Λ ( t − 1 ) \ker\Lambda^{(t)}\subset\ker\Lambda^{(t-1)} ker Λ ( t ) ⊂ ker Λ ( t − 1 ) Ξ = Π ′ − Π t \Xi=\Pi'-\Pi_{t} Ξ = Π ′ − Π t Π t \Pi_{t} Π t Π ′ \Pi' Π ′ Π ′ \Pi' Π ′ Π t \Pi_{t} Π t Ξ \Xi Ξ Λ \Lambda Λ F t − 1 \mathcal{F}_{t-1} F t − 1
Λ Π t = 0 , Λ Π ′ = Λ Ξ , Π t ⊥ Π ′ ⊥ = Π ′ ⊥ , Π t ⊥ Π ′ = Ξ . (83) \Lambda\Pi_{t}=0,\qquad\Lambda\Pi'=\Lambda\Xi,\qquad\Pi_{t}^{\perp}\Pi'^{\perp}=\Pi'^{\perp},\qquad\Pi_{t}^{\perp}\Pi'=\Xi. \tag*{(83)} Λ Π t = 0 , Λ Π ′ = ΛΞ , Π t ⊥ Π ′ ⊥ = Π ′ ⊥ , Π t ⊥ Π ′ = Ξ. (83) The events A ′ ∩ { W x t ∈ C } A'\cap\{Wx^{t}\in C\} A ′ ∩ { W x t ∈ C } A ′ ∈ F t − 1 A'\in\mathcal{F}_{t-1} A ′ ∈ F t − 1 C ⊂ R V C\subset\mathbb{R}^{V} C ⊂ R V π \pi π F t \mathcal{F}_{t} F t π \pi π λ \lambda λ
E [ 1 A ′ 1 C ( Λ g ) f ( g ) ] = E [ Ψ ( ⋅ , g ) ] . (84) \mathbb{E}\left[1_{A'}1_{C}(\Lambda g)f(g)\right]=\mathbb{E}\left[\Psi(\cdot,g)\right]. \tag*{(84)} E [ 1 A ′ 1 C ( Λ g ) f ( g ) ] = E [ Ψ ( ⋅ , g ) ] . (84) where
Ψ ( ω , w ) = 1 A ′ ( ω ) 1 C ( Λ w ) ∫ f ( Π t ⊥ w + Π t w ′ ) P 0 ( d w ′ ) . \Psi(\omega,w)=\mathbf{1}_{A'}(\omega)\mathbf{1}_{C}(\Lambda w)\int f(\Pi_t^\perp w+\Pi_t w')\,P_0(\mathrm{d}w'). Ψ ( ω , w ) = 1 A ′ ( ω ) 1 C ( Λ w ) ∫ f ( Π t ⊥ w + Π t w ′ ) P 0 ( d w ′ ) . and where we used W x t = Λ g Wx^t=\Lambda g W x t = Λ g Ψ \Psi Ψ F t − 1 ⊗ B ( R E ) \mathcal{F}_{t-1}\otimes\mathcal{B}(\mathbb{R}^{E}) F t − 1 ⊗ B ( R E ) ( b t − 1 ) (b_{t-1}) ( b t − 1 ) (83) , we have
E [ Ψ ( ⋅ , g ) ] = E ∫ Ψ ( ⋅ , Π ′ ⊥ g + Π ′ w ) P 0 ( d w ) = E [ 1 A ′ ∬ 1 C ( Λ Π ′ ⊥ g + Λ Ξ w ) f ( Π ′ ⊥ g + Ξ w + Π t w ′ ) P 0 ( d w ) P 0 ( d w ′ ) ] . \begin{aligned}
\mathbb{E}\left[\Psi(\cdot,g)\right]=\mathbb{E}\int\Psi(\cdot,\Pi'^\perp g+\Pi'w)\,P_0(\mathrm{d}w) \\
&=\mathbb{E}\left[\mathbf{1}_{A'}\iint\mathbf{1}_{C}(\Lambda\Pi'^\perp g+\Lambda\Xi w)f(\Pi'^\perp g+\Xi w+\Pi_t w')\,P_0(\mathrm{d}w)P_0(\mathrm{d}w')\right].
\end{aligned} E [ Ψ ( ⋅ , g ) ] = E ∫ Ψ ( ⋅ , Π ′ ⊥ g + Π ′ w ) P 0 ( d w ) = E [ 1 A ′ ∬ 1 C ( Λ Π ′ ⊥ g + ΛΞ w ) f ( Π ′ ⊥ g + Ξ w + Π t w ′ ) P 0 ( d w ) P 0 ( d w ′ ) ] . For fixed ω \omega ω Ξ w \Xi w Ξ w Π t w \Pi_t w Π t w P 0 P_0 P 0 Ξ Π t = 0 \Xi\Pi_t=0 Ξ Π t = 0 ( Ξ w , Ξ w + Π t w ′ ) (\Xi w,\Xi w+\Pi_t w') ( Ξ w , Ξ w + Π t w ′ ) P 0 ⊗ P 0 P_0\otimes P_0 P 0 ⊗ P 0 ( Ξ w , Π ′ w ) (\Xi w,\Pi'w) ( Ξ w , Π ′ w ) P 0 P_0 P 0 Λ Ξ w = Λ Π ′ w \Lambda\Xi w=\Lambda\Pi'w ΛΞ w = Λ Π ′ w
∫ 1 C ( Λ ( Π ′ ⊥ g + Π ′ w ) ) f ( Π ′ ⊥ g + Π ′ w ) P 0 ( d w ) . \int\mathbf{1}_{C}(\Lambda(\Pi'^\perp g+\Pi'w))f(\Pi'^\perp g+\Pi'w)\,P_0(\mathrm{d}w). ∫ 1 C ( Λ ( Π ′ ⊥ g + Π ′ w )) f ( Π ′ ⊥ g + Π ′ w ) P 0 ( d w ) . By ( b t − 1 ) (b_{t-1}) ( b t − 1 ) ( ω , w ) ↦ 1 A ′ ( ω ) 1 C ( Λ ( ω ) w ) f ( w ) (\omega,w)\mapsto\mathbf{1}_{A'}(\omega)\mathbf{1}_{C}(\Lambda(\omega)w)f(w) ( ω , w ) ↦ 1 A ′ ( ω ) 1 C ( Λ ( ω ) w ) f ( w ) 1 A ′ \mathbf{1}_{A'} 1 A ′ (84) . This proves ( b t ) (b_t) ( b t ) t = T t=T t = T
(c) By (b) with F ( ω , w ) = ∥ w ∥ 2 F(\omega,w)=\lVert w\rVert^2 F ( ω , w ) = ∥ w ∥ 2 g ˉ \bar g g ˉ Π \Pi Π
E [ ∥ g ∥ 2 ∣ H ] = ∫ ∥ g ˉ + Π w ∥ 2 P 0 ( d w ) = ∥ g ˉ ∥ 2 + tr Π = ∥ g ˉ ∥ 2 + ∣ E ∣ − rank Π ⊥ . \mathbb{E}\left[\lVert g\rVert^2\mid\mathcal{H}\right]=\int\lVert\bar g+\Pi w\rVert^2P_0(\mathrm{d}w)=\lVert\bar g\rVert^2+\operatorname{tr}\Pi=\lVert\bar g\rVert^2+|E|-\operatorname{rank}\Pi^\perp. E [ ∥ g ∥ 2 ∣ H ] = ∫ ∥ g ˉ + Π w ∥ 2 P 0 ( d w ) = ∥ g ˉ ∥ 2 + tr Π = ∥ g ˉ ∥ 2 + ∣ E ∣ − rank Π ⊥ . Taking expectations and using E ∥ g ∥ 2 = ∣ E ∣ \mathbb{E}\lVert g\rVert^2=|E| E ∥ g ∥ 2 = ∣ E ∣ E ∥ g ˉ ∥ 2 = E rank Π ⊥ ≤ n T \mathbb{E}\lVert\bar g\rVert^2=\mathbb{E}\operatorname{rank}\Pi^\perp\le nT E ∥ g ˉ ∥ 2 = E rank Π ⊥ ≤ n T W ˉ \bar W W ˉ ∑ x , y W ˉ x y 2 = 2 D − 1 ∥ g ˉ ∥ 2 \sum_{x,y}\bar W_{xy}^2=2D^{-1}\lVert\bar g\rVert^2 ∑ x , y W ˉ x y 2 = 2 D − 1 ∥ g ˉ ∥ 2 W ˉ \bar W W ˉ n n n ∥ W ˉ ∥ t r ≤ ( 2 n / D ) 1 / 2 ∥ g ˉ ∥ \lVert\bar W\rVert_{\mathrm{tr}}\le(2n/D)^{1/2}\lVert\bar g\rVert ∥ W ˉ ∥ tr ≤ ( 2 n / D ) 1/2 ∥ g ˉ ∥
E ∥ W ˉ ∥ t r ≤ ( 2 n D ) 1 / 2 ( E ∥ g ˉ ∥ 2 ) 1 / 2 ≤ n ( 2 T D ) 1 / 2 . \mathbb{E}\lVert\bar W\rVert_{\mathrm{tr}}\le\left(\frac{2n}{D}\right)^{1/2}\left(\mathbb{E}\lVert\bar g\rVert^2\right)^{1/2}\le n\left(\frac{2T}{D}\right)^{1/2}. E ∥ W ˉ ∥ tr ≤ ( D 2 n ) 1/2 ( E ∥ g ˉ ∥ 2 ) 1/2 ≤ n ( D 2 T ) 1/2 . This completes the proof.
Expansion around the center
The following identity is deterministic. It holds for every realization of the disorder and every orthogonal projection Π \Pi Π R E \mathbb{R}^{E} R E g ˉ = Π ⊥ g \bar g=\Pi^\perp g g ˉ = Π ⊥ g W ˉ \bar W W ˉ g ˉ \bar g g ˉ § 7.2 . For m ∈ ( − 1 , 1 ) V m\in(-1,1)^V m ∈ ( − 1 , 1 ) V r ∈ R V r\in\mathbb{R}^V r ∈ R V
r x = β ( W m ) x + h − atanh ( m x ) − β 2 m x ( S κ ) x ( x ∈ V ) . (85) r_x=\beta(Wm)_x+h-\operatorname{atanh}(m_x)-\beta^2m_x(S\kappa)_x\qquad(x\in V). \tag*{(85)} r x = β ( W m ) x + h − atanh ( m x ) − β 2 m x ( S κ ) x ( x ∈ V ) . (85) Lemma 6.5. Let G ∈ G D G\in\mathcal{G}_D G ∈ G D β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R m ∈ ( − 1 , 1 ) V m\in(-1,1)^V m ∈ ( − 1 , 1 ) V Π \Pi Π R E \mathbb{R}^{E} R E
Z G = e Θ G ( m ) E π m exp ( ⟨ r , v ⟩ + β 2 ⟨ v , W ˉ v ⟩ + β 2 ⟨ m ⊙ v , S ( m ⊙ v ) ⟩ − β 2 2 ∥ Π ⊥ a ( v ) ∥ 2 + β ⟨ g , Π a ( v ) ⟩ − β 2 2 ∥ Π a ( v ) ∥ 2 ) . (86) Z_G=e^{\Theta_G(m)}\mathbb{E}_{\pi_m}\exp\left(\langle r,v\rangle+\frac{\beta}{2}\langle v,\bar Wv\rangle+\beta^2\langle m\odot v,S(m\odot v)\rangle-\frac{\beta^2}{2}\lVert\Pi^\perp a(v)\rVert^2+\beta\langle g,\Pi a(v)\rangle-\frac{\beta^2}{2}\lVert\Pi a(v)\rVert^2\right). \tag*{(86)} Z G = e Θ G ( m ) E π m exp ( ⟨ r , v ⟩ + 2 β ⟨ v , W ˉ v ⟩ + β 2 ⟨ m ⊙ v , S ( m ⊙ v )⟩ − 2 β 2 ∥ Π ⊥ a ( v ) ∥ 2 + β ⟨ g , Π a ( v )⟩ − 2 β 2 ∥ Π a ( v ) ∥ 2 ) . (86) Proof. Let y x = atanh ( m x ) y_x = \operatorname{atanh}(m_x) y x = atanh ( m x ) π m ( σ ) = ∏ x e y x σ x / ( 2 cosh y x ) \pi_m(\sigma) = \prod_x e^{y_x\sigma_x}/(2\cosh y_x) π m ( σ ) = ∏ x e y x σ x / ( 2 cosh y x ) σ \sigma σ f f f { − 1 , 1 } V \{-1,1\}^{V} { − 1 , 1 } V
∑ σ f ( σ ) = E π m [ f ( σ ) ∏ x ∈ V 2 cosh ( y x ) e − y x σ x ] . \sum_{\sigma} f(\sigma) = \mathbb{E}_{\pi_m}\left[f(\sigma)\prod_{x\in V}2\cosh(y_x)e^{-y_x\sigma_x}\right]. σ ∑ f ( σ ) = E π m [ f ( σ ) x ∈ V ∏ 2 cosh ( y x ) e − y x σ x ] . By (6) , we have log 2 cosh y x − y x m x = ent ( m x ) \log2\cosh y_x-y_xm_x=\operatorname{ent}(m_x) log 2 cosh y x − y x m x = ent ( m x ) σ = m + v \sigma=m+v σ = m + v exp ( ∑ x ent ( m x ) − ⟨ y , v ⟩ ) \exp(\sum_x\operatorname{ent}(m_x)-\langle y,v\rangle) exp ( ∑ x ent ( m x ) − ⟨ y , v ⟩) W W W H G ( m + v ) = H G ( m ) + ⟨ W m , v ⟩ + H G ( v ) H_G(m+v)=H_G(m)+\langle Wm,v\rangle+H_G(v) H G ( m + v ) = H G ( m ) + ⟨ W m , v ⟩ + H G ( v ) f ( σ ) = exp ( β H G ( σ ) + h ∑ x σ x ) f(\sigma)=\exp(\beta H_G(\sigma)+h\sum_x\sigma_x) f ( σ ) = exp ( β H G ( σ ) + h ∑ x σ x )
Z G = exp ( β H G ( m ) + h ∑ x m x + ∑ x ent ( m x ) ) E π m exp ( ⟨ β W m + h 1 − y , v ⟩ + β H G ( v ) ) . (87) Z_G=\exp\left(\beta H_G(m)+h\sum_xm_x+\sum_x\operatorname{ent}(m_x)\right)\mathbb{E}_{\pi_m}\exp\left(\langle\beta Wm+h\mathbf{1}-y,v\rangle+\beta H_G(v)\right). \tag*{(87)} Z G = exp ( β H G ( m ) + h x ∑ m x + x ∑ ent ( m x ) ) E π m exp ( ⟨ β W m + h 1 − y , v ⟩ + β H G ( v ) ) . (87) Next, H G ( v ) = ∑ x y ∈ E D − 1 / 2 g x y v x v y = ⟨ g , a ( v ) ⟩ H_G(v)=\sum_{xy\in E}D^{-1/2}g_{xy}v_xv_y=\langle g,a(v)\rangle H G ( v ) = ∑ x y ∈ E D − 1/2 g x y v x v y = ⟨ g , a ( v )⟩ g = g ˉ + Π g g=\bar g+\Pi g g = g ˉ + Π g ⟨ g ˉ , a ( v ) ⟩ = ∑ x y ∈ E D − 1 / 2 g ˉ x y v x v y = ⟨ v , W ˉ v ⟩ / 2 \langle\bar g,a(v)\rangle=\sum_{xy\in E}D^{-1/2}\bar g_{xy}v_xv_y=\langle v,\bar Wv\rangle/2 ⟨ g ˉ , a ( v )⟩ = ∑ x y ∈ E D − 1/2 g ˉ x y v x v y = ⟨ v , W ˉ v ⟩ /2 ⟨ Π g , a ( v ) ⟩ = ⟨ g , Π a ( v ) ⟩ \langle\Pi g,a(v)\rangle=\langle g,\Pi a(v)\rangle ⟨ Π g , a ( v )⟩ = ⟨ g , Π a ( v )⟩
β H G ( v ) = β 2 ⟨ v , W ˉ v ⟩ + β ⟨ g , Π a ( v ) ⟩ . (88) \beta H_G(v)=\frac{\beta}{2}\langle v,\bar Wv\rangle+\beta\langle g,\Pi a(v)\rangle. \tag*{(88)} β H G ( v ) = 2 β ⟨ v , W ˉ v ⟩ + β ⟨ g , Π a ( v )⟩ . (88) Finally, we use the identity v x 2 = κ x − 2 m x v x v_x^2=\kappa_x-2m_xv_x v x 2 = κ x − 2 m x v x σ x 2 = 1 \sigma_x^2=1 σ x 2 = 1 S S S
∥ a ( v ) ∥ 2 = ∑ x y ∈ E S x y ( κ x − 2 m x v x ) ( κ y − 2 m y v y ) = 1 2 ⟨ κ , S κ ⟩ − 2 ⟨ m ⊙ S κ , v ⟩ + 2 ⟨ m ⊙ v , S ( m ⊙ v ) ⟩ . \begin{aligned}
\|a(v)\|^2
&=\sum_{xy\in E}S_{xy}(\kappa_x-2m_xv_x)(\kappa_y-2m_yv_y)\\
&=\frac{1}{2}\langle\kappa,S\kappa\rangle-2\langle m\mathbin{\odot}S\kappa,v\rangle+2\langle m\mathbin{\odot}v,S(m\mathbin{\odot}v)\rangle.
\end{aligned} ∥ a ( v ) ∥ 2 = x y ∈ E ∑ S x y ( κ x − 2 m x v x ) ( κ y − 2 m y v y ) = 2 1 ⟨ κ , S κ ⟩ − 2 ⟨ m ⊙ S κ , v ⟩ + 2 ⟨ m ⊙ v , S ( m ⊙ v )⟩ . Since ∥ Π a ( v ) ∥ 2 = ∥ a ( v ) ∥ 2 − ∥ Π ⊥ a ( v ) ∥ 2 \|\Pi a(v)\|^2=\|a(v)\|^2-\|\Pi^\perp a(v)\|^2 ∥Π a ( v ) ∥ 2 = ∥ a ( v ) ∥ 2 − ∥ Π ⊥ a ( v ) ∥ 2
β 2 2 ∥ Π a ( v ) ∥ 2 = β 2 4 ⟨ κ , S κ ⟩ − β 2 ⟨ m ⊙ S κ , v ⟩ + β 2 ⟨ m ⊙ v , S ( m ⊙ v ) ⟩ − β 2 2 ∥ Π ⊥ a ( v ) ∥ 2 . (89) \frac{\beta^2}{2}\|\Pi a(v)\|^2=\frac{\beta^2}{4}\langle\kappa,S\kappa\rangle-\beta^2\langle m\mathbin{\odot}S\kappa,v\rangle+\beta^2\langle m\mathbin{\odot}v,S(m\mathbin{\odot}v)\rangle-\frac{\beta^2}{2}\|\Pi^\perp a(v)\|^2. \tag*{(89)} 2 β 2 ∥Π a ( v ) ∥ 2 = 4 β 2 ⟨ κ , S κ ⟩ − β 2 ⟨ m ⊙ S κ , v ⟩ + β 2 ⟨ m ⊙ v , S ( m ⊙ v )⟩ − 2 β 2 ∥ Π ⊥ a ( v ) ∥ 2 . (89) We insert (88) into (87) , and we add and subtract ( β 2 / 2 ) ∥ Π a ( v ) ∥ 2 (\beta^2/2)\|\Pi a(v)\|^2 ( β 2 /2 ) ∥Π a ( v ) ∥ 2 (89) for the added term, moving the constant ( β 2 / 4 ) ⟨ κ , S κ ⟩ (\beta^2/4)\langle\kappa,S\kappa\rangle ( β 2 /4 ) ⟨ κ , S κ ⟩ (85) , we obtain (86) .
A band of typical configurations
We begin with a moment bound for the disorder. It holds on every G ∈ G D G\in\mathcal{G}_D G ∈ G D W W W
Lemma 6.6. Let G ∈ G D G \in\mathcal{G}_D G ∈ G D
E max σ ∥ W σ ∥ 2 ≤ 8 n ( log 2 + 1 4 ) ≤ 8 n . \mathbb{E}\max_{\sigma}\|W\sigma\|^2 \le8n\left(\log2+\frac{1}{4}\right)\le8n. E σ max ∥ W σ ∥ 2 ≤ 8 n ( log 2 + 4 1 ) ≤ 8 n . Further, for every realization of the disorder and every w ∈ [ − 1 , 1 ] V w\in[-1,1]^V w ∈ [ − 1 , 1 ] V ∥ W w ∥ ≤ max σ ∥ W σ ∥ \|Ww\|\le\max_{\sigma}\|W\sigma\| ∥ W w ∥ ≤ max σ ∥ W σ ∥
Proof. Fix σ ∈ { − 1 , 1 } V \sigma\in\{-1,1\}^V σ ∈ { − 1 , 1 } V W σ W\sigma W σ ( W σ ) x = D − 1 / 2 ∑ y : x y ∈ E g x y σ y (W\sigma)_x=D^{-1/2}\sum_{y:xy\in E}g_{xy}\sigma_y ( W σ ) x = D − 1/2 ∑ y : x y ∈ E g x y σ y 1 1 1 x ≠ y x\ne y x = y ( W σ ) x (W\sigma)_x ( W σ ) x ( W σ ) y (W\sigma)_y ( W σ ) y g x y g_{xy} g x y x y ∈ E xy\in E x y ∈ E W σ W\sigma W σ C σ = I + Δ σ S Δ σ C_\sigma=I+\Delta_\sigma S\Delta_\sigma C σ = I + Δ σ S Δ σ Δ σ \Delta_\sigma Δ σ σ x \sigma_x σ x (4) , the eigenvalues c 1 , … , c n c_1,\ldots,c_n c 1 , … , c n C σ C_\sigma C σ [ 0 , 2 ] [0,2] [ 0 , 2 ] ∑ j c j = tr C σ = n \sum_jc_j=\operatorname{tr}C_\sigma=n ∑ j c j = tr C σ = n C σ C_\sigma C σ
E exp ( 1 8 ∥ W σ ∥ 2 ) = ∏ j = 1 n ( 1 − c j 4 ) − 1 / 2 ≤ exp ( ∑ j = 1 n c j 4 ) = e n / 4 . \mathbb{E}\exp\left(\frac{1}{8}\|W\sigma\|^2\right)=\prod_{j=1}^n\left(1-\frac{c_j}{4}\right)^{-1/2}\le\exp\left(\sum_{j=1}^n\frac{c_j}{4}\right)=e^{n/4}. E exp ( 8 1 ∥ W σ ∥ 2 ) = j = 1 ∏ n ( 1 − 4 c j ) − 1/2 ≤ exp ( j = 1 ∑ n 4 c j ) = e n /4 . where we used that − log ( 1 − s ) ≤ 2 s -\log(1-s)\le2s − log ( 1 − s ) ≤ 2 s s ∈ [ 0 , 1 / 2 ] s\in[0,1/2] s ∈ [ 0 , 1/2 ]
E max σ ∥ W σ ∥ 2 ≤ 8 log E max σ e ∥ W σ ∥ 2 / 8 ≤ 8 log ∑ σ E e ∥ W σ ∥ 2 / 8 ≤ 8 ( n log 2 + n 4 ) . \mathbb{E}\max_{\sigma}\|W\sigma\|^2\le8\log\mathbb{E}\max_{\sigma}e^{\|W\sigma\|^2/8}\le8\log\sum_{\sigma}\mathbb{E}e^{\|W\sigma\|^2/8}\le8\left(n\log2+\frac{n}{4}\right). E σ max ∥ W σ ∥ 2 ≤ 8 log E σ max e ∥ W σ ∥ 2 /8 ≤ 8 log σ ∑ E e ∥ W σ ∥ 2 /8 ≤ 8 ( n log 2 + 4 n ) . For the final claim, let w ∈ [ − 1 , 1 ] V w\in[-1,1]^V w ∈ [ − 1 , 1 ] V W W W W w = E π w W σ Ww=\mathbb{E}_{\pi_w}W\sigma W w = E π w W σ ∥ W w ∥ ≤ E π w ∥ W σ ∥ ≤ max σ ∥ W σ ∥ \|Ww\|\le\mathbb{E}_{\pi_w}\|W\sigma\|\le\max_{\sigma}\|W\sigma\| ∥ W w ∥ ≤ E π w ∥ W σ ∥ ≤ max σ ∥ W σ ∥
We now define the band. For the rest of this section, we write
C β , h = 4 ( 1 + h 2 + 8 β 2 + β 4 ) , ε D ( η ) = C β , h η 2 D + 1 + ( 2 T ) 1 / 2 η D 1 / 2 + T η D ( η > 0 ) . (90) C_{\beta,h}=4(1+h^2+8\beta^2+\beta^4),\qquad\varepsilon_D(\eta)=\frac{C_{\beta,h}}{\eta^2D}+\frac{1+(2T)^{1/2}}{\eta D^{1/2}}+\frac{T}{\eta D}\qquad(\eta>0). \tag*{(90)} C β , h = 4 ( 1 + h 2 + 8 β 2 + β 4 ) , ε D ( η ) = η 2 D C β , h + η D 1/2 1 + ( 2 T ) 1/2 + η D T ( η > 0 ) . (90) and we note that ε D ( η ) \varepsilon_D(\eta) ε D ( η ) β \beta β h h h T T T D D D η \eta η
Lemma 6.7. Let G ∈ G D G\in\mathcal{G}_D G ∈ G D β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R T T T G G G m m m Π \Pi Π W ˉ \bar W W ˉ § 7.2 , and let r r r (85) . For η > 0 \eta>0 η > 0 B η B_\eta B η σ ∈ { − 1 , 1 } V \sigma\in\{-1,1\}^V σ ∈ { − 1 , 1 } V v = σ − m v=\sigma-m v = σ − m
∣ ⟨ r , v ⟩ ∣ ≤ η n , ⟨ v , ∣ W ˉ ∣ v ⟩ ≤ η n , ⟨ m ⊙ v , ∣ S ∣ ( m ⊙ v ) ⟩ ≤ η n , ∥ Π ⊥ a ( v ) ∥ 2 ≤ η n . (91) |\langle r,v\rangle|\le\eta n,\qquad\langle v,|\bar W|v\rangle\le\eta n,\qquad\langle m\odot v,|S|(m\odot v)\rangle\le\eta n,\qquad\|\Pi^\perp a(v)\|^2\le\eta n. \tag*{(91)} ∣ ⟨ r , v ⟩ ∣ ≤ η n , ⟨ v , ∣ W ˉ ∣ v ⟩ ≤ η n , ⟨ m ⊙ v , ∣ S ∣ ( m ⊙ v )⟩ ≤ η n , ∥ Π ⊥ a ( v ) ∥ 2 ≤ η n . (91) Then { σ ∈ B η } ∈ H \{\sigma\in B_\eta\}\in\mathcal{H} { σ ∈ B η } ∈ H σ \sigma σ p η = π m ( B η ) p_\eta=\pi_m(B_\eta) p η = π m ( B η ) E ( 1 − p η ) ≤ ε D ( η ) \mathbb{E}(1-p_\eta)\le\varepsilon_D(\eta) E ( 1 − p η ) ≤ ε D ( η )
Proof. By Lemma 6.4(a), the vectors m m m W m Wm W m r r r Π \Pi Π W ˉ \bar W W ˉ H \mathcal{H} H ∣ W ˉ ∣ |\bar W| ∣ W ˉ ∣ W ˉ \bar W W ˉ p η p_\eta p η H \mathcal{H} H
Fix a realization. Under π m \pi_m π m v v v v x v_x v x κ x \kappa_x κ x E π m ⟨ r , v ⟩ 2 = ∑ x κ x r x 2 \mathbb{E}_{\pi_m}\langle r,v\rangle^2=\sum_x\kappa_x r_x^2 E π m ⟨ r , v ⟩ 2 = ∑ x κ x r x 2 ∑ x κ x r x 2 / ( η n ) 2 \sum_x\kappa_x r_x^2/(\eta n)^2 ∑ x κ x r x 2 / ( η n ) 2 ∣ W ‾ ∣ |\overline{W}| ∣ W ∣
E π m ⟨ v , ∣ W ‾ ∣ v ⟩ = ∑ x ∣ W ‾ ∣ x x κ x ≤ tr ∣ W ‾ ∣ = ∥ W ‾ ∥ t r , \mathbb{E}_{\pi_m}\langle v,|\overline{W}|v\rangle=\sum_x|\overline{W}|_{xx}\kappa_x\leq\operatorname{tr}|\overline{W}|=\|\overline{W}\|_{\mathrm{tr}}, E π m ⟨ v , ∣ W ∣ v ⟩ = x ∑ ∣ W ∣ xx κ x ≤ tr ∣ W ∣ = ∥ W ∥ tr , and Markov’s inequality bounds the probability of the second failure by ∥ W ‾ ∥ t r / ( η n ) \|\overline{W}\|_{\mathrm{tr}}/(\eta n) ∥ W ∥ tr / ( η n )
E π m ⟨ m ⊙ v , ∣ S ∣ ( m ⊙ v ) ⟩ = ∑ x ∣ S ∣ x x m x 2 κ x ≤ tr ∣ S ∣ ≤ ( n tr S 2 ) 1 / 2 = n D 1 / 2 , \mathbb{E}_{\pi_m}\langle m\odot v,|S|(m\odot v)\rangle=\sum_x|S|_{xx}m_x^2\kappa_x\leq\operatorname{tr}|S|\leq\left(n\operatorname{tr}S^2\right)^{1/2}=\frac{n}{D^{1/2}}, E π m ⟨ m ⊙ v , ∣ S ∣ ( m ⊙ v )⟩ = x ∑ ∣ S ∣ xx m x 2 κ x ≤ tr ∣ S ∣ ≤ ( n tr S 2 ) 1/2 = D 1/2 n , where we used the Cauchy–Schwarz inequality for the eigenvalues of S S S (4) . Fourth, the entries of a ( v ) a(v) a ( v ) π m \pi_m π m e , e ′ e,e' e , e ′ x x x v x v_x v x a ( v ) e a ( v ) e ′ a(v)_e a(v)_{e'} a ( v ) e a ( v ) e ′ E π m a ( v ) x y 2 = S x y κ x κ y ≤ D − 1 \mathbb{E}_{\pi_m}a(v)_{xy}^2=S_{xy}\kappa_x\kappa_y\leq D^{-1} E π m a ( v ) x y 2 = S x y κ x κ y ≤ D − 1 a ( v ) a(v) a ( v ) D − 1 D^{-1} D − 1
E π m ∥ Π ⊥ a ( v ) ∥ 2 = ∑ e ∈ E ( Π ⊥ ) e e E π m a ( v ) e 2 ≤ tr Π ⊥ D = rank Π ⊥ D . \mathbb{E}_{\pi_m}\|\Pi^\perp a(v)\|^2=\sum_{e\in E}(\Pi^\perp)_{ee}\mathbb{E}_{\pi_m}a(v)_e^2\leq\frac{\operatorname{tr}\Pi^\perp}{D}=\frac{\operatorname{rank}\Pi^\perp}{D}. E π m ∥ Π ⊥ a ( v ) ∥ 2 = e ∈ E ∑ ( Π ⊥ ) ee E π m a ( v ) e 2 ≤ D tr Π ⊥ = D rank Π ⊥ . By the union bound, we conclude that
1 − p η ≤ ∑ x κ x r x 2 η 2 n 2 + ∥ W ‾ ∥ t r η n + 1 η D 1 / 2 + rank Π ⊥ η n D . (92) 1-p_\eta\leq\frac{\sum_x\kappa_xr_x^2}{\eta^2n^2}+\frac{\|\overline{W}\|_{\mathrm{tr}}}{\eta n}+\frac{1}{\eta D^{1/2}}+\frac{\operatorname{rank}\Pi^\perp}{\eta nD}. \tag*{(92)} 1 − p η ≤ η 2 n 2 ∑ x κ x r x 2 + η n ∥ W ∥ tr + η D 1/2 1 + η n D rank Π ⊥ . (92) It remains to bound the expectation of the first term. Let s x = atanh ( m x ) s_x=\operatorname{atanh}(m_x) s x = atanh ( m x ) κ x = sech 2 s x ≤ 4 e − 2 ∣ s x ∣ \kappa_x=\operatorname{sech}^2s_x\leq4e^{-2|s_x|} κ x = sech 2 s x ≤ 4 e − 2∣ s x ∣ κ x s x 2 ≤ 4 sup s ≥ 0 s 2 e − 2 s = 4 e − 2 ≤ 1 \kappa_xs_x^2\leq4\sup_{s\geq0}s^2e^{-2s}=4e^{-2}\leq1 κ x s x 2 ≤ 4 sup s ≥ 0 s 2 e − 2 s = 4 e − 2 ≤ 1 m x ∈ ( − 1 , 1 ) m_x\in(-1,1) m x ∈ ( − 1 , 1 ) ( c 1 + c 2 + c 3 + c 4 ) 2 ≤ 4 ( c 1 2 + c 2 2 + c 3 2 + c 4 2 ) (c_1+c_2+c_3+c_4)^2\leq4(c_1^2+c_2^2+c_3^2+c_4^2) ( c 1 + c 2 + c 3 + c 4 ) 2 ≤ 4 ( c 1 2 + c 2 2 + c 3 2 + c 4 2 ) κ x ≤ 1 \kappa_x\leq1 κ x ≤ 1 ∣ m x ∣ ≤ 1 |m_x|\leq1 ∣ m x ∣ ≤ 1 0 ≤ ( S κ ) x ≤ 1 0\leq(S\kappa)_x\leq1 0 ≤ ( S κ ) x ≤ 1 (85) that
κ x r x 2 ≤ 4 ( β 2 ( W m ) x 2 + h 2 + κ x s x 2 + β 4 ) ≤ 4 ( β 2 ( W m ) x 2 + h 2 + 1 + β 4 ) . \kappa_xr_x^2\leq4\left(\beta^2(Wm)_x^2+h^2+\kappa_xs_x^2+\beta^4\right)\leq4\left(\beta^2(Wm)_x^2+h^2+1+\beta^4\right). κ x r x 2 ≤ 4 ( β 2 ( W m ) x 2 + h 2 + κ x s x 2 + β 4 ) ≤ 4 ( β 2 ( W m ) x 2 + h 2 + 1 + β 4 ) . Summing over x x x Lemma 6.6 , which gives E ∥ W m ∥ 2 ≤ E max σ ∥ W σ ∥ 2 ≤ 8 n \mathbb{E}\|Wm\|^2\leq\mathbb{E}\max_\sigma\|W\sigma\|^2\leq8n E ∥ W m ∥ 2 ≤ E max σ ∥ W σ ∥ 2 ≤ 8 n
E ∑ x κ x r x 2 ≤ 32 β 2 n + 4 n ( h 2 + 1 + β 4 ) = C β , h n . \mathbb{E}\sum_x\kappa_xr_x^2\leq32\beta^2n+4n(h^2+1+\beta^4)=C_{\beta,h}n. E x ∑ κ x r x 2 ≤ 32 β 2 n + 4 n ( h 2 + 1 + β 4 ) = C β , h n . We take expectations in (92) , and we use this bound together with Lemma 6.4 (c). Since every D D D n ≥ D + 1 n\geq D+1 n ≥ D + 1 E ( 1 − p η ) ≤ ϵ D ( η ) \mathbb{E}(1-p_\eta)\leq\epsilon_D(\eta) E ( 1 − p η ) ≤ ϵ D ( η )
On the band, the error terms in Lemma 6.5 cost at most a multiple of η n \eta n η n
c β = 1 + β 2 + 3 β 2 2 . (93) c_\beta=1+\frac{\beta}{2}+\frac{3\beta^2}{2}. \tag*{(93)} c β = 1 + 2 β + 2 3 β 2 . (93) and on the event { p η > 0 } \{p_\eta> 0\} { p η > 0 }
L η ( y ) = E π m [ exp ( β ⟨ y , Π a ( v ) ⟩ − β 2 2 ∥ Π a ( v ) ∥ 2 ) ∣ B η ] ( y ∈ R E ) . (94) L_\eta(y) = \mathbb{E}_{\pi_m}\left[\left.\exp\left(\beta\langle y,\Pi a(v)\rangle- \frac{\beta^2}{2}\|\Pi a(v)\|^2\right)\right|B_\eta\right] \qquad(y \in\mathbb{R}^{E}). \tag*{(94)} L η ( y ) = E π m [ exp ( β ⟨ y , Π a ( v )⟩ − 2 β 2 ∥Π a ( v ) ∥ 2 ) B η ] ( y ∈ R E ) . (94) On { p η = 0 } \{p_\eta= 0\} { p η = 0 } L η ( y ) = 1 L_\eta(y) = 1 L η ( y ) = 1 y y y y y y Π y \Pi y Π y ( ω , y ) ↦ L η ( y ) (\omega,y)\mapsto L_\eta(y) ( ω , y ) ↦ L η ( y ) H ⊗ B ( R E ) \mathcal{H}\otimes\mathcal{B}(\mathbb{R}^{E}) H ⊗ B ( R E ) { p η > 0 } × R E \{p_\eta> 0\}\times\mathbb{R}^{E} { p η > 0 } × R E ( m , Π , y ) (m,\Pi,y) ( m , Π , y ) H \mathcal{H} H
Lemma 6.8. Under the assumptions of Lemma 6.7 , for every η > 0 \eta> 0 η > 0 { p η > 0 } \{p_\eta> 0\} { p η > 0 }
log Z G ≥ Θ G ( m ) − c β η n + log p η + log L η ( g ) . \log Z_G \ge\Theta_G(m) - c_\beta\eta n + \log p_\eta+ \log L_\eta(g). log Z G ≥ Θ G ( m ) − c β η n + log p η + log L η ( g ) . Proof. We apply Lemma 6.5 with the projection Π \Pi Π § 7.2 , and we restrict the expectation in (86) to σ ∈ B η \sigma\in B_\eta σ ∈ B η K K K − ∣ K ∣ ≤ K ≤ ∣ K ∣ -|K| \le K \le|K| − ∣ K ∣ ≤ K ≤ ∣ K ∣ σ ∈ B η \sigma\in B_\eta σ ∈ B η
⟨ r , v ⟩ ≥ − η n , β 2 ⟨ v , W ‾ v ⟩ ≥ − β 2 η n , \langle r,v\rangle\ge-\eta n,\qquad\frac{\beta}{2}\langle v,\overline{W}v\rangle\ge-\frac{\beta}{2}\eta n, ⟨ r , v ⟩ ≥ − η n , 2 β ⟨ v , W v ⟩ ≥ − 2 β η n , β 2 ⟨ m ⊙ v , S ( m ⊙ v ) ⟩ ≥ − β 2 η n , − β 2 2 ∥ Π ⊥ a ( v ) ∥ 2 ≥ − β 2 2 η n . \beta^2\langle m\odot v,S(m\odot v)\rangle\ge-\beta^2\eta n,\qquad-\frac{\beta^2}{2}\|\Pi^\perp a(v)\|^2 \ge-\frac{\beta^2}{2}\eta n. β 2 ⟨ m ⊙ v , S ( m ⊙ v )⟩ ≥ − β 2 η n , − 2 β 2 ∥ Π ⊥ a ( v ) ∥ 2 ≥ − 2 β 2 η n . The sum of the right sides is − c β η n -c_\beta\eta n − c β η n
Z G ≥ e Θ G ( m ) − c β η n E π m [ 1 B η ( σ ) exp ( β ⟨ g , Π a ( v ) ⟩ − β 2 2 ∥ Π a ( v ) ∥ 2 ) ] = e Θ G ( m ) − c β η n p η L η ( g ) . Z_G \ge e^{\Theta_G(m)-c_\beta\eta n}\mathbb{E}_{\pi_m}\left[1_{B_\eta}(\sigma)\exp\left(\beta\langle g,\Pi a(v)\rangle-\frac{\beta^2}{2}\|\Pi a(v)\|^2\right)\right] = e^{\Theta_G(m)-c_\beta\eta n}p_\eta L_\eta(g). Z G ≥ e Θ G ( m ) − c β η n E π m [ 1 B η ( σ ) exp ( β ⟨ g , Π a ( v )⟩ − 2 β 2 ∥Π a ( v ) ∥ 2 ) ] = e Θ G ( m ) − c β η n p η L η ( g ) . Taking logarithms completes the proof.
Relative entropy of the restricted and projected model
In this subsection, S S S E E E β \beta β n n n Definition 6.1 . Since S S S ∑ { x , y } ∈ E S x y = n / 2 \sum_{\{x,y\}\in E}S_{xy}=n/2 ∑ { x , y } ∈ E S x y = n /2 ∣ v x ∣ ≤ 1 + ∣ m x ∣ ≤ 2 |v_x|\le1+|m_x|\le2 ∣ v x ∣ ≤ 1 + ∣ m x ∣ ≤ 2 v = σ − m v=\sigma-m v = σ − m
∥ a ( v ) ∥ 2 = ∑ { x , y } ∈ E S x y v x 2 v y 2 ≤ 16 ∑ { x , y } ∈ E S x y = 8 n (95) \|a(v)\|^2=\sum_{\{x,y\}\in E}S_{xy}v_x^2v_y^2\le16\sum_{\{x,y\}\in E}S_{xy}=8n \tag*{(95)} ∥ a ( v ) ∥ 2 = { x , y } ∈ E ∑ S x y v x 2 v y 2 ≤ 16 { x , y } ∈ E ∑ S x y = 8 n (95) for every m ∈ [ − 1 , 1 ] n m\in[-1,1]^n m ∈ [ − 1 , 1 ] n σ ∈ { − 1 , 1 } n \sigma\in\{-1,1\}^n σ ∈ { − 1 , 1 } n
Lemma 6.9. The function m ↦ D ( m ) m\mapsto D(m) m ↦ D ( m ) [ − 1 , 1 ] n [-1,1]^n [ − 1 , 1 ] n m ∈ [ − 1 , 1 ] n m\in[-1,1]^n m ∈ [ − 1 , 1 ] n Π \Pi Π R E \mathbb{R}^{E} R E B ⊂ { − 1 , 1 } n B\subset\{-1,1\}^n B ⊂ { − 1 , 1 } n p = π m ( B ) > 0 p=\pi_m(B)>0 p = π m ( B ) > 0
L B , Π ( y ) = E π m [ exp ( β ⟨ y , Π a ( v ) ⟩ − β 2 2 ∥ Π a ( v ) ∥ 2 ) ∣ B ] ( y ∈ R E ) . L_{B,\Pi}(y)=\mathbb{E}_{\pi_m}\left[\left.\exp\left(\beta\langle y,\Pi a(v)\rangle-\frac{\beta^2}{2}\|\Pi a(v)\|^2\right)\right|B\right]\qquad(y\in\mathbb{R}^{E}). L B , Π ( y ) = E π m [ exp ( β ⟨ y , Π a ( v )⟩ − 2 β 2 ∥Π a ( v ) ∥ 2 ) B ] ( y ∈ R E ) . Then Q B , Π = L B , Π P 0 Q_{B,\Pi}=L_{B,\Pi}P_0 Q B , Π = L B , Π P 0 β Π a ( v ∗ ) + Z \beta\Pi a(v^*)+Z β Π a ( v ∗ ) + Z σ ∗ \sigma^* σ ∗ π m ( ⋅ ∣ B ) \pi_m(\cdot\mid B) π m ( ⋅ ∣ B ) v ∗ = σ ∗ − m v^*=\sigma^*-m v ∗ = σ ∗ − m Z Z Z R E \mathbb{R}^{E} R E σ ∗ \sigma^* σ ∗
p KL ( Q B , Π ∥ P 0 ) ≤ D ( m ) − p log p − ( 1 − p ) log ( 1 − p ) ≤ D ( m ) + log 2 , (96) p\,\operatorname{KL}(Q_{B,\Pi}\mathbin\Vert P_0)\leq\mathcal{D}(m)-p\log p-(1-p)\log(1-p)\leq\mathcal{D}(m)+\log2, \tag*{(96)} p KL ( Q B , Π ∥ P 0 ) ≤ D ( m ) − p log p − ( 1 − p ) log ( 1 − p ) ≤ D ( m ) + log 2 , (96) with the convention 0 log 0 = 0 0\log0=0 0 log 0 = 0
Proof. The identification of Q B , Π Q_{B,\Pi} Q B , Π (78) . By (95) , we have
exp ( − β ( 8 n ) 1 / 2 ∥ y ∥ − 4 β 2 n ) ≤ L B , Π ( y ) ≤ exp ( β ( 8 n ) 1 / 2 ∥ y ∥ ) , (97) \exp\left(-\beta(8n)^{1/2}\lVert y\rVert-4\beta^2n\right)\leq L_{B,\Pi}(y)\leq\exp\left(\beta(8n)^{1/2}\lVert y\rVert\right), \tag*{(97)} exp ( − β ( 8 n ) 1/2 ∥ y ∥ − 4 β 2 n ) ≤ L B , Π ( y ) ≤ exp ( β ( 8 n ) 1/2 ∥ y ∥ ) , (97) and in particular all relative entropies in this proof are finite. For the continuity claim, the density L m L_m L m Q m Q_m Q m (78) , is continuous in ( m , y ) (m,y) ( m , y ) (97) with B = { − 1 , 1 } n B=\{-1,1\}^n B = { − 1 , 1 } n Π = I \Pi=I Π = I
∣ L m ( y ) log L m ( y ) ∣ ≤ ( β ( 8 n ) 1 / 2 ∥ y ∥ + 4 β 2 n ) e β ( 8 n ) 1 / 2 ∥ y ∥ . \lvert L_m(y)\log L_m(y)\rvert\leq\left(\beta(8n)^{1/2}\lVert y\rVert+4\beta^2n\right)e^{\beta(8n)^{1/2}\lVert y\rVert}. ∣ L m ( y ) log L m ( y )∣ ≤ ( β ( 8 n ) 1/2 ∥ y ∥ + 4 β 2 n ) e β ( 8 n ) 1/2 ∥ y ∥ . The right side does not depend on m m m P 0 P_0 P 0 D ( m ) = ∫ L m log L m d P 0 \mathcal{D}(m)=\int L_m\log L_m\,\mathrm{d}P_0 D ( m ) = ∫ L m log L m d P 0 m m m
We now prove (96) , first for Π = I \Pi=I Π = I L B = L B , I L_B=L_{B,I} L B = L B , I Q B = Q B , I Q_B=Q_{B,I} Q B = Q B , I p = 1 p=1 p = 1 Q B = Q m Q_B=Q_m Q B = Q m p < 1 p<1 p < 1 L m = p L B + ( 1 − p ) L B c L_m=pL_B+(1-p)L_{B^c} L m = p L B + ( 1 − p ) L B c L B c L_{B^c} L B c L m ≥ p L B L_m\geq pL_B L m ≥ p L B log L B ≤ log L m − log p \log L_B\leq\log L_m-\log p log L B ≤ log L m − log p
KL ( Q B ∥ P 0 ) = ∫ L B log L B d P 0 ≤ ∫ L B log L m d P 0 − log p . \operatorname{KL}(Q_B\mathbin\Vert P_0)=\int L_B\log L_B\,\mathrm{d}P_0\leq\int L_B\log L_m\,\mathrm{d}P_0-\log p. KL ( Q B ∥ P 0 ) = ∫ L B log L B d P 0 ≤ ∫ L B log L m d P 0 − log p . Since L m ≥ ( 1 − p ) L B c L_m\geq(1-p)L_{B^c} L m ≥ ( 1 − p ) L B c
∫ L B c log L m d P 0 ≥ log ( 1 − p ) + ∫ L B c log L B c d P 0 ≥ log ( 1 − p ) , \int L_{B^c}\log L_m\,\mathrm{d}P_0\geq\log(1-p)+\int L_{B^c}\log L_{B^c}\,\mathrm{d}P_0\geq\log(1-p), ∫ L B c log L m d P 0 ≥ log ( 1 − p ) + ∫ L B c log L B c d P 0 ≥ log ( 1 − p ) , because the last integral is a relative entropy and is nonnegative. Then
D ( m ) = ∫ L m log L m d P 0 = p ∫ L B log L m d P 0 + ( 1 − p ) ∫ L B c log L m d P 0 ≥ p ∫ L B log L m d P 0 + ( 1 − p ) log ( 1 − p ) . \begin{aligned}
\mathcal{D}(m)&=\int L_m\log L_m\,\mathrm{d}P_0\\
&=p\int L_B\log L_m\,\mathrm{d}P_0+(1-p)\int L_{B^c}\log L_m\,\mathrm{d}P_0\\
&\geq p\int L_B\log L_m\,\mathrm{d}P_0+(1-p)\log(1-p).
\end{aligned} D ( m ) = ∫ L m log L m d P 0 = p ∫ L B log L m d P 0 + ( 1 − p ) ∫ L B c log L m d P 0 ≥ p ∫ L B log L m d P 0 + ( 1 − p ) log ( 1 − p ) . Combining the last three displays, we obtain (96) for Π = I \Pi=I Π = I (96) holds because the binary entropy is at most log 2 \log2 log 2
For a general Π \Pi Π KL ( Q B , Π ∥ P 0 ) ≤ KL ( Q B ∥ P 0 ) \operatorname{KL}(Q_{B,\Pi}\mathbin\Vert P_0)\leq\operatorname{KL}(Q_B\mathbin\Vert P_0) KL ( Q B , Π ∥ P 0 ) ≤ KL ( Q B ∥ P 0 ) y ↦ Π y + Π ⊥ z ′ y\mapsto\Pi y+\Pi^\perp z' y ↦ Π y + Π ⊥ z ′ z ′ z' z ′ P 0 P_0 P 0 Q B Q_B Q B Q B , Π Q_{B,\Pi} Q B , Π P 0 P_0 P 0 Π y \Pi y Π y Π ⊥ y \Pi^\perp y Π ⊥ y ⟨ Π ⊥ y , a ⟩ \langle\Pi^\perp y,a\rangle ⟨ Π ⊥ y , a ⟩ ∥ Π ⊥ a ∥ 2 \lVert\Pi^\perp a\rVert^2 ∥ Π ⊥ a ∥ 2 a ∈ R E a\in\mathbb{R}^{E} a ∈ R E ⟨ y , a ⟩ = \langle y,a\rangle= ⟨ y , a ⟩ = ⟨ Π y , Π a ⟩ + ⟨ Π ⊥ y , Π ⊥ a ⟩ \langle\Pi y,\Pi a\rangle+\langle\Pi^\perp y,\Pi^\perp a\rangle ⟨ Π y , Π a ⟩ + ⟨ Π ⊥ y , Π ⊥ a ⟩ ∥ a ∥ 2 = ∥ Π a ∥ 2 + ∥ Π ⊥ a ∥ 2 \|a\|^2=\|\Pi a\|^2+\|\Pi^\perp a\|^2 ∥ a ∥ 2 = ∥Π a ∥ 2 + ∥ Π ⊥ a ∥ 2 Π ⊥ y \Pi^\perp y Π ⊥ y L B L_B L B
E P 0 [ L B ( y ) ∣ Π y ] = L B , Π ( y ) . \mathbb{E}_{P_0}[L_B(y)\mid\Pi y]=L_{B,\Pi}(y). E P 0 [ L B ( y ) ∣ Π y ] = L B , Π ( y ) . By the convexity of s ↦ s log s s\mapsto s\log s s ↦ s log s [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
KL ( Q B , Π ∥ P 0 ) = E P 0 [ L B , Π log L B , Π ] ≤ E P 0 [ L B log L B ] = KL ( Q B ∥ P 0 ) . \operatorname{KL}(Q_{B,\Pi}\mathbin\Vert P_0)=\mathbb{E}_{P_0}[L_{B,\Pi}\log L_{B,\Pi}]\leq\mathbb{E}_{P_0}[L_B\log L_B]=\operatorname{KL}(Q_B\mathbin\Vert P_0). KL ( Q B , Π ∥ P 0 ) = E P 0 [ L B , Π log L B , Π ] ≤ E P 0 [ L B log L B ] = KL ( Q B ∥ P 0 ) . This completes the proof.
The next lemma converts the relative entropy of a Gaussian mixture with respect to the Gaussian law into the relative entropy in the reverse direction. We recall (F5): for every k ≥ 1 k\geq1 k ≥ 1 ρ \rho ρ R k \mathbb{R}^k R k W 2 ( ρ , γ k ) 2 ≤ 2 KL ( ρ ∥ γ k ) W_2(\rho,\gamma_k)^2\leq2\operatorname{KL}(\rho\mathbin\Vert\gamma_k) W 2 ( ρ , γ k ) 2 ≤ 2 KL ( ρ ∥ γ k ) γ k \gamma_k γ k R k \mathbb{R}^k R k W 2 W_2 W 2
Lemma 6.10. Let k ≥ 1 k\geq1 k ≥ 1 ℓ ≥ 0 \ell\geq0 ℓ ≥ 0 ν \nu ν { b ∈ R k : ∥ b ∥ ≤ ℓ } \{b\in\mathbb{R}^k:\|b\|\leq\ell\} { b ∈ R k : ∥ b ∥ ≤ ℓ }
L ( y ) = ∫ exp ( ⟨ y , b ⟩ − 1 2 ∥ b ∥ 2 ) ν ( d b ) ( y ∈ R k ) , L(y)=\int\exp\left(\langle y,b\rangle-\frac{1}{2}\|b\|^2\right)\nu(db)\qquad(y\in\mathbb{R}^k), L ( y ) = ∫ exp ( ⟨ y , b ⟩ − 2 1 ∥ b ∥ 2 ) ν ( d b ) ( y ∈ R k ) , and let Q = L γ k Q=L\gamma_k Q = L γ k
0 ≤ − ∫ log L d γ k = KL ( γ k ∥ Q ) ≤ ℓ ( 2 KL ( Q ∥ γ k ) ) 1 / 2 . 0\leq-\int\log L\,d\gamma_k=\operatorname{KL}(\gamma_k\mathbin\Vert Q)\leq\ell\left(2\operatorname{KL}(Q\mathbin\Vert\gamma_k)\right)^{1/2}. 0 ≤ − ∫ log L d γ k = KL ( γ k ∥ Q ) ≤ ℓ ( 2 KL ( Q ∥ γ k ) ) 1/2 . Proof. The function L L L ∇ log L ( y ) = ∫ b ν y ( d b ) \nabla\log L(y)=\int b\,\nu_y(db) ∇ log L ( y ) = ∫ b ν y ( d b ) ν y \nu_y ν y exp ( ⟨ y , b ⟩ − ∥ b ∥ 2 / 2 ) ν ( d b ) \exp(\langle y,b\rangle-\|b\|^2/2)\nu(db) exp (⟨ y , b ⟩ − ∥ b ∥ 2 /2 ) ν ( d b ) ν y \nu_y ν y ℓ \ell ℓ ∥ ∇ log L ∥ ≤ ℓ \|\nabla\log L\|\leq\ell ∥∇ log L ∥ ≤ ℓ log L \log L log L ℓ \ell ℓ Q Q Q b + Z b+Z b + Z b ∼ ν b\sim\nu b ∼ ν Z ∼ γ k Z\sim\gamma_k Z ∼ γ k Q Q Q γ k \gamma_k γ k log L \log L log L KL ( Q ∥ γ k ) = ∫ log L d Q \operatorname{KL}(Q\mathbin\Vert\gamma_k)=\int\log L\,dQ KL ( Q ∥ γ k ) = ∫ log L d Q KL ( γ k ∥ Q ) = − ∫ log L d γ k ≥ 0 \operatorname{KL}(\gamma_k\mathbin\Vert Q)=-\int\log L\,d\gamma_k\geq0 KL ( γ k ∥ Q ) = − ∫ log L d γ k ≥ 0 ( Y , Y ′ ) (Y,Y') ( Y , Y ′ ) Q Q Q γ k \gamma_k γ k
KL ( Q ∥ γ k ) + KL ( γ k ∥ Q ) = E [ log L ( Y ) − log L ( Y ′ ) ] ≤ ℓ E ∥ Y − Y ′ ∥ ≤ ℓ ( E ∥ Y − Y ′ ∥ 2 ) 1 / 2 . \operatorname{KL}(Q\mathbin\Vert\gamma_k)+\operatorname{KL}(\gamma_k\mathbin\Vert Q)=\mathbb{E}[\log L(Y)-\log L(Y')]\leq\ell\mathbb{E}\|Y-Y'\|\leq\ell\left(\mathbb{E}\|Y-Y'\|^2\right)^{1/2}. KL ( Q ∥ γ k ) + KL ( γ k ∥ Q ) = E [ log L ( Y ) − log L ( Y ′ )] ≤ ℓ E ∥ Y − Y ′ ∥ ≤ ℓ ( E ∥ Y − Y ′ ∥ 2 ) 1/2 . Taking the infimum over couplings and using (F5), we find that the left side is at most ℓ W 2 ( Q , γ k ) ≤ ℓ ( 2 KL ( Q ∥ γ k ) ) 1 / 2 \ell W_2(Q,\gamma_k)\leq\ell(2\operatorname{KL}(Q\mathbin\Vert\gamma_k))^{1/2} ℓ W 2 ( Q , γ k ) ≤ ℓ ( 2 KL ( Q ∥ γ k ) ) 1/2 KL ( Q ∥ γ k ) ≥ 0 \operatorname{KL}(Q\mathbin\Vert\gamma_k)\geq0 KL ( Q ∥ γ k ) ≥ 0
Corollary 6.11. Under the assumptions of Lemma 6.9 , we have
∫ log L B , Π d P 0 ≥ − 4 β ( n KL ( Q B , Π ∥ P 0 ) ) 1 / 2 ≥ − 4 β ( n ( D ( m ) + log 2 ) p ) 1 / 2 . \int\log L_{B,\Pi}\,dP_0\geq-4\beta\left(n\operatorname{KL}(Q_{B,\Pi}\mathbin\Vert P_0)\right)^{1/2}\geq-4\beta\left(\frac{n(\mathcal{D}(m)+\log2)}{p}\right)^{1/2}. ∫ log L B , Π d P 0 ≥ − 4 β ( n KL ( Q B , Π ∥ P 0 ) ) 1/2 ≥ − 4 β ( p n ( D ( m ) + log 2 ) ) 1/2 . Proof. By Lemma 6.9 , L B , Π L_{B,\Pi} L B , Π Lemma 6.10 with k = ∣ E ∣ k=|E| k = ∣ E ∣ ν \nu ν β Π a ( v ∗ ) \beta\Pi a(v^*) β Π a ( v ∗ ) (95) and ∥ Π a ∥ ≤ ∥ a ∥ \|\Pi a\|\leq\|a\| ∥Π a ∥ ≤ ∥ a ∥ ℓ = β ( 8 n ) 1 / 2 \ell=\beta(8n)^{1/2} ℓ = β ( 8 n ) 1/2 Lemma 6.10 implies the first inequality, since β ( 8 n ) 1 / 2 ⋅ 2 1 / 2 = 4 β n 1 / 2 \beta(8n)^{1/2}\cdot2^{1/2}=4\beta n^{1/2} β ( 8 n ) 1/2 ⋅ 2 1/2 = 4 β n 1/2 (96) .
Proof of the main result
We first prove a nonasymptotic version of Theorem 6.2 . Recall the constants ε D ( η ) \varepsilon_D(\eta) ε D ( η ) c β c_\beta c β
Proposition 6.12. Let β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R D ≥ 1 D \ge1 D ≥ 1 T ≥ 1 T \ge1 T ≥ 1 G ∈ G D G \in\mathcal{G}_D G ∈ G D n n n D \mathcal{D} D Definition 6.1 with S = A G / D S = A_G/D S = A G / D T T T G G G m m m η > 0 \eta> 0 η > 0 ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 )
p G ( β , h ) ≥ 1 n E Θ G ( m ) − c β η − C β , h ′ ( ε D ( η ) ε ) 1 / 2 − 4 β ( n − 1 E D ( m ) + n − 1 log 2 1 − ε ) 1 / 2 + log ( 1 − ε ) n , (98) p_G(\beta,h) \ge\frac{1}{n}\mathbb{E}\,\Theta_G(m) - c_\beta\eta- C'_{\beta,h}\left(\frac{\varepsilon_D(\eta)}{\varepsilon}\right)^{1/2} - 4\beta\left(\frac{n^{-1}\mathbb{E}\,\mathcal{D}(m)+n^{-1}\log2}{1-\varepsilon}\right)^{1/2} + \frac{\log(1-\varepsilon)}{n}, \tag*{(98)} p G ( β , h ) ≥ n 1 E Θ G ( m ) − c β η − C β , h ′ ( ε ε D ( η ) ) 1/2 − 4 β ( 1 − ε n − 1 E D ( m ) + n − 1 log 2 ) 1/2 + n log ( 1 − ε ) , (98) where C β , h ′ = β C 2 1 / 2 + ∣ h ∣ + log 2 + β 2 / 4 C'_{\beta,h}=\beta C_2^{1/2}+|h|+\log2+\beta^2/4 C β , h ′ = β C 2 1/2 + ∣ h ∣ + log 2 + β 2 /4 C 2 C_2 C 2 p = 2 p=2 p = 2
Proof. We have ∣ log Z G ∣ ≤ n log 2 + β max σ ∣ H G ( σ ) ∣ + ∣ h ∣ n |\log Z_G| \le n\log2+\beta\max_\sigma|H_G(\sigma)|+|h|n ∣ log Z G ∣ ≤ n log 2 + β max σ ∣ H G ( σ ) ∣ + ∣ h ∣ n ∣ ent ( m x ) ∣ ≤ log 2 |\operatorname{ent}(m_x)| \le\log2 ∣ ent ( m x ) ∣ ≤ log 2 0 ≤ ⟨ κ , S κ ⟩ ≤ ∑ x , y S x y = n 0 \le\langle\kappa,S\kappa\rangle\le\sum_{x,y}S_{xy}=n 0 ≤ ⟨ κ , S κ ⟩ ≤ ∑ x , y S x y = n κ ∈ [ 0 , 1 ] V \kappa\in[0,1]^V κ ∈ [ 0 , 1 ] V
∣ Θ G ( m ) ∣ ≤ β max σ ∣ H G ( σ ) ∣ + n ( ∣ h ∣ + log 2 + β 2 4 ) . |\Theta_G(m)| \le\beta\max_\sigma|H_G(\sigma)|+n\left(|h|+\log2+\frac{\beta^2}{4}\right). ∣ Θ G ( m ) ∣ ≤ β σ max ∣ H G ( σ ) ∣ + n ( ∣ h ∣ + log 2 + 4 β 2 ) . and, by the Minkowski inequality,
( E Θ G ( m ) 2 ) 1 / 2 ≤ β ( E max σ H G ( σ ) 2 ) 1 / 2 + n ( ∣ h ∣ + log 2 + β 2 4 ) ≤ C β , h ′ n . (99) \left(\mathbb{E}\,\Theta_G(m)^2\right)^{1/2}\le\beta\left(\mathbb{E}\max_\sigma H_G(\sigma)^2\right)^{1/2}+n\left(|h|+\log2+\frac{\beta^2}{4}\right)\le C'_{\beta,h}n. \tag*{(99)} ( E Θ G ( m ) 2 ) 1/2 ≤ β ( E σ max H G ( σ ) 2 ) 1/2 + n ( ∣ h ∣ + log 2 + 4 β 2 ) ≤ C β , h ′ n . (99) Let G = { p η ≥ 1 − ε } \mathcal{G}=\{p_\eta\ge1-\varepsilon\} G = { p η ≥ 1 − ε } p η = π m ( B η ) p_\eta=\pi_m(B_\eta) p η = π m ( B η ) Lemma 6.7 . Then G ∈ H \mathcal{G}\in\mathcal{H} G ∈ H Lemma 6.7 , we have
P ( G c ) = P ( 1 − p η > ε ) ≤ ε D ( η ) ε . (100) \mathbb{P}(\mathcal{G}^{\mathsf c})=\mathbb{P}(1-p_\eta>\varepsilon)\le\frac{\varepsilon_D(\eta)}{\varepsilon}. \tag*{(100)} P ( G c ) = P ( 1 − p η > ε ) ≤ ε ε D ( η ) . (100) On G \mathcal{G} G Lemma 6.8 gives
1 G log Z G ≥ 1 G ( Θ G ( m ) − c β η n + log ( 1 − ε ) ) + 1 G log L η ( g ) . (101) \mathbf{1}_{\mathcal{G}}\log Z_G\ge\mathbf{1}_{\mathcal{G}}\left(\Theta_G(m)-c_\beta\eta n+\log(1-\varepsilon)\right)+\mathbf{1}_{\mathcal{G}}\log L_\eta(g). \tag*{(101)} 1 G log Z G ≥ 1 G ( Θ G ( m ) − c β η n + log ( 1 − ε ) ) + 1 G log L η ( g ) . (101) We compute the expectation of the last term by conditioning on H \mathcal{H} H F ( ω , y ) = 1 G ( ω ) log L η ( y ) F(\omega,y)=\mathbf{1}_{\mathcal{G}}(\omega)\log L_\eta(y) F ( ω , y ) = 1 G ( ω ) log L η ( y ) F F F H ⊗ B ( R E ) \mathcal{H}\otimes\mathcal{B}(\mathbb{R}^E) H ⊗ B ( R E ) ∣ F ( ω , y ) ∣ ≤ β ( 8 n ) 1 / 2 ∥ y ∥ + 4 β 2 n |F(\omega,y)|\le\beta(8n)^{1/2}\lVert y\rVert+4\beta^2n ∣ F ( ω , y ) ∣ ≤ β ( 8 n ) 1/2 ∥ y ∥ + 4 β 2 n F ( ⋅ , g ) F(\cdot,g) F ( ⋅ , g ) Π g ˉ = 0 \Pi\bar{g}=0 Π g ˉ = 0 L η ( y ) L_\eta(y) L η ( y ) y y y Π y \Pi y Π y L η ( g ˉ + Π w ) = L η ( w ) L_\eta(\bar{g}+\Pi w)=L_\eta(w) L η ( g ˉ + Π w ) = L η ( w ) Lemma 6.4 (b) to the positive and negative parts of F F F
E [ 1 G log L η ( g ) ] = E [ 1 G ∫ log L η d P 0 ] . \mathbb{E}\left[\mathbf{1}_{\mathcal{G}}\log L_\eta(g)\right]=\mathbb{E}\left[\mathbf{1}_{\mathcal{G}}\int\log L_\eta\,\mathrm{d}P_0\right]. E [ 1 G log L η ( g ) ] = E [ 1 G ∫ log L η d P 0 ] . For each fixed ω ∈ G \omega\in\mathcal{G} ω ∈ G L η L_\eta L η L B , Π L_{B,\Pi} L B , Π m ( ω ) m(\omega) m ( ω ) Π ( ω ) \Pi(\omega) Π ( ω ) B = B η ( ω ) B = B_\eta(\omega) B = B η ( ω ) p = p η ( ω ) ≥ 1 − ε p = p_\eta(\omega) \ge1-\varepsilon p = p η ( ω ) ≥ 1 − ε D ( m ) \mathcal{D}(m) D ( m ) D \mathcal{D} D m ( ω ) m(\omega) m ( ω )
1 G ∫ log L η d P 0 ≥ − 4 β ( n ( D ( m ) + log 2 ) 1 − ε ) 1 / 2 . \mathbf{1}_{\mathcal{G}}\int\log L_\eta\,\mathrm{d}P_0 \ge-4\beta\left(\frac{n(\mathcal{D}(m)+\log2)}{1-\varepsilon}\right)^{1/2}. 1 G ∫ log L η d P 0 ≥ − 4 β ( 1 − ε n ( D ( m ) + log 2 ) ) 1/2 . Taking expectations and using Jensen’s inequality for the square root, we find
E [ 1 G log L η ( g ) ] ≥ − 4 β ( n ( E D ( m ) + log 2 ) 1 − ε ) 1 / 2 . (102) \mathbb{E}\left[\mathbf{1}_{\mathcal{G}}\log L_\eta(g)\right] \ge-4\beta\left(\frac{n(\mathbb{E}\mathcal{D}(m)+\log2)}{1-\varepsilon}\right)^{1/2}. \tag*{(102)} E [ 1 G log L η ( g ) ] ≥ − 4 β ( 1 − ε n ( E D ( m ) + log 2 ) ) 1/2 . (102) On G c \mathcal{G}^{c} G c W W W β H G ( σ ) + h ∑ x σ x \beta H_G(\sigma)+h\sum_x \sigma_x β H G ( σ ) + h ∑ x σ x { − 1 , 1 } V \{-1,1\}^{V} { − 1 , 1 } V log Z G ≥ n log 2 ≥ 0 \log Z_G \ge n\log2 \ge0 log Z G ≥ n log 2 ≥ 0 (101) and (102) , and using P ( G ) ≤ 1 \mathbb{P}(\mathcal{G}) \le1 P ( G ) ≤ 1 log ( 1 − ε ) < 0 \log(1-\varepsilon)<0 log ( 1 − ε ) < 0
E log Z G ≥ E [ 1 G Θ G ( m ) ] − c β η n + log ( 1 − ε ) − 4 β ( n ( E D ( m ) + log 2 ) 1 − ε ) 1 / 2 . \mathbb{E}\log Z_G \ge\mathbb{E}\left[\mathbf{1}_{\mathcal{G}}\Theta_G(m)\right]-c_\beta\eta n+\log(1-\varepsilon)-4\beta\left(\frac{n(\mathbb{E}\mathcal{D}(m)+\log2)}{1-\varepsilon}\right)^{1/2}. E log Z G ≥ E [ 1 G Θ G ( m ) ] − c β η n + log ( 1 − ε ) − 4 β ( 1 − ε n ( E D ( m ) + log 2 ) ) 1/2 . Finally, by the Cauchy–Schwarz inequality, (99) , and (100) ,
E [ 1 G Θ G ( m ) ] ≥ E Θ G ( m ) − P ( G c ) 1 / 2 ( E Θ G ( m ) 2 ) 1 / 2 ≥ E Θ G ( m ) − C β , h , n ′ ( ε D ( η ) ε ) 1 / 2 . \mathbb{E}\left[\mathbf{1}_{\mathcal{G}}\Theta_G(m)\right]\ge\mathbb{E}\Theta_G(m)-\mathbb{P}(\mathcal{G}^{c})^{1/2}\left(\mathbb{E}\Theta_G(m)^2\right)^{1/2}\ge\mathbb{E}\Theta_G(m)-C'_{\beta,h,n}\left(\frac{\varepsilon_D(\eta)}{\varepsilon}\right)^{1/2}. E [ 1 G Θ G ( m ) ] ≥ E Θ G ( m ) − P ( G c ) 1/2 ( E Θ G ( m ) 2 ) 1/2 ≥ E Θ G ( m ) − C β , h , n ′ ( ε ε D ( η ) ) 1/2 . Dividing by n n n (98) .
Proof of Theorem 6.2. Fix η > 0 \eta>0 η > 0 ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) b k b_k b k (98) for G = G k G=G_k G = G k p G k ( β , h ) ≥ n k − 1 E Θ G k ( m ( k ) ) − b k p_{G_k}(\beta,h)\ge n_k^{-1}\mathbb{E}\Theta_{G_k}(m^{(k)})-b_k p G k ( β , h ) ≥ n k − 1 E Θ G k ( m ( k ) ) − b k k → ∞ k\to\infty k → ∞ ε D k ( η ) → 0 \varepsilon_D{}_k(\eta)\to0 ε D k ( η ) → 0 D k → ∞ D_k\to\infty D k → ∞ β \beta β h h h T T T η \eta η n k − 1 log 2 n_k^{-1}\log2 n k − 1 log 2 n k − 1 log ( 1 − ε ) n_k^{-1}\log(1-\varepsilon) n k − 1 log ( 1 − ε ) n k ≥ D k + 1 → ∞ n_k\ge D_k+1\to\infty n k ≥ D k + 1 → ∞ (79) and the continuity of the square root imply that lim sup k b k ≤ c β η + 4 β ( δ / ( 1 − ε ) ) 1 / 2 \limsup_k b_k\le c_\beta\eta+4\beta(\delta/(1-\varepsilon))^{1/2} lim sup k b k ≤ c β η + 4 β ( δ / ( 1 − ε ) ) 1/2 (99) , the sequence n k − 1 E Θ G k ( m ( k ) ) n_k^{-1}\mathbb{E}\Theta_{G_k}(m^{(k)}) n k − 1 E Θ G k ( m ( k ) ) lim inf k ( a k − b k ) ≥ lim inf k a k − lim sup k b k \liminf_k(a_k-b_k)\ge\liminf_k a_k-\limsup_k b_k lim inf k ( a k − b k ) ≥ lim inf k a k − lim sup k b k
lim inf k → ∞ p G k ( β , h ) ≥ lim inf k → ∞ 1 n k E Θ G k ( m ( k ) ) − c β η − 4 β ( δ 1 − ε ) 1 / 2 . \liminf_{k\to\infty}p_{G_k}(\beta,h)\ge\liminf_{k\to\infty}\frac{1}{n_k}\mathbb{E}\Theta_{G_k}(m^{(k)})-c_\beta\eta-4\beta\left(\frac{\delta}{1-\varepsilon}\right)^{1/2}. k → ∞ lim inf p G k ( β , h ) ≥ k → ∞ lim inf n k 1 E Θ G k ( m ( k ) ) − c β η − 4 β ( 1 − ε δ ) 1/2 . Letting η ↓ 0 \eta\downarrow0 η ↓ 0 ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 (80) .
Remark 6.13. The requirement that the center is itself a query is used through Lemma 6.4(a): the vector W m Wm W m r r r H \mathcal{H} H B η B_\eta B η L η L_\eta L η H \mathcal{H} H W m Wm W m ⟨ β W m , v ⟩ \langle\beta Wm,v\rangle ⟨ β W m , v ⟩ H \mathcal{H} H L η L_\eta L η
Relative entropy of the planted product model
The goal of this section is to prove Theorem 7.5 , which states that the relative entropy D ( m ) D(m) D ( m ) n n n ψ m x \psi_{m_x} ψ m x ψ μ \psi_\mu ψ μ μ \mu μ μ \mu μ Γ μ ≤ 0 \Gamma_\mu\le0 Γ μ ≤ 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] μ + \mu_+ μ + Γ μ + ( s ) ≤ 0 \Gamma_{\mu_+}(s) \le0 Γ μ + ( s ) ≤ 0 s ≥ 0 s \ge0 s ≥ 0
The planted product model is a spiked Wigner model with a sparse variance profile and a prior that depends on the vertex. For the spiked Wigner model with independent and identically distributed noise, the limit of the relative entropy, or equivalently of the mutual information between the spike and the observation, was computed in [98 ] , where earlier work is discussed, and for bounded centered unit-variance priors, El Alaoui, Krzakala, and Jordan determined the region in which the spiked and the unspiked models are contiguous [59 ] . Guionnet, Ko, Krzakala, and Zdeborová computed this limit for inhomogeneous noise whose inverse variances form a positive-semidefinite profile with a block structure or a regular limit [78 ] . Behne and Reeves computed it when the signal-to-noise ratio is constant on each of finitely many blocks, for every nonnegative matrix of block ratios [23 ] . On the Nishimori line, multispecies spin glasses are spiked models with binary signals and a block variance profile. Alberici, Camilli, Contucci, and Mingione computed their free energy when the interaction is elliptic [4 ] , and for the deep Boltzmann machine, whose interaction matrix is not positive semidefinite [3 ] . For general variance profiles and certain priors with independent and identically distributed symmetric coordinates of variance at most one, De and Kunisky proved that strong detection is impossible below a spectral threshold; their assumptions include a 1-moment-subgaussian condition on the product of two independent samples of the prior and, in our setting, require D D D n n n [52 ] , Theorem 1.11.
Interpolations of Guerra type [76 ] between a matrix channel and scalar channels were used by Korada and Macris [94 ] for p p p [97 ] for spiked models. Barbier and Macris made the interpolation path adaptive [14 ] . Our proof of Theorem 7.5 follows the Hamilton–Jacobi approach of Mourrat [104 , 105 ] , in which the relative entropy of a family of interpolating channels is compared with the solution of a Hamilton–Jacobi equation given by a Hopf–Lax formula. For inference of matrix tensor products, Chen and Xia showed that the solution of the corresponding Hamilton–Jacobi equation is an upper bound on the limit of the free energy for every interaction matrix, also when the nonlinearity is not convex [35 ] . We need only an upper bound, the entries of S S S 1 / D 1/D 1/ D S S S Γ μ \Gamma_\mu Γ μ [41 ] ; see the discussion after (110) .
Section 7.1 collects identities for Gaussian channels. Section 7.2 defines the scalar transform and proves Lemma 7.2 . Section 7.3 proves Proposition 7.4, and Section 7.4 states Theorem 7.5 and Corollary 7.6. The remaining subsections prove Theorem 7.5 . In Section 7.5, we interpolate between independent scalar channels at the vertices and the planted edge channel. We compare the normalized relative entropy F ( t , q ) F(t,q) F ( t , q ) F F F F F F F F F
Throughout this section, S S S n × n n \times n n × n Definition 6.1 . We index its rows and columns by a set V V V ∣ V ∣ = n |V| = n ∣ V ∣ = n E = { { x , y } : S x y > 0 } E = \{\{x,y\}: S_{xy} > 0\} E = {{ x , y } : S x y > 0 } S S S w ∈ R V w \in\mathbb{R}^{V} w ∈ R V ∣ w ∣ |w| ∣ w ∣ ∥ w ∥ n = n − 1 / 2 ∣ w ∣ \|w\|_{n} = n^{-1/2}|w| ∥ w ∥ n = n − 1/2 ∣ w ∣ dist n ( w , A ) = inf z ∈ A ∥ w − z ∥ n \operatorname{dist}_{n}(w,A) = \inf_{z \in A}\|w-z\|_{n} dist n ( w , A ) = inf z ∈ A ∥ w − z ∥ n A ⊂ R V A \subset\mathbb{R}^{V} A ⊂ R V 1 \mathbf{1} 1 w ⋅ w ′ w \cdot w' w ⋅ w ′ w ⊙ w ′ w \odot w' w ⊙ w ′ w ⊙ 2 = w ⊙ w w^{\odot2} = w \odot w w ⊙ 2 = w ⊙ w A ⊂ R V A \subset\mathbb{R}^{V} A ⊂ R V c ∈ R c \in\mathbb{R} c ∈ R c A = { c w : w ∈ A } cA = \{cw : w \in A\} c A = { c w : w ∈ A } S A = { S w : w ∈ A } SA = \{Sw : w \in A\} S A = { S w : w ∈ A } max x , y S x y ≤ 1 / D \max_{x,y} S_{xy} \le1/D max x , y S x y ≤ 1/ D D ≥ 1 D \ge1 D ≥ 1
∥ S ∥ ≤ 1 , tr S 2 ≤ n D , max x ∈ V ( S 2 ) x x ≤ 1 D , n ≥ D + 1. (103) \|S\| \le1,\qquad\operatorname{tr} S^{2} \le\frac{n}{D},\qquad\max_{x \in V}(S^{2})_{xx} \le\frac{1}{D},\qquad n \ge D+1. \tag*{(103)} ∥ S ∥ ≤ 1 , tr S 2 ≤ D n , x ∈ V max ( S 2 ) xx ≤ D 1 , n ≥ D + 1. (103) Indeed, the bound on the operator norm follows from Jensen’s inequality as in (4) , and ( S 2 ) x x = ∑ y S x y 2 ≤ D − 1 ∑ y S x y = D − 1 (S^{2})_{xx} = \sum_y S_{xy}^{2} \le D^{-1}\sum_y S_{xy} = D^{-1} ( S 2 ) xx = ∑ y S x y 2 ≤ D − 1 ∑ y S x y = D − 1 x x x S S S 1 / D 1/D 1/ D D D D
Gaussian channels
We begin with identities relating relative entropy and posterior moments for Gaussian channels. The first derivative formula in Lemma 7.1 is the relation of Guo, Shamai, and Verdú [79 ] between mutual information and the minimum mean-square error, written for relative entropy. The second derivatives in Lemma 7.1 (c) are a special case of the Hessian formula of Payaró and Palomar [120 Theorem 5] . We include the short proofs for completeness.
Let N ≥ 1 N \ge1 N ≥ 1 θ \theta θ R N \mathbb{R}^{N} R N Z Z Z R N \mathbb{R}^{N} R N θ \theta θ λ ∈ [ 0 , ∞ ) N \lambda\in[0,\infty)^{N} λ ∈ [ 0 , ∞ ) N Y λ = ( λ j θ j + Z j ) j ≤ N Y^{\lambda} = (\sqrt{\lambda_{j}}\theta_{j} + Z_{j})_{j \le N} Y λ = ( λ j θ j + Z j ) j ≤ N θ ′ \theta^{\prime} θ ′ θ \theta θ E ′ \mathbb{E}^{\prime} E ′ θ ′ \theta^{\prime} θ ′ Y λ Y^{\lambda} Y λ
L λ ( y ) = E ′ exp ( ∑ j ≤ N ( λ j y j θ j ′ − λ j ( θ j ′ ) 2 / 2 ) ) ( y ∈ R N ) (104) L_{\lambda}(y) = \mathbb{E}^{\prime}\exp\left(\sum_{j \le N}\left(\sqrt{\lambda_{j}}y_{j}\theta_{j}^{\prime} - \lambda_{j}(\theta_{j}^{\prime})^{2}/2\right)\right) \qquad(y \in\mathbb{R}^{N}) \tag*{(104)} L λ ( y ) = E ′ exp j ≤ N ∑ ( λ j y j θ j ′ − λ j ( θ j ′ ) 2 /2 ) ( y ∈ R N ) (104) with respect to the standard Gaussian law N ( 0 , I N ) N(0,I_{N}) N ( 0 , I N )
I ( λ ) = KL ( Law ( Y λ ) ∥ N ( 0 , I N ) ) = E log L λ ( Y λ ) . I(\lambda) = \operatorname{KL}(\operatorname{Law}(Y^{\lambda}) \,\|\, N(0,I_{N})) = \mathbb{E}\log L_{\lambda}(Y^{\lambda}). I ( λ ) = KL ( Law ( Y λ ) ∥ N ( 0 , I N )) = E log L λ ( Y λ ) . By Bayes’s rule, the conditional law of θ \theta θ Y λ = y Y^{\lambda}=y Y λ = y θ ′ \theta^{\prime} θ ′ (104) . Given Y λ Y^{\lambda} Y λ θ 1 , θ 2 , … \theta^{1},\theta^{2},\ldots θ 1 , θ 2 , … ⟨ ⋅ ⟩ λ \langle\cdot\rangle_{\lambda} ⟨ ⋅ ⟩ λ ⟨ θ j 1 ⟩ λ = E [ θ j ∣ Y λ ] \langle\theta_{j}^{1}\rangle_{\lambda}=\mathbb{E}[\theta_{j}\mid Y^{\lambda}] ⟨ θ j 1 ⟩ λ = E [ θ j ∣ Y λ ] θ \theta θ θ 1 \theta^{1} θ 1 Y λ Y^{\lambda} Y λ [110 ]
E ⟨ f ( θ , θ 1 , … , θ k , Y λ ) ⟩ λ = E ⟨ f ( θ k + 1 , θ 1 , … , θ k , Y λ ) ⟩ λ (105) \mathbb{E}\left\langle f(\theta,\theta^{1},\ldots,\theta^{k},Y^{\lambda})\right\rangle_{\lambda}=\mathbb{E}\left\langle f(\theta^{k+1},\theta^{1},\ldots,\theta^{k},Y^{\lambda})\right\rangle_{\lambda} \tag*{(105)} E ⟨ f ( θ , θ 1 , … , θ k , Y λ ) ⟩ λ = E ⟨ f ( θ k + 1 , θ 1 , … , θ k , Y λ ) ⟩ λ (105) for all k ≥ 0 k \ge0 k ≥ 0 f f f (105) only for deterministic λ \lambda λ Y λ = ( λ j θ j + Z j ) j Y^{\lambda}=(\sqrt{\lambda_{j}}\theta_{j}+Z_{j})_{j} Y λ = ( λ j θ j + Z j ) j
I ( λ ) = E log E ′ exp ( ∑ j ≤ N ( λ j Z j θ j ′ + λ j θ j θ j ′ − λ j ( θ j ′ ) 2 / 2 ) ) (106) I(\lambda)=\mathbb{E}\log\mathbb{E}^{\prime}\exp\left(\sum_{j \le N}\left(\sqrt{\lambda_{j}}Z_{j}\theta_{j}^{\prime}+\lambda_{j}\theta_{j}\theta_{j}^{\prime}-\lambda_{j}(\theta_{j}^{\prime})^{2}/2\right)\right) \tag*{(106)} I ( λ ) = E log E ′ exp j ≤ N ∑ ( λ j Z j θ j ′ + λ j θ j θ j ′ − λ j ( θ j ′ ) 2 /2 ) (106) and ⟨ ⋅ ⟩ λ \langle\cdot\rangle_{\lambda} ⟨ ⋅ ⟩ λ (106) , with θ ′ \theta^{\prime} θ ′ ( θ , Z ) (\theta,Z) ( θ , Z ) f f f
∂ Z j ⟨ f ( θ 1 ) ⟩ λ = λ j ( ⟨ f ( θ 1 ) θ j 1 ⟩ λ − ⟨ f ( θ 1 ) ⟩ λ ⟨ θ j 1 ⟩ λ ) , ∂ λ j ⟨ f ( θ 1 ) ⟩ λ = ⟨ f ( θ 1 ) A j ( θ 1 ) ⟩ λ − ⟨ f ( θ 1 ) ⟩ λ ⟨ A j ( θ 1 ) ⟩ λ , (107) \begin{aligned}
\partial_{Z_{j}}\langle f(\theta^{1})\rangle_{\lambda} &= \sqrt{\lambda_{j}}\left(\langle f(\theta^{1})\theta_{j}^{1}\rangle_{\lambda}-\langle f(\theta^{1})\rangle_{\lambda}\langle\theta_{j}^{1}\rangle_{\lambda}\right), \\
\partial_{\lambda_{j}}\langle f(\theta^{1})\rangle_{\lambda} &= \langle f(\theta^{1})A_{j}(\theta^{1})\rangle_{\lambda}-\langle f(\theta^{1})\rangle_{\lambda}\langle A_{j}(\theta^{1})\rangle_{\lambda}, \tag*{(107)}
\end{aligned} ∂ Z j ⟨ f ( θ 1 ) ⟩ λ ∂ λ j ⟨ f ( θ 1 ) ⟩ λ = λ j ( ⟨ f ( θ 1 ) θ j 1 ⟩ λ − ⟨ f ( θ 1 ) ⟩ λ ⟨ θ j 1 ⟩ λ ) , = ⟨ f ( θ 1 ) A j ( θ 1 ) ⟩ λ − ⟨ f ( θ 1 ) ⟩ λ ⟨ A j ( θ 1 ) ⟩ λ , (107) where the second identity holds for λ j > 0 \lambda_{j}>0 λ j > 0
A j ( θ 1 ) = Z j θ j 1 2 λ j + θ j θ j 1 − ( θ j 1 ) 2 2 . A_{j}(\theta^{1})=\frac{Z_{j}\theta_{j}^{1}}{2\sqrt{\lambda_{j}}}+\theta_{j}\theta_{j}^{1}-\frac{(\theta_{j}^{1})^{2}}{2}. A j ( θ 1 ) = 2 λ j Z j θ j 1 + θ j θ j 1 − 2 ( θ j 1 ) 2 . Lemma 7.1. Let N N N θ \theta θ I I I c θ = max j ∥ θ j ∥ ∞ c_\theta= \max_j \lVert\theta_j\rVert_\infty c θ = max j ∥ θ j ∥ ∞
(a) The function I I I [ 0 , ∞ ) N [0,\infty)^N [ 0 , ∞ ) N I ( 0 ) = 0 I(0)=0 I ( 0 ) = 0 I I I
(b) Let j ≤ N j\leq N j ≤ N λ ∈ [ 0 , ∞ ) N \lambda\in[0,\infty)^N λ ∈ [ 0 , ∞ ) N λ j > 0 \lambda_j>0 λ j > 0 ∂ λ j I ( λ ) \partial_{\lambda_j}I(\lambda) ∂ λ j I ( λ )
∂ λ j I ( λ ) = 1 2 E ⟨ θ j 1 ⟩ λ 2 ∈ [ 0 , 1 2 E θ j 2 ] . \partial_{\lambda_j}I(\lambda)=\frac{1}{2}\mathbb{E}\langle\theta_j^1\rangle_\lambda^2\in\left[0,\frac{1}{2}\mathbb{E}\theta_j^2\right]. ∂ λ j I ( λ ) = 2 1 E ⟨ θ j 1 ⟩ λ 2 ∈ [ 0 , 2 1 E θ j 2 ] . The function λ ↦ E ⟨ θ j 1 ⟩ λ 2 \lambda\mapsto\mathbb{E}\langle\theta_j^1\rangle_\lambda^2 λ ↦ E ⟨ θ j 1 ⟩ λ 2 [ 0 , ∞ ) N [0,\infty)^N [ 0 , ∞ ) N
(c) Let j , k ≤ N j,k\leq N j , k ≤ N λ ∈ [ 0 , ∞ ) N \lambda\in[0,\infty)^N λ ∈ [ 0 , ∞ ) N λ j > 0 \lambda_j>0 λ j > 0 λ k > 0 \lambda_k>0 λ k > 0
∂ λ k ∂ λ j I ( λ ) = 1 2 E ( ⟨ θ j 1 θ k 1 ⟩ λ − ⟨ θ j 1 ⟩ λ ⟨ θ k 1 ⟩ λ ) 2 . \partial_{\lambda_k}\partial_{\lambda_j}I(\lambda)=\frac{1}{2}\mathbb{E}\left(\langle\theta_j^1\theta_k^1\rangle_\lambda-\langle\theta_j^1\rangle_\lambda\langle\theta_k^1\rangle_\lambda\right)^2. ∂ λ k ∂ λ j I ( λ ) = 2 1 E ( ⟨ θ j 1 θ k 1 ⟩ λ − ⟨ θ j 1 ⟩ λ ⟨ θ k 1 ⟩ λ ) 2 . and the right side is continuous on [ 0 , ∞ ) N [0,\infty)^N [ 0 , ∞ ) N
Proof. (a) The exponent in (106) is bounded in absolute value by ∑ j ( c θ λ j ∣ Z j ∣ + 3 c θ 2 λ j / 2 ) \sum_j(c_\theta\sqrt{\lambda_j}|Z_j|+3c_\theta^2\lambda_j/2) ∑ j ( c θ λ j ∣ Z j ∣ + 3 c θ 2 λ j /2 ) θ ′ \theta' θ ′ λ \lambda λ λ \lambda λ I I I I ≥ 0 I\geq0 I ≥ 0 I ( 0 ) = 0 I(0)=0 I ( 0 ) = 0
(b) Let λ j > 0 \lambda_j>0 λ j > 0 λ ′ \lambda' λ ′ λ \lambda λ λ j ′ ≥ λ j / 2 \lambda'_j\geq\lambda_j/2 λ j ′ ≥ λ j /2 λ j \lambda_j λ j (106) is ⟨ A j ( θ 1 ) ⟩ λ ′ \langle A_j(\theta^1)\rangle_{\lambda'} ⟨ A j ( θ 1 ) ⟩ λ ′ C ( 1 + ∣ Z j ∣ ) C(1+|Z_j|) C ( 1 + ∣ Z j ∣ ) C C C c θ c_\theta c θ λ j \lambda_j λ j ∂ λ j I = E ⟨ A j ( θ 1 ) ⟩ λ \partial_{\lambda_j}I=\mathbb{E}\langle A_j(\theta^1)\rangle_\lambda ∂ λ j I = E ⟨ A j ( θ 1 ) ⟩ λ (107) and Gaussian integration by parts (F1) in Z j Z_j Z j θ \theta θ ( Z k ) k ≠ j (Z_k)_{k\ne j} ( Z k ) k = j
E [ Z j ⟨ θ j 1 ⟩ λ ] = λ j E [ ⟨ ( θ j 1 ) 2 ⟩ λ − ⟨ θ j 1 ⟩ λ 2 ] . \mathbb{E}\left[Z_j\langle\theta_j^1\rangle_\lambda\right]=\sqrt{\lambda_j}\mathbb{E}\left[\langle(\theta_j^1)^2\rangle_\lambda-\langle\theta_j^1\rangle_\lambda^2\right]. E [ Z j ⟨ θ j 1 ⟩ λ ] = λ j E [ ⟨( θ j 1 ) 2 ⟩ λ − ⟨ θ j 1 ⟩ λ 2 ] . Inserting this identity into the expression for the derivative above, we obtain
∂ λ j I = 1 2 E ⟨ ( θ j 1 ) 2 ⟩ λ − 1 2 E ⟨ θ j 1 ⟩ λ 2 + E [ θ j ⟨ θ j 1 ⟩ λ ] − 1 2 E ⟨ ( θ j 1 ) 2 ⟩ λ = E [ θ j ⟨ θ j 1 ⟩ λ ] − 1 2 E ⟨ θ j 1 ⟩ λ 2 . \partial_{\lambda_j}I=\frac{1}{2}\mathbb{E}\langle(\theta_j^1)^2\rangle_\lambda-\frac{1}{2}\mathbb{E}\langle\theta_j^1\rangle_\lambda^2+\mathbb{E}\left[\theta_j\langle\theta_j^1\rangle_\lambda\right]-\frac{1}{2}\mathbb{E}\langle(\theta_j^1)^2\rangle_\lambda=\mathbb{E}\left[\theta_j\langle\theta_j^1\rangle_\lambda\right]-\frac{1}{2}\mathbb{E}\langle\theta_j^1\rangle_\lambda^2. ∂ λ j I = 2 1 E ⟨( θ j 1 ) 2 ⟩ λ − 2 1 E ⟨ θ j 1 ⟩ λ 2 + E [ θ j ⟨ θ j 1 ⟩ λ ] − 2 1 E ⟨( θ j 1 ) 2 ⟩ λ = E [ θ j ⟨ θ j 1 ⟩ λ ] − 2 1 E ⟨ θ j 1 ⟩ λ 2 . By (105) , E [ θ j ⟨ θ j 1 ⟩ λ ] = E ⟨ θ j 2 θ j 1 ⟩ λ = E ⟨ θ j 1 ⟩ λ 2 \mathbb{E}[\theta_j\langle\theta_j^1\rangle_\lambda]=\mathbb{E}\langle\theta_j^2\theta_j^1\rangle_\lambda=\mathbb{E}\langle\theta_j^1\rangle_\lambda^2 E [ θ j ⟨ θ j 1 ⟩ λ ] = E ⟨ θ j 2 θ j 1 ⟩ λ = E ⟨ θ j 1 ⟩ λ 2 ⟨ θ j 1 ⟩ λ = E [ θ j ∣ Y λ ] \langle\theta_j^1\rangle_\lambda=\mathbb{E}[\theta_j\mid Y^\lambda] ⟨ θ j 1 ⟩ λ = E [ θ j ∣ Y λ ] E ⟨ θ j 1 ⟩ λ 2 ≤ E θ j 2 \mathbb{E}\langle\theta_j^1\rangle_\lambda^2\leq\mathbb{E}\theta_j^2 E ⟨ θ j 1 ⟩ λ 2 ≤ E θ j 2 ( θ , Z ) (\theta,Z) ( θ , Z ) ⟨ θ j 1 ⟩ λ \langle\theta_j^1\rangle_\lambda ⟨ θ j 1 ⟩ λ λ ∈ [ 0 , ∞ ) N \lambda\in[0,\infty)^N λ ∈ [ 0 , ∞ ) N c θ c_\theta c θ
(c) Let λ j , λ k > 0 \lambda_j,\lambda_k>0 λ j , λ k > 0 (105) , 2 ∂ λ j I ( λ ′ ) = E [ θ j ⟨ θ j 1 ⟩ λ ′ ] 2\partial_{\lambda_j}I(\lambda')=\mathbb{E}[\theta_j\langle\theta_j^1\rangle_{\lambda'}] 2 ∂ λ j I ( λ ′ ) = E [ θ j ⟨ θ j 1 ⟩ λ ′ ] λ ′ \lambda' λ ′ λ \lambda λ λ k \lambda_k λ k b l = ⟨ θ l 1 ⟩ λ b_l=\langle\theta_l^1\rangle_\lambda b l = ⟨ θ l 1 ⟩ λ c j k = ⟨ θ j 1 θ k 1 ⟩ λ c_{jk}=\langle\theta_j^1\theta_k^1\rangle_\lambda c j k = ⟨ θ j 1 θ k 1 ⟩ λ c j k k = ⟨ θ j 1 ( θ k 1 ) 2 ⟩ λ c_{jkk}=\langle\theta_j^1(\theta_k^1)^2\rangle_\lambda c j k k = ⟨ θ j 1 ( θ k 1 ) 2 ⟩ λ (107) , we have 2 ∂ λ k ∂ λ j I = T 1 + T 2 + T 3 2\partial_{\lambda_k}\partial_{\lambda_j}I=T_1+T_2+T_3 2 ∂ λ k ∂ λ j I = T 1 + T 2 + T 3
T 1 = 1 2 λ k E [ θ j Z k ( c j k − b j b k ) ] , T 2 = E [ θ j θ k ( c j k − b j b k ) ] , T 3 = − 1 2 E [ θ j ( c j k k − b j c k k ) ] . \begin{aligned}
T_1&=\frac{1}{2\sqrt{\lambda_k}}\mathbb{E}\left[\theta_j Z_k(c_{jk}-b_jb_k)\right],\qquad T_2=\mathbb{E}\left[\theta_j\theta_k(c_{jk}-b_jb_k)\right],\\
T_3&=-\frac{1}{2}\mathbb{E}\left[\theta_j(c_{jkk}-b_jc_{kk})\right].
\end{aligned} T 1 T 3 = 2 λ k 1 E [ θ j Z k ( c j k − b j b k ) ] , T 2 = E [ θ j θ k ( c j k − b j b k ) ] , = − 2 1 E [ θ j ( c j k k − b j c k k ) ] . By (107) , we have ∂ Z k ( c j k − b j b k ) = λ k ( c j k k − b j c k k − 2 c j k b k + 2 b j b k 2 ) \partial_{Z_k}(c_{jk}-b_jb_k)=\sqrt{\lambda_k}(c_{jkk}-b_jc_{kk}-2c_{jk}b_k+2b_jb_k^2) ∂ Z k ( c j k − b j b k ) = λ k ( c j k k − b j c k k − 2 c j k b k + 2 b j b k 2 ) Z k Z_k Z k
T 1 = 1 2 E [ θ j ( c j k k − b j c k k ) ] − E [ θ j ( c j k b k − b j b k 2 ) ] . T_1=\frac{1}{2}\mathbb{E}\left[\theta_j(c_{jkk}-b_jc_{kk})\right]-\mathbb{E}\left[\theta_j(c_{jk}b_k-b_jb_k^2)\right]. T 1 = 2 1 E [ θ j ( c j k k − b j c k k ) ] − E [ θ j ( c j k b k − b j b k 2 ) ] . The quantities b l b_l b l c j k c_{jk} c j k c j k k c_{jkk} c j k k Y λ Y^\lambda Y λ E [ θ j ∣ Y λ ] = b j \mathbb{E}[\theta_j\mid Y^\lambda]=b_j E [ θ j ∣ Y λ ] = b j E [ θ j θ k ∣ Y λ ] = c j k \mathbb{E}[\theta_j\theta_k\mid Y^\lambda]=c_{jk} E [ θ j θ k ∣ Y λ ] = c j k
T 1 + T 3 = − E [ b j c j k b k − b j 2 b k 2 ] , T 2 = E [ c j k 2 − c j k b j b k ] , T_1+T_3=-\mathbb{E}\left[b_jc_{jk}b_k-b_j^2b_k^2\right],\qquad T_2=\mathbb{E}\left[c_{jk}^2-c_{jk}b_jb_k\right], T 1 + T 3 = − E [ b j c j k b k − b j 2 b k 2 ] , T 2 = E [ c j k 2 − c j k b j b k ] , and T 1 + T 2 + T 3 = E ( c j k − b j b k ) 2 T_1+T_2+T_3=\mathbb{E}(c_{jk}-b_jb_k)^2 T 1 + T 2 + T 3 = E ( c j k − b j b k ) 2
The scalar transform
For m ∈ [ − 1 , 1 ] m\in[-1,1] m ∈ [ − 1 , 1 ] ν m \nu_m ν m σ − m \sigma-m σ − m σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 } m m m ν m \nu_m ν m ( 1 + m ) / 2 (1+m)/2 ( 1 + m ) /2 1 − m 1-m 1 − m ( 1 − m ) / 2 (1-m)/2 ( 1 − m ) /2 − 1 − m -1-m − 1 − m κ ( m ) = 1 − m 2 \kappa(m)=1-m^2 κ ( m ) = 1 − m 2 m = ± 1 m=\pm1 m = ± 1 r ≥ 0 r\ge0 r ≥ 0
ψ m ( r ) = E log E τ ∼ ν m exp ( r Z τ + r τ τ ∗ − r 2 τ 2 ) , (108) \psi_m(r)=\mathbb{E}\log\mathbb{E}_{\tau\sim\nu_m}\exp\left(\sqrt{r}Z\tau+r\tau\tau^*-\frac{r}{2}\tau^2\right), \tag*{(108)} ψ m ( r ) = E log E τ ∼ ν m exp ( r Z τ + r τ τ ∗ − 2 r τ 2 ) , (108) where the outer expectation is over independent τ ∗ ∼ ν m \tau^*\sim\nu_m τ ∗ ∼ ν m Z ∼ N ( 0 , 1 ) Z\sim N(0,1) Z ∼ N ( 0 , 1 ) (106) , ψ m ( r ) \psi_m(r) ψ m ( r ) I I I N = 1 N=1 N = 1 θ = τ ∗ \theta=\tau^* θ = τ ∗ ψ m ( r ) = KL ( Law ( r τ ∗ + Z ) ∥ N ( 0 , 1 ) ) \psi_m(r)=\operatorname{KL}(\operatorname{Law}(\sqrt{r}\tau^*+Z)\mathbin\Vert N(0,1)) ψ m ( r ) = KL ( Law ( r τ ∗ + Z ) ∥ N ( 0 , 1 )) μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
ψ μ ( r ) = ∫ ψ m ( r ) μ ( d m ) , Γ μ ( s ) = 2 [ ψ μ ( β 2 s ) − β 2 s 2 4 ] ( r , s ≥ 0 ) . \psi_\mu(r)=\int\psi_m(r)\,\mu(\mathrm{d}m),\qquad\Gamma_\mu(s)=2\left[\psi_\mu(\beta^2s)-\frac{\beta^2s^2}{4}\right]\qquad(r,s\ge0). ψ μ ( r ) = ∫ ψ m ( r ) μ ( d m ) , Γ μ ( s ) = 2 [ ψ μ ( β 2 s ) − 4 β 2 s 2 ] ( r , s ≥ 0 ) . Lemma 7.2. Let m ∈ [ − 1 , 1 ] m\in[-1,1] m ∈ [ − 1 , 1 ] r ≥ 0 r\ge0 r ≥ 0 b m , r = E [ τ ∗ ∣ r τ ∗ + Z ] b_{m,r}=\mathbb{E}[\tau^*\mid\sqrt{r}\tau^*+Z] b m , r = E [ τ ∗ ∣ r τ ∗ + Z ] τ ∗ \tau^* τ ∗ Z Z Z (108) .
(a) We have ψ m ≥ 0 \psi_m\ge0 ψ m ≥ 0 ψ m ( 0 ) = 0 \psi_m(0)=0 ψ m ( 0 ) = 0
(b) The function ψ m \psi_m ψ m [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
ψ m ′ ( r ) = 1 2 E b m , r 2 ∈ [ 0 , κ ( m ) 2 ] . \psi_m'(r)=\frac{1}{2}\mathbb{E}b_{m,r}^2\in\left[0,\frac{\kappa(m)}{2}\right]. ψ m ′ ( r ) = 2 1 E b m , r 2 ∈ [ 0 , 2 κ ( m ) ] . (c) The function ψ m \psi_m ψ m 1 / 2 1/2 1/2
(d) The function ( m , r ) ↦ ψ m ( r ) (m,r) \mapsto\psi_m(r) ( m , r ) ↦ ψ m ( r ) [ − 1 , 1 ] × [ 0 , ∞ ) [-1,1] \times[0,\infty) [ − 1 , 1 ] × [ 0 , ∞ )
Proof. Parts (a) and (b) are Lemma 7.1(a,b) with N = 1 N=1 N = 1 θ = τ ∗ \theta=\tau^* θ = τ ∗ E ( τ ∗ ) 2 = κ ( m ) \mathbb{E}(\tau^*)^2=\kappa(m) E ( τ ∗ ) 2 = κ ( m ) r = 0 r=0 r = 0 lim r ↓ 0 ψ m ′ ( r ) \lim_{r\downarrow0}\psi_m'(r) lim r ↓ 0 ψ m ′ ( r ) κ ( m ) ≤ 1 \kappa(m)\le1 κ ( m ) ≤ 1
For (d), let r ≥ 0 r\ge0 r ≥ 0
c ± = ± r ( 1 ∓ m ) , ℓ m , r ( w ) = 1 + m 2 e c + w − c + 2 / 2 + 1 − m 2 e c − w − c − 2 / 2 ( w ∈ R ) . c_{\pm}=\pm\sqrt{r}(1\mp m), \qquad\ell_{m,r}(w)=\frac{1+m}{2}e^{c_+w-c_+^2/2}+\frac{1-m}{2}e^{c_-w-c_-^2/2}\qquad(w\in\mathbb{R}). c ± = ± r ( 1 ∓ m ) , ℓ m , r ( w ) = 2 1 + m e c + w − c + 2 /2 + 2 1 − m e c − w − c − 2 /2 ( w ∈ R ) . By (104) , the function ℓ m , r \ell_{m,r} ℓ m , r r τ ∗ + Z \sqrt{r}\tau^*+Z r τ ∗ + Z N ( 0 , 1 ) N(0,1) N ( 0 , 1 ) ψ m ( r ) = E [ ℓ m , r ( Z ) log ℓ m , r ( Z ) ] \psi_m(r)=\mathbb{E}[\ell_{m,r}(Z)\log\ell_{m,r}(Z)] ψ m ( r ) = E [ ℓ m , r ( Z ) log ℓ m , r ( Z )] ( m , r , w ) ↦ ℓ m , r ( w ) (m,r,w)\mapsto\ell_{m,r}(w) ( m , r , w ) ↦ ℓ m , r ( w ) ℓ m , r ( w ) \ell_{m,r}(w) ℓ m , r ( w ) e c ± w − c ± 2 / 2 e^{c_{\pm}w-c_{\pm}^2/2} e c ± w − c ± 2 /2 ∣ c ± ∣ ≤ 2 r |c_{\pm}|\le2\sqrt{r} ∣ c ± ∣ ≤ 2 r ∣ log ℓ m , r ( w ) ∣ ≤ 2 r ∣ w ∣ + 2 r |\log\ell_{m,r}(w)|\le2\sqrt{r}|w|+2r ∣ log ℓ m , r ( w ) ∣ ≤ 2 r ∣ w ∣ + 2 r
∣ ℓ m , r ( w ) log ℓ m , r ( w ) ∣ ≤ ( 2 r ∣ w ∣ + 2 r ) e 2 r ∣ w ∣ . |\ell_{m,r}(w)\log\ell_{m,r}(w)|\le(2\sqrt{r}|w|+2r)e^{2\sqrt{r}|w|}. ∣ ℓ m , r ( w ) log ℓ m , r ( w ) ∣ ≤ ( 2 r ∣ w ∣ + 2 r ) e 2 r ∣ w ∣ . For r r r w w w ( m , r ) (m,r) ( m , r ) N ( 0 , 1 ) N(0,1) N ( 0 , 1 )
By Lemma 7.2, for each probability measure μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ψ μ \psi_\mu ψ μ ψ μ ( 0 ) = 0 \psi_\mu(0)=0 ψ μ ( 0 ) = 0 ψ μ \psi_\mu ψ μ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
ψ μ ′ ( r ) = ∫ ψ m ′ ( r ) μ ( d m ) ∈ [ 0 , 1 2 ∫ κ ( m ) μ ( d m ) ] ⊂ [ 0 , 1 2 ] . (109) \psi_\mu'(r)=\int\psi_m'(r)\,\mu(dm)\in\left[0,\frac{1}{2}\int\kappa(m)\,\mu(dm)\right]\subset\left[0,\frac{1}{2}\right]. \tag*{(109)} ψ μ ′ ( r ) = ∫ ψ m ′ ( r ) μ ( d m ) ∈ [ 0 , 2 1 ∫ κ ( m ) μ ( d m ) ] ⊂ [ 0 , 2 1 ] . (109) Indeed, m ↦ ψ m ′ ( u ) m\mapsto\psi_m'(u) m ↦ ψ m ′ ( u ) m m m ψ μ ( r ) = ∫ 0 r ∫ ψ m ′ ( u ) μ ( d m ) d u \psi_\mu(r)=\int_0^r\int\psi_m'(u)\,\mu(dm)\,du ψ μ ( r ) = ∫ 0 r ∫ ψ m ′ ( u ) μ ( d m ) d u u u u Γ μ ( 0 ) = 0 \Gamma_\mu(0)=0 Γ μ ( 0 ) = 0
Γ μ ( s ) = β 2 ∫ 0 s ( 2 ψ μ ′ ( β 2 u ) − u ) d u ( s ≥ 0 ) . (110) \Gamma_\mu(s)=\beta^2\int_0^s\left(2\psi_\mu'(\beta^2u)-u\right)\,du\qquad(s\ge0). \tag*{(110)} Γ μ ( s ) = β 2 ∫ 0 s ( 2 ψ μ ′ ( β 2 u ) − u ) d u ( s ≥ 0 ) . (110) Since ψ − m = ψ m \psi_{-m}=\psi_m ψ − m = ψ m Γ μ \Gamma_\mu Γ μ ∣ m ∣ |m| ∣ m ∣ μ \mu μ Γ μ \Gamma_\mu Γ μ [41 (3.43)–(3.44)] , applied to this law; a short computation with (113) below identifies the two definitions. Let q = ∫ m 2 μ ( d m ) q=\int m^2\mu(dm) q = ∫ m 2 μ ( d m ) q < 1 q<1 q < 1 Γ μ ≤ 0 \Gamma_\mu\le0 Γ μ ≤ 0 [ 0 , 1 − q ] [0,1-q] [ 0 , 1 − q ] ∣ m ∣ |m| ∣ m ∣ [41 Proposition 13] . In this case, the generalized TAP correction is replica symmetric and equals the classical TAP correction, and Plefka’s condition β 2 ∫ ( 1 − m 2 ) 2 μ ( d m ) ≤ 1 \beta^2\int(1-m^2)^2\mu(dm)\le1 β 2 ∫ ( 1 − m 2 ) 2 μ ( d m ) ≤ 1 [41 Propositions 13 and 14] .
The scalar criterion for the Parisi center
Let β > 0 \beta> 0 β > 0 h ≥ 0 h \ge0 h ≥ 0 § 3.3 : ν = ν β , h \nu= \nu_{\beta,h} ν = ν β , h ν ( t ) = ν ( [ 0 , t ] ) \nu(t) = \nu([0,t]) ν ( t ) = ν ([ 0 , t ]) Φ = Φ ν \Phi= \Phi_\nu Φ = Φ ν q + = max supp ν q_+ = \max\operatorname{supp}\nu q + = max supp ν X X X (11) , M t = ∂ x Φ ( t , X t ) M_t = \partial_x\Phi(t,X_t) M t = ∂ x Φ ( t , X t ) μ + \mu_+ μ + M q + M_{q_+} M q + Ψ \Psi Ψ (14) . By (P4), ∂ x Φ ( t , x ) = tanh x \partial_x\Phi(t,x) = \tanh x ∂ x Φ ( t , x ) = tanh x t ∈ [ q + , 1 ] t \in[q_+,1] t ∈ [ q + , 1 ] ν ( t ) = 1 \nu(t) = 1 ν ( t ) = 1 t ≥ q + t \ge q_+ t ≥ q + (11) becomes
X t = X q + + β 2 ∫ q + t tanh ( X u ) d u + β ( B t − B q + ) , M t = tanh X t ( t ∈ [ q + , 1 ] ) . (111) X_t = X_{q_+} + \beta^2 \int_{q_+}^{t} \tanh(X_u)\,\mathrm{d}u + \beta(B_t-B_{q_+}), \qquad M_t = \tanh X_t \qquad(t \in[q_+,1]). \tag*{(111)} X t = X q + + β 2 ∫ q + t tanh ( X u ) d u + β ( B t − B q + ) , M t = tanh X t ( t ∈ [ q + , 1 ]) . (111) In particular, μ + \mu_+ μ + tanh X q + \tanh X_{q_+} tanh X q + μ + ( ( − 1 , 1 ) ) = 1 \mu_+((-1,1)) = 1 μ + (( − 1 , 1 )) = 1 q + < 1 q_+ < 1 q + < 1 E M q + 2 = q + \mathbb{E}M_{q_+}^2 = q_+ E M q + 2 = q +
∫ κ ( m ) μ + ( d m ) = 1 − E M q + 2 = 1 − q + . (112) \int\kappa(m)\,\mu_+(\mathrm{d}m) = 1-\mathbb{E}M_{q_+}^2 = 1-q_+. \tag*{(112)} ∫ κ ( m ) μ + ( d m ) = 1 − E M q + 2 = 1 − q + . (112) The next lemma identifies the law of the Parisi diffusion above q + q_+ q + [ q + , 1 ] [q_+,1] [ q + , 1 ] β 2 tanh x \beta^2\tanh x β 2 tanh x β 2 \beta^2 β 2 cosh \cosh cosh
Lemma 7.3. Let β > 0 \beta> 0 β > 0 h ≥ 0 h \ge0 h ≥ 0 s ∈ [ 0 , 1 − q + ] s \in[0,1-q_+] s ∈ [ 0 , 1 − q + ] g : R → R g:\mathbb{R}\to\mathbb{R} g : R → R y ∈ R y \in\mathbb{R} y ∈ R
k s ( y ) = ∑ σ ∈ { − 1 , 1 } e σ y 2 cosh y E g ( y + β 2 s σ + β s Z ) , k_s(y) = \sum_{\sigma\in\{-1,1\}} \frac{e^{\sigma y}}{2\cosh y}\,\mathbb{E}g(y+\beta^2s\sigma+\beta\sqrt{s}Z), k s ( y ) = σ ∈ { − 1 , 1 } ∑ 2 cosh y e σ y E g ( y + β 2 s σ + β s Z ) , where Z ∼ N ( 0 , 1 ) Z \sim N(0,1) Z ∼ N ( 0 , 1 ) E g ( X q + + s ) = E k s ( X q + ) \mathbb{E}g(X_{q_++s}) = \mathbb{E}k_s(X_{q_+}) E g ( X q + + s ) = E k s ( X q + )
Proof. For s = 0 s = 0 s = 0 s > 0 s > 0 s > 0 T ′ = q + + s ≤ 1 T' = q_+ + s \le1 T ′ = q + + s ≤ 1 ( F t ) (\mathcal{F}_t) ( F t ) B B B X X X t ∈ [ q + , T ′ ] t \in[q_+,T'] t ∈ [ q + , T ′ ]
Λ t = exp ( − β ∫ q + t tanh ( X u ) d B u − β 2 2 ∫ q + t tanh 2 ( X u ) d u ) . \Lambda_t = \exp\left(-\beta\int_{q_+}^{t}\tanh(X_u)\,\mathrm{d}B_u-\frac{\beta^2}{2}\int_{q_+}^{t}\tanh^2(X_u)\,\mathrm{d}u\right). Λ t = exp ( − β ∫ q + t tanh ( X u ) d B u − 2 β 2 ∫ q + t tanh 2 ( X u ) d u ) . Since the integrand β tanh ( X u ) \beta\tanh(X_u) β tanh ( X u ) [92 Chapter 3] for this condition and for Girsanov’s theorem), and Λ \Lambda Λ [ q + , T ′ ] [q_+,T'] [ q + , T ′ ] Λ q + = 1 \Lambda_{q_+}=1 Λ q + = 1 R \mathbb{R} R Λ T ′ \Lambda_{T'} Λ T ′ P \mathbb{P} P E [ Λ T ′ ∣ F q + ] = 1 \mathbb{E}[\Lambda_{T'}\mid\mathcal{F}_{q_+}] = 1 E [ Λ T ′ ∣ F q + ] = 1 R \mathbb{R} R P \mathbb{P} P F q + \mathcal{F}_{q_+} F q + B ~ t = B t − B q + + β ∫ q + t tanh ( X u ) d u \widetilde{B}_t = B_t-B_{q_+}+\beta\int_{q_+}^{t}\tanh(X_u)\,\mathrm{d}u B t = B t − B q + + β ∫ q + t tanh ( X u ) d u t ∈ [ q + , T ′ ] t \in[q_+,T'] t ∈ [ q + , T ′ ] R \mathbb{R} R F q + \mathcal{F}_{q_+} F q + (111) , we have X t = X q + + β B ~ t X_t = X_{q_+}+\beta\widetilde{B}_t X t = X q + + β B t t ∈ [ q + , T ′ ] t \in[q_+,T'] t ∈ [ q + , T ′ ]
Next, (111) and Itô’s formula give
log cosh X T ′ − log cosh X q + = β ∫ q + T ′ tanh ( X u ) d B u + β 2 2 ∫ q + T ′ ( 1 + tanh 2 ( X u ) ) d u = − log Λ T ′ + β 2 s 2 . \begin{aligned}
\log\cosh X_{T'}-\log\cosh X_{q_+}
= \beta\int_{q_+}^{T'}\tanh(X_u)\,\mathrm{d}B_u+\frac{\beta^2}{2}\int_{q_+}^{T'}\left(1+\tanh^2(X_u)\right)\,\mathrm{d}u \\
&= -\log\Lambda_{T'}+\frac{\beta^2s}{2}.
\end{aligned} log cosh X T ′ − log cosh X q + = β ∫ q + T ′ tanh ( X u ) d B u + 2 β 2 ∫ q + T ′ ( 1 + tanh 2 ( X u ) ) d u = − log Λ T ′ + 2 β 2 s . Since Λ T ′ > 0 \Lambda_{T'} > 0 Λ T ′ > 0
E g ( X T ′ ) = E R [ g ( X T ′ ) Λ T ′ − 1 ] = e − β 2 s / 2 E R [ ( g cosh ) ( X q + + β B ~ T ′ ) cosh X q + ] . \mathbb{E}_{\mathsf{g}}(X_{T'}) = \mathbb{E}_{\mathsf{R}}\left[\mathsf{g}(X_{T'})\Lambda_{T'}^{-1}\right] = e^{-\beta^{2}s/2}\mathbb{E}_{\mathsf{R}}\left[\frac{(\mathsf{g}\cosh)(X_{q_+}+\beta\widetilde{B}_{T'})}{\cosh X_{q_+}}\right]. E g ( X T ′ ) = E R [ g ( X T ′ ) Λ T ′ − 1 ] = e − β 2 s /2 E R [ cosh X q + ( g cosh ) ( X q + + β B T ′ ) ] . Under R \mathsf{R} R B ~ T ′ \widetilde{B}_{T'} B T ′ N ( 0 , s ) N(0,s) N ( 0 , s ) X q + X_{q_+} X q + R \mathsf{R} R P \mathsf{P} P cosh ( x + z ) ≤ e ∣ z ∣ cosh x \cosh(x+z)\le e^{|z|}\cosh x cosh ( x + z ) ≤ e ∣ z ∣ cosh x ∥ g ∥ ∞ e β ∣ B ~ T ′ ∣ \|\mathsf{g}\|_\infty e^{\beta|\widetilde{B}_{T'}|} ∥ g ∥ ∞ e β ∣ B T ′ ∣ E g ( X T ′ ) = E k ( X q + ) \mathbb{E}_{\mathsf{g}}(X_{T'})=\mathbb{E}_{k}(X_{q_+}) E g ( X T ′ ) = E k ( X q + )
k ( y ) = e − β 2 s / 2 cosh y E [ ( g cosh ) ( y + β s Z ) ] . k(y)=\frac{e^{-\beta^{2}s/2}}{\cosh y}\mathbb{E}\left[(\mathsf{g}\cosh)(y+\beta\sqrt{s}Z)\right]. k ( y ) = cosh y e − β 2 s /2 E [ ( g cosh ) ( y + β s Z ) ] . Finally, we write cosh \cosh cosh e ± x e^{\pm x} e ± x E [ f ( Z ) e c Z ] = e c 2 / 2 E f ( Z + c ) \mathbb{E}[f(Z)e^{cZ}]=e^{c^{2}/2}\mathbb{E}f(Z+c) E [ f ( Z ) e c Z ] = e c 2 /2 E f ( Z + c ) c = ± β s c=\pm\beta\sqrt{s} c = ± β s k = k s k=k_s k = k s
Proposition 7.4. Let β > 0 \beta>0 β > 0 h ≥ 0 h\ge0 h ≥ 0 μ + \mu_+ μ + M q + = tanh X q + M_{q_+}=\tanh X_{q_+} M q + = tanh X q + Γ μ + ( s ) ≤ 0 \Gamma_{\mu_+}(s)\le0 Γ μ + ( s ) ≤ 0 s ≥ 0 s\ge0 s ≥ 0
Proof. Let m = tanh y ∈ ( − 1 , 1 ) m=\tanh y\in(-1,1) m = tanh y ∈ ( − 1 , 1 ) r ≥ 0 r\ge0 r ≥ 0 σ ∗ \sigma^* σ ∗ Z ∼ N ( 0 , 1 ) Z\sim N(0,1) Z ∼ N ( 0 , 1 ) P ( σ ∗ = ± 1 ) = ( 1 ± m ) / 2 = e ± y / ( 2 cosh y ) \mathbb{P}(\sigma^*=\pm1)=(1\pm m)/2=e^{\pm y}/(2\cosh y) P ( σ ∗ = ± 1 ) = ( 1 ± m ) /2 = e ± y / ( 2 cosh y )
2 ψ m ′ ( r ) = E tanh 2 ( y + r Z + r σ ∗ ) − m 2 . (113) 2\psi'_{m}(r)=\mathbb{E}\tanh^{2}(y+\sqrt{r}Z+r\sigma^*)-m^{2}. \tag*{(113)} 2 ψ m ′ ( r ) = E tanh 2 ( y + r Z + r σ ∗ ) − m 2 . (113) Indeed, τ ∗ = σ ∗ − m \tau^*=\sigma^*-m τ ∗ = σ ∗ − m ν m \nu_m ν m W = r τ ∗ + Z W=\sqrt{r}\tau^*+Z W = r τ ∗ + Z σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 } τ = σ − m \tau=\sigma-m τ = σ − m
r W τ − r 2 τ 2 = σ ( r W + r m ) − r W m − r 2 ( 1 + m 2 ) , \sqrt{r}W\tau-\frac{r}{2}\tau^{2}=\sigma(\sqrt{r}W+rm)-\sqrt{r}Wm-\frac{r}{2}(1+m^{2}), r W τ − 2 r τ 2 = σ ( r W + r m ) − r W m − 2 r ( 1 + m 2 ) , where we used σ 2 = 1 \sigma^{2}=1 σ 2 = 1 σ \sigma σ P ( σ ∗ = σ ∣ W ) ∝ e σ ( y + r W + r m ) \mathbb{P}(\sigma^*=\sigma\mid W)\propto e^{\sigma(y+\sqrt{r}W+rm)} P ( σ ∗ = σ ∣ W ) ∝ e σ ( y + r W + r m ) E [ σ ∗ ∣ W ] = tanh ( y + r W + r m ) = tanh ( y + r Z + r σ ∗ ) \mathbb{E}[\sigma^*\mid W]=\tanh(y+\sqrt{r}W+rm)=\tanh(y+\sqrt{r}Z+r\sigma^*) E [ σ ∗ ∣ W ] = tanh ( y + r W + r m ) = tanh ( y + r Z + r σ ∗ ) b m , r = E [ σ ∗ ∣ W ] − m b_{m,r}=\mathbb{E}[\sigma^*\mid W]-m b m , r = E [ σ ∗ ∣ W ] − m E E [ σ ∗ ∣ W ] = m \mathbb{E}\mathbb{E}[\sigma^*\mid W]=m EE [ σ ∗ ∣ W ] = m E b m , r 2 = E tanh 2 ( y + r Z + r σ ∗ ) − m 2 \mathbb{E}b_{m,r}^{2}=\mathbb{E}\tanh^{2}(y+\sqrt{r}Z+r\sigma^*)-m^{2} E b m , r 2 = E tanh 2 ( y + r Z + r σ ∗ ) − m 2 Lemma 7.2 (b) gives (7.11).
Let s ∈ [ 0 , 1 − q + ] s\in[0,1-q_+] s ∈ [ 0 , 1 − q + ] Lemma 7.3 with g = tanh 2 g=\tanh^{2} g = tanh 2 r = β 2 s r=\beta^{2}s r = β 2 s k s k_s k s Lemma 7.3 is k s ( y ) = 2 ψ tanh y ′ ( β 2 s ) + tanh 2 y k_s(y)=2\psi'_{\tanh y}(\beta^{2}s)+\tanh^{2}y k s ( y ) = 2 ψ t a n h y ′ ( β 2 s ) + tanh 2 y
E M q + + s 2 = E tanh 2 X q + + s = E [ 2 ψ M q + ′ ( β 2 s ) ] + E M q + 2 = 2 ψ μ + ′ ( β 2 s ) + q + . \mathbb{E}M_{q_++s}^{2}=\mathbb{E}\tanh^{2}X_{q_++s}=\mathbb{E}\left[2\psi'_{M_{q_+}}(\beta^{2}s)\right]+\mathbb{E}M_{q_+}^{2}=2\psi'_{\mu_+}(\beta^{2}s)+q_+. E M q + + s 2 = E tanh 2 X q + + s = E [ 2 ψ M q + ′ ( β 2 s ) ] + E M q + 2 = 2 ψ μ + ′ ( β 2 s ) + q + . Inserting this into (7.8), we find for s ∈ [ 0 , 1 − q + ] s\in[0,1-q_+] s ∈ [ 0 , 1 − q + ]
Γ μ + ( s ) = β 2 ∫ 0 s ( E M q + + u 2 − q + − u ) d u = β 2 ∫ q + q + + s ( E M t 2 − t ) d t \Gamma_{\mu_+}(s)=\beta^{2}\int_{0}^{s}\left(\mathbb{E}M_{q_++u}^{2}-q_+-u\right)\,du=\beta^{2}\int_{q_+}^{q_++s}\left(\mathbb{E}M_{t}^{2}-t\right)\,dt Γ μ + ( s ) = β 2 ∫ 0 s ( E M q + + u 2 − q + − u ) d u = β 2 ∫ q + q + + s ( E M t 2 − t ) d t = 2 ( Ψ ( q + ) − Ψ ( q + + s ) ) ≤ 0 , =2\left(\Psi(q_+)-\Psi(q_++s)\right)\le0, = 2 ( Ψ ( q + ) − Ψ ( q + + s ) ) ≤ 0 , where the inequality holds by Theorem 2.3 , since q + ∈ supp ν q_+\in\operatorname{supp}\nu q + ∈ supp ν s ≥ 1 − q + s\ge1-q_+ s ≥ 1 − q +
Γ μ + ′ ( s ) = β 2 ( 2 ψ μ + ′ ( β 2 s ) − s ) ≤ β 2 ( 1 − q + − s ) ≤ 0 , \Gamma'_{\mu_+}(s)=\beta^{2}\left(2\psi'_{\mu_+}(\beta^{2}s)-s\right)\le\beta^{2}(1-q_+-s)\le0, Γ μ + ′ ( s ) = β 2 ( 2 ψ μ + ′ ( β 2 s ) − s ) ≤ β 2 ( 1 − q + − s ) ≤ 0 , which gives $\Gamma_{\mu_+}(s)\le\Gamma_{\mu_+}(1-q_+)\le 0.
Since ∫ m 2 μ + ( d m ) = q + < 1 \int m^2\mu_+(\mathrm{d}m)=q_+<1 ∫ m 2 μ + ( d m ) = q + < 1 [41 Proposition 13] imply that, for the SK model, the generalized TAP correction of Chen, Panchenko, and Subag at the law of ∣ M q + ∣ |M_{q_+}| ∣ M q + ∣ p p p [40 Remark 6] . By [41 Remark 15] , for some β \beta β h > 0 h>0 h > 0 q + q_+ q +
The relative-entropy bound
We recall the planted product model of Definition 6.1. For m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V v ∗ = σ ∗ − m v^*=\sigma^*-m v ∗ = σ ∗ − m v x ∗ v_x^* v x ∗ ν m x \nu_{m_x} ν m x Q m Q_m Q m Y = β a ( v ∗ ) + Z Y=\beta a(v^*)+Z Y = β a ( v ∗ ) + Z R E \mathbb{R}^E R E a ( v ) x y = S x y v x v y a(v)_{xy}=\sqrt{S_{xy}}v_xv_y a ( v ) x y = S x y v x v y Z Z Z D ( m ) = KL ( Q m ∥ P 0 ) \mathcal{D}(m)=\operatorname{KL}(Q_m\mathbin\Vert P_0) D ( m ) = KL ( Q m ∥ P 0 ) a ≥ 0 a\geq0 a ≥ 0
A S ( a ) = a 1 + 2 β 2 S [ 0 , 1 ] V = { a 1 + 2 β 2 S y : y ∈ [ 0 , 1 ] V } . \mathcal{A}_S(a)=a\mathbf{1}+2\beta^2S[0,1]^V=\{a\mathbf{1}+2\beta^2Sy:y\in[0,1]^V\}. A S ( a ) = a 1 + 2 β 2 S [ 0 , 1 ] V = { a 1 + 2 β 2 S y : y ∈ [ 0 , 1 ] V } . Since S S S A S ( a ) ⊂ [ a , a + 2 β 2 ] V \mathcal{A}_S(a)\subset[a,a+2\beta^2]^V A S ( a ) ⊂ [ a , a + 2 β 2 ] V μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V
Δ S , a ( m ; μ ) = sup r ∈ A S ( a ) 1 n ∑ x ∈ V ( ψ m x ( r x ) − ψ μ ( r x ) ) . (114) \Delta_{S,a}(m;\mu)=\sup_{r\in\mathcal{A}_S(a)}\frac{1}{n}\sum_{x\in V}\left(\psi_{m_x}(r_x)-\psi_\mu(r_x)\right). \tag*{(114)} Δ S , a ( m ; μ ) = r ∈ A S ( a ) sup n 1 x ∈ V ∑ ( ψ m x ( r x ) − ψ μ ( r x ) ) . (114) Since the function inside the supremum is continuous in ( m , r ) (m,r) ( m , r ) A S ( a ) \mathcal{A}_S(a) A S ( a ) Δ S , a ( ⋅ ; μ ) \Delta_{S,a}(\cdot\,;\mu) Δ S , a ( ⋅ ; μ ) [ − 1 , 1 ] V [-1,1]^V [ − 1 , 1 ] V
Theorem 7.5. Let β ≥ 0 \beta\geq0 β ≥ 0
(a) For every a > 0 a>0 a > 0 e β , a : [ 1 , ∞ ) → [ 0 , ∞ ) e_{\beta,a}:[1,\infty)\to[0,\infty) e β , a : [ 1 , ∞ ) → [ 0 , ∞ ) β \beta β a a a lim D → ∞ e β , a ( D ) = 0 \lim_{D\to\infty}e_{\beta,a}(D)=0 lim D → ∞ e β , a ( D ) = 0 D ≥ 1 D\geq1 D ≥ 1 S S S max x , y S x y ≤ 1 / D \max_{x,y}S_{xy}\leq1/D max x , y S x y ≤ 1/ D m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
1 n D ( m ) ≤ 1 2 sup 0 ≤ s ≤ 1 Γ μ ( s ) + 2 a + Δ S , a ( m ; μ ) + e β , a ( D ) . \frac{1}{n}\mathcal{D}(m)\leq\frac{1}{2}\sup_{0\leq s\leq1}\Gamma_\mu(s)+2a+\Delta_{S,a}(m;\mu)+e_{\beta,a}(D). n 1 D ( m ) ≤ 2 1 0 ≤ s ≤ 1 sup Γ μ ( s ) + 2 a + Δ S , a ( m ; μ ) + e β , a ( D ) . (b) For each k ≥ 1 k\geq1 k ≥ 1 D k ≥ 1 D_k\geq1 D k ≥ 1 S k S_k S k n k × n k n_k\times n_k n k × n k max x , y ( S k ) x y ≤ 1 / D k \max_{x,y}(S_k)_{xy}\leq1/D_k max x , y ( S k ) x y ≤ 1/ D k m ( k ) m^{(k)} m ( k ) [ − 1 , 1 ] n k [-1,1]^{n_k} [ − 1 , 1 ] n k μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] D k → ∞ D_k\to\infty D k → ∞ a > 0 a>0 a > 0
lim k → ∞ E sup r ∈ A S k ( a ) ∣ 1 n k ∑ x ψ m x ( k ) ( r x ) − 1 n k ∑ x ψ μ ( r x ) ∣ = 0. (115) \lim_{k\to\infty}\mathbb{E}\sup_{r\in\mathcal{A}_{S_k}(a)}
\left|\frac{1}{n_k}\sum_x\psi_{m_x^{(k)}}(r_x)-\frac{1}{n_k}\sum_x\psi_\mu(r_x)\right|=0. \tag*{(115)} k → ∞ lim E r ∈ A S k ( a ) sup n k 1 x ∑ ψ m x ( k ) ( r x ) − n k 1 x ∑ ψ μ ( r x ) = 0. (115) Then
lim sup k → ∞ 1 n k E D ( m ( k ) ) ≤ 1 2 sup 0 ≤ s ≤ 1 Γ μ ( s ) . \limsup_{k\to\infty}\frac{1}{n_k}\mathbb{E}\,D(m^{(k)}) \le\frac{1}{2}\sup_{0\le s\le1}\Gamma_\mu(s). k → ∞ lim sup n k 1 E D ( m ( k ) ) ≤ 2 1 0 ≤ s ≤ 1 sup Γ μ ( s ) . Corollary 7.6. Let β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 ( μ k ) (\mu_k) ( μ k ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] μ + \mu_+ μ + lim sup k → ∞ sup 0 ≤ s ≤ 1 Γ μ k ( s ) ≤ 0 \limsup_{k\to\infty}\sup_{0\le s\le1}\Gamma_{\mu_k}(s)\le0 lim sup k → ∞ sup 0 ≤ s ≤ 1 Γ μ k ( s ) ≤ 0
Remark 7.7. The matrix S S S G ∈ G D G\in\mathcal{G}_D G ∈ G D T L d \mathbb{T}_L^d T L d L L L Q d Q_d Q d − 1 -1 − 1 A G / D A_G/D A G / D Theorem 7.5 uses three properties of S S S S ⪯ I S\preceq I S ⪯ I § 8.10 . Second, S S S S [ 0 , 1 ] V ⊂ [ 0 , 1 ] V S[0,1]^V\subset[0,1]^V S [ 0 , 1 ] V ⊂ [ 0 , 1 ] V Lemma 7.16 . Third, ∥ S ∥ ≤ 1 \lVert S\rVert\le1 ∥ S ∥ ≤ 1 ∣ w ⋅ S w ∣ ≤ ∣ w ∣ ∣ S w ∣ \lvert w\cdot Sw\rvert\le\lvert w\rvert\lvert Sw\rvert ∣ w ⋅ S w ∣ ≤ ∣ w ∣ ∣ S w ∣ w ⋅ S w w\cdot Sw w ⋅ S w
Interpolating channels
In this subsection and the next three, we fix β > 0 \beta>0 β > 0 m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V v ∗ = ( v x ∗ ) x ∈ V v^*=(v_x^*)_{x\in V} v ∗ = ( v x ∗ ) x ∈ V ν m x \nu_{m_x} ν m x Z = ( Z e ) e ∈ E Z=(Z_e)_{e\in E} Z = ( Z e ) e ∈ E ζ = ( ζ x ) x ∈ V \zeta=(\zeta_x)_{x\in V} ζ = ( ζ x ) x ∈ V v ∗ v^* v ∗ Z Z Z ζ \zeta ζ t ≥ 0 t\ge0 t ≥ 0 q ∈ [ 0 , ∞ ) V q\in[0,\infty)^V q ∈ [ 0 , ∞ ) V
Y e = β ( t S x y ) 1 / 2 v x ∗ v y ∗ + Z e ( e = { x , y } ∈ E ) , Y x ′ = q x 1 / 2 v x ∗ + ζ x ( x ∈ V ) . (116) Y_e=\beta(tS_{xy})^{1/2}v_x^*v_y^*+Z_e\quad(e=\{x,y\}\in E),\qquad Y_x'=q_x^{1/2}v_x^*+\zeta_x\quad(x\in V). \tag*{(116)} Y e = β ( t S x y ) 1/2 v x ∗ v y ∗ + Z e ( e = { x , y } ∈ E ) , Y x ′ = q x 1/2 v x ∗ + ζ x ( x ∈ V ) . (116) This is the setting of § 8.1 with N = ∣ E ∣ + n N=\lvert E\rvert+n N = ∣ E ∣ + n θ = ( ( v x ∗ v y ∗ ) { x , y } ∈ E , ( v x ∗ ) x ∈ V ) \theta=((v_x^*v_y^*)_{\{x,y\}\in E},(v_x^*)_{x\in V}) θ = (( v x ∗ v y ∗ ) { x , y } ∈ E , ( v x ∗ ) x ∈ V ) λ = λ ( t , q ) = ( ( β 2 t S e ) e ∈ E , ( q x ) x ∈ V ) \lambda=\lambda(t,q)=((\beta^2tS_e)_{e\in E},(q_x)_{x\in V}) λ = λ ( t , q ) = (( β 2 t S e ) e ∈ E , ( q x ) x ∈ V ) S e = S x y S_e=S_{xy} S e = S x y e = { x , y } e=\{x,y\} e = { x , y }
F ( t , q ) = 1 n I ( λ ( t , q ) ) . F(t,q)=\frac{1}{n}I(\lambda(t,q)). F ( t , q ) = n 1 I ( λ ( t , q )) . Let ν m = ⨂ x ν m x \nu_m=\bigotimes_x\nu_{m_x} ν m = ⨂ x ν m x v ∗ v^* v ∗ v v v
E t , q ( v ) = ∑ { x , y } ∈ E ( β ( t S x y ) 1 / 2 Z x y v x v y + β 2 t S x y v x ∗ v y ∗ v x v y − β 2 t S x y 2 v x 2 v y 2 ) + ∑ x ∈ V ( q x 1 / 2 ζ x v x + q x v x ∗ v x − q x 2 v x 2 ) , \mathcal{E}_{t,q}(v)=\sum_{\{x,y\}\in E}\left(\beta(tS_{xy})^{1/2}Z_{xy}v_xv_y+\beta^2tS_{xy}v_x^*v_y^*v_xv_y-\frac{\beta^2tS_{xy}}{2}v_x^2v_y^2\right)
+\sum_{x\in V}\left(q_x^{1/2}\zeta_xv_x+q_xv_x^*v_x-\frac{q_x}{2}v_x^2\right), E t , q ( v ) = { x , y } ∈ E ∑ ( β ( t S x y ) 1/2 Z x y v x v y + β 2 t S x y v x ∗ v y ∗ v x v y − 2 β 2 t S x y v x 2 v y 2 ) + x ∈ V ∑ ( q x 1/2 ζ x v x + q x v x ∗ v x − 2 q x v x 2 ) , and let
F ^ ( t , q ) = 1 n log ∑ v ν m ( v ) e E t , q ( v ) . \widehat{F}(t,q)=\frac{1}{n}\log\sum_v\nu_m(v)e^{\mathcal{E}_{t,q}(v)}. F ( t , q ) = n 1 log v ∑ ν m ( v ) e E t , q ( v ) . By (7.4), we have F ( t , q ) = E F ^ ( t , q ) F(t,q)=\mathbb{E}\widehat{F}(t,q) F ( t , q ) = E F ( t , q ) E \mathbb{E} E ( v ∗ , Z , ζ ) (v^*,Z,\zeta) ( v ∗ , Z , ζ ) ⟨ ⋅ ⟩ t , q \langle\cdot\rangle_{t,q} ⟨ ⋅ ⟩ t , q § 8.1 is the average over independent samples v 1 , v 2 , … v^1,v^2,\ldots v 1 , v 2 , … ν m ( v ) e E t , q ( v ) \nu_m(v)e^{\mathcal{E}_{t,q}(v)} ν m ( v ) e E t , q ( v ) ⟨ ⋅ ⟩ t , q \langle\cdot\rangle_{t,q} ⟨ ⋅ ⟩ t , q v v v v ∗ v^* v ∗ ( t , q ) (t,q) ( t , q ) ( v ∗ , Z , ζ ) (v^*, Z, \zeta) ( v ∗ , Z , ζ )
b x = ⟨ v x ⟩ t , q , Σ x y = ⟨ v x v y ⟩ t , q − b x b y , p x = E b x 2 . b_x=\langle v_x\rangle_{t,q},\qquad\Sigma_{xy}=\langle v_xv_y\rangle_{t,q}-b_xb_y,\qquad p_x=\mathbb{E}b_x^2. b x = ⟨ v x ⟩ t , q , Σ x y = ⟨ v x v y ⟩ t , q − b x b y , p x = E b x 2 . These depend on ( t , q ) (t,q) ( t , q ) ∣ v x ∣ ≤ 1 + ∣ m x ∣ ≤ 2 |v_x|\le1+|m_x|\le2 ∣ v x ∣ ≤ 1 + ∣ m x ∣ ≤ 2 ∣ v x ∗ ∣ ≤ 2 |v_x^*|\le2 ∣ v x ∗ ∣ ≤ 2 ∣ b x ∣ ≤ 2 |b_x|\le2 ∣ b x ∣ ≤ 2 ∣ Σ x y ∣ ≤ 8 |\Sigma_{xy}|\le8 ∣ Σ x y ∣ ≤ 8
Lemma 7.8. Let β > 0 \beta>0 β > 0 m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V
(a) We have F ( 1 , 0 ) = n − 1 D ( m ) F(1,0)=n^{-1}\mathcal{D}(m) F ( 1 , 0 ) = n − 1 D ( m ) F ( 0 , q ) = n − 1 ∑ x ψ m x ( q x ) F(0,q)=n^{-1}\sum_x\psi_{m_x}(q_x) F ( 0 , q ) = n − 1 ∑ x ψ m x ( q x ) q ∈ [ 0 , ∞ ) V q\in[0,\infty)^V q ∈ [ 0 , ∞ ) V
(b) The function F F F [ 0 , ∞ ) × [ 0 , ∞ ) V [0,\infty)\times[0,\infty)^V [ 0 , ∞ ) × [ 0 , ∞ ) V t t t q x q_x q x ∣ F ( t , q ) − F ( t , q ′ ) ∣ ≤ ∥ q − q ′ ∥ n / 2 |F(t,q)-F(t,q')|\le\|q-q'\|_n/2 ∣ F ( t , q ) − F ( t , q ′ ) ∣ ≤ ∥ q − q ′ ∥ n /2
(c) Let q ∈ ( 0 , ∞ ) V q\in(0,\infty)^V q ∈ ( 0 , ∞ ) V t ≥ 0 t\ge0 t ≥ 0 F ( t , ⋅ ) F(t,\cdot) F ( t , ⋅ ) q q q
2 n ∂ q x F ( t , q ) = p x ∈ [ 0 , κ x ] , 2 n ∂ q x ∂ q y F ( t , q ) = E Σ x y 2 . 2n\,\partial_{q_x}F(t,q)=p_x\in[0,\kappa_x],\qquad2n\,\partial_{q_x}\partial_{q_y}F(t,q)=\mathbb{E}\Sigma_{xy}^{2}. 2 n ∂ q x F ( t , q ) = p x ∈ [ 0 , κ x ] , 2 n ∂ q x ∂ q y F ( t , q ) = E Σ x y 2 . For every t > 0 t>0 t > 0 F ( ⋅ , q ) F(\cdot,q) F ( ⋅ , q ) t t t
∂ t F ( t , q ) = β 2 4 n ∑ x , y ∈ V S x y E ⟨ v x v y ⟩ t , q 2 . \partial_tF(t,q)=\frac{\beta^2}{4n}\sum_{x,y\in V}S_{xy}\mathbb{E}\langle v_xv_y\rangle_{t,q}^{2}. ∂ t F ( t , q ) = 4 n β 2 x , y ∈ V ∑ S x y E ⟨ v x v y ⟩ t , q 2 . Proof. (a) At q = 0 q=0 q = 0 β S x y 1 / 2 v x ∗ v y ∗ = β a ( v ∗ ) x y \beta S_{xy}^{1/2}v_x^*v_y^*=\beta a(v^*)_{xy} β S x y 1/2 v x ∗ v y ∗ = β a ( v ∗ ) x y (116) at ( 1 , 0 ) (1,0) ( 1 , 0 ) Q m ⊗ N ( 0 , I V ) Q_m\otimes N(0,I_V) Q m ⊗ N ( 0 , I V ) P 0 ⊗ N ( 0 , I V ) P_0\otimes N(0,I_V) P 0 ⊗ N ( 0 , I V ) D ( m ) \mathcal{D}(m) D ( m ) t = 0 t=0 t = 0 x x x N ( 0 , I E ) ⊗ ⨂ x Law ( q x 1 / 2 v x ∗ + ζ x ) N(0,I_E)\otimes\bigotimes_x\operatorname{Law}(q_x^{1/2}v_x^*+\zeta_x) N ( 0 , I E ) ⊗ ⨂ x Law ( q x 1/2 v x ∗ + ζ x ) Law ( q x 1 / 2 v x ∗ + ζ x ) \operatorname{Law}(q_x^{1/2}v_x^*+\zeta_x) Law ( q x 1/2 v x ∗ + ζ x ) N ( 0 , 1 ) N(0,1) N ( 0 , 1 ) ψ m x ( q x ) \psi_{m_x}(q_x) ψ m x ( q x )
(b) and (c) The map ( t , q ) ↦ λ ( t , q ) (t,q)\mapsto\lambda(t,q) ( t , q ) ↦ λ ( t , q ) Lemma 7.1 (a). If t > 0 t>0 t > 0 q ∈ ( 0 , ∞ ) V q\in(0,\infty)^V q ∈ ( 0 , ∞ ) V λ ( t , q ) \lambda(t,q) λ ( t , q ) Lemma 7.1 (b,c) the function I I I λ ( t , q ) \lambda(t,q) λ ( t , q ) λ \lambda λ E E E q q q q q q t = 0 t=0 t = 0 Lemma 7.1 (b,c) and the chain rule, together with E ( v x ∗ ) 2 = κ x \mathbb{E}(v_x^*)^2=\kappa_x E ( v x ∗ ) 2 = κ x
∑ e ∈ E β 2 S e 1 2 E ⟨ v x v y ⟩ t , q 2 = β 2 4 ∑ x , y ∈ V S x y E ⟨ v x v y ⟩ t , q 2 , \sum_{e\in E}\beta^2S_e\frac{1}{2}\mathbb{E}\langle v_xv_y\rangle_{t,q}^{2}
=\frac{\beta^2}{4}\sum_{x,y\in V}S_{xy}\mathbb{E}\langle v_xv_y\rangle_{t,q}^{2}, e ∈ E ∑ β 2 S e 2 1 E ⟨ v x v y ⟩ t , q 2 = 4 β 2 x , y ∈ V ∑ S x y E ⟨ v x v y ⟩ t , q 2 , which holds because S S S S x y = 0 S_{xy}=0 S x y = 0 { x , y } ∉ E \{x,y\}\notin E { x , y } ∈ / E 0 ≤ ∂ q x F ≤ 1 / ( 2 n ) 0\le\partial_{q_x}F\le1/(2n) 0 ≤ ∂ q x F ≤ 1/ ( 2 n ) ( 0 , ∞ ) V (0,\infty)^V ( 0 , ∞ ) V ∣ F ( t , q ) − F ( t , q ′ ) ∣ ≤ ( 2 n ) − 1 ∑ x ∣ q x − q x ′ ∣ ≤ ∥ q − q ′ ∥ n / 2 |F(t,q)-F(t,q')|\le(2n)^{-1}\sum_x|q_x-q'_x|\le\|q-q'\|_n/2 ∣ F ( t , q ) − F ( t , q ′ ) ∣ ≤ ( 2 n ) − 1 ∑ x ∣ q x − q x ′ ∣ ≤ ∥ q − q ′ ∥ n /2 q , q ′ ∈ ( 0 , ∞ ) V q,q'\in(0,\infty)^V q , q ′ ∈ ( 0 , ∞ ) V q , q ′ ∈ [ 0 , ∞ ) V q,q'\in[0,\infty)^V q , q ′ ∈ [ 0 , ∞ ) V
The score covariance identity
Fix t ≥ 0 t \ge0 t ≥ 0 q ∈ ( 0 , ∞ ) V q \in(0,\infty)^V q ∈ ( 0 , ∞ ) V F ^ ( t , ⋅ ) \widehat{F}(t,\cdot) F ( t , ⋅ ) F ^ ( t , ⋅ ) \widehat{F}(t,\cdot) F ( t , ⋅ ) q q q 2 n ∇ q F ^ ( t , q ) = χ 2n\nabla_q\widehat{F}(t,q)=\chi 2 n ∇ q F ( t , q ) = χ
χ x = ζ x b x q x + 2 v x ∗ b x − ⟨ ( v x ∗ ) 2 ⟩ t , q . (117) \chi_x=\frac{\zeta_xb_x}{\sqrt{q_x}}+2v_x^*b_x-\langle(v_x^*)^2\rangle_{t,q}. \tag*{(117)} χ x = q x ζ x b x + 2 v x ∗ b x − ⟨( v x ∗ ) 2 ⟩ t , q . (117) Differentiating once more, we find
n ∂ q x ∂ q y F ^ ( t , q ) = ⟨ A x A y ⟩ t , q − ⟨ A x ⟩ t , q ⟨ A y ⟩ t , q − 1 x = y ζ x b x 4 q x 3 / 2 , A x ( v ) = ζ x v x 2 q x + v x ∗ v x − v x 2 2 . (118) n\partial_{q_x}\partial_{q_y}\widehat{F}(t,q)=\langle A_xA_y\rangle_{t,q}-\langle A_x\rangle_{t,q}\langle A_y\rangle_{t,q}-\mathbf{1}_{x=y}\frac{\zeta_xb_x}{4q_x^{3/2}},\qquad A_x(v)=\frac{\zeta_xv_x}{2\sqrt{q_x}}+v_x^*v_x-\frac{v_x^2}{2}. \tag*{(118)} n ∂ q x ∂ q y F ( t , q ) = ⟨ A x A y ⟩ t , q − ⟨ A x ⟩ t , q ⟨ A y ⟩ t , q − 1 x = y 4 q x 3/2 ζ x b x , A x ( v ) = 2 q x ζ x v x + v x ∗ v x − 2 v x 2 . (118) The first two terms form a covariance matrix, which is positive semidefinite. Since ∣ b x ∣ ≤ 2 |b_x|\le2 ∣ b x ∣ ≤ 2
∇ q 2 F ^ ( t , q ) ⪰ − 1 n diag ( ∣ ζ x ∣ 2 q x 3 / 2 ) x ∈ V . (119) \nabla_q^2\widehat{F}(t,q)\succeq-\frac{1}{n}\operatorname{diag}\left(\frac{|\zeta_x|}{2q_x^{3/2}}\right)_{x\in V}. \tag*{(119)} ∇ q 2 F ( t , q ) ⪰ − n 1 diag ( 2 q x 3/2 ∣ ζ x ∣ ) x ∈ V . (119) Lemma 7.9. Let t ≥ 0 t\ge0 t ≥ 0 q ∈ ( 0 , ∞ ) V q\in(0,\infty)^V q ∈ ( 0 , ∞ ) V U = χ − b ⊙ 2 U=\chi-b^{\odot2} U = χ − b ⊙ 2 E χ = p \mathbb{E}\chi=p E χ = p E U = 0 \mathbb{E}U=0 E U = 0
E [ U x U y ] = 1 x = y p x q x + E Σ x y 2 ( x , y ∈ V ) . \mathbb{E}[U_xU_y]=\mathbf{1}_{x=y}\frac{p_x}{q_x}+\mathbb{E}\Sigma_{xy}^2\qquad(x,y\in V). E [ U x U y ] = 1 x = y q x p x + E Σ x y 2 ( x , y ∈ V ) . Proof. By Lemma 7.8(c), E χ x = 2 n ∂ q x F ( t , q ) = p x \mathbb{E}\chi_x=2n\partial_{q_x}F(t,q)=p_x E χ x = 2 n ∂ q x F ( t , q ) = p x E U = 0 \mathbb{E}U=0 E U = 0 ⟨ v x 2 ⟩ t , q = Σ x x + b x 2 \langle v_x^2\rangle_{t,q}=\Sigma_{xx}+b_x^2 ⟨ v x 2 ⟩ t , q = Σ xx + b x 2 U = U 1 + U 2 U=U^1+U^2 U = U 1 + U 2
U x 1 = 2 b x ( v x ∗ − b x ) , U x 2 = ζ x b x q x − Σ x x . U_x^1=2b_x(v_x^*-b_x),\qquad U_x^2=\frac{\zeta_xb_x}{\sqrt{q_x}}-\Sigma_{xx}. U x 1 = 2 b x ( v x ∗ − b x ) , U x 2 = q x ζ x b x − Σ xx . We first compute E [ U x 2 U y 2 ] \mathbb{E}[U_x^2U_y^2] E [ U x 2 U y 2 ] ( v ∗ , Z ) (v^*,Z) ( v ∗ , Z ) b b b ζ \zeta ζ ∂ ζ y b x = q y Σ x y \partial_{\zeta_y}b_x=\sqrt{q_y}\Sigma_{xy} ∂ ζ y b x = q y Σ x y f x = b x / q x f_x=b_x/\sqrt{q_x} f x = b x / q x ∂ y = ∂ ζ y \partial_y=\partial_{\zeta_y} ∂ y = ∂ ζ y ∂ y f x = ( q y / q x ) 1 / 2 Σ x y \partial_yf_x=(q_y/q_x)^{1/2}\Sigma_{xy} ∂ y f x = ( q y / q x ) 1/2 Σ x y U x 2 = ζ x f x − ∂ x f x U_x^2=\zeta_xf_x-\partial_xf_x U x 2 = ζ x f x − ∂ x f x f x f_x f x ζ \zeta ζ ζ \zeta ζ
E [ ζ x f x ζ y f y ] = 1 x = y E f x 2 + E [ ∂ y ∂ x f x f y + ∂ x f x ∂ y f y + ∂ y f x ∂ x f y + f x ∂ x ∂ y f y ] , \mathbb{E}[\zeta_xf_x\zeta_yf_y]=\mathbf{1}_{x=y}\mathbb{E}f_x^2+\mathbb{E}\left[\partial_y\partial_xf_xf_y+\partial_xf_x\partial_yf_y+\partial_yf_x\partial_xf_y+f_x\partial_x\partial_yf_y\right], E [ ζ x f x ζ y f y ] = 1 x = y E f x 2 + E [ ∂ y ∂ x f x f y + ∂ x f x ∂ y f y + ∂ y f x ∂ x f y + f x ∂ x ∂ y f y ] , E [ ζ x f x ∂ y f y ] = E [ ∂ x f x ∂ y f y + f x ∂ x ∂ y f y ] , E [ ∂ x f x ζ y f y ] = E [ ∂ y ∂ x f x f y + ∂ x f x ∂ y f y ] , \mathbb{E}[\zeta_xf_x\partial_yf_y]=\mathbb{E}\left[\partial_xf_x\partial_yf_y+f_x\partial_x\partial_yf_y\right],\qquad\mathbb{E}[\partial_xf_x\zeta_yf_y]=\mathbb{E}\left[\partial_y\partial_xf_xf_y+\partial_xf_x\partial_yf_y\right], E [ ζ x f x ∂ y f y ] = E [ ∂ x f x ∂ y f y + f x ∂ x ∂ y f y ] , E [ ∂ x f x ζ y f y ] = E [ ∂ y ∂ x f x f y + ∂ x f x ∂ y f y ] , where the expectations are conditional on ( v ∗ , Z ) (v^*,Z) ( v ∗ , Z ) U x 2 U y 2 U_x^2U_y^2 U x 2 U y 2
E [ U x 2 U y 2 ] = 1 x = y E f x 2 + E [ ∂ y f x ∂ x f y ] = 1 x = y E b x 2 q x + E Σ x y 2 , \mathbb{E}[U_x^2U_y^2]=\mathbf{1}_{x=y}\mathbb{E}f_x^2+\mathbb{E}\left[\partial_yf_x\partial_xf_y\right]=\mathbf{1}_{x=y}\frac{\mathbb{E}b_x^2}{q_x}+\mathbb{E}\Sigma_{xy}^2, E [ U x 2 U y 2 ] = 1 x = y E f x 2 + E [ ∂ y f x ∂ x f y ] = 1 x = y q x E b x 2 + E Σ x y 2 , first conditionally on ( v ∗ , Z ) (v^*,Z) ( v ∗ , Z ) Y \mathcal{Y} Y σ \sigma σ (116) . The quantities b b b Σ \Sigma Σ Y \mathcal{Y} Y v ∗ v^* v ∗ Y \mathcal{Y} Y E [ v x ∗ − b x ∣ Y ] = 0 \mathbb{E}[v_x^*-b_x\mid\mathcal{Y}]=0 E [ v x ∗ − b x ∣ Y ] = 0 E [ ( v x ∗ − b x ) ( v y ∗ − b y ) ∣ Y ] = Σ x y \mathbb{E}[(v_x^*-b_x)(v_y^*-b_y)\mid\mathcal{Y}]=\Sigma_{xy} E [( v x ∗ − b x ) ( v y ∗ − b y ) ∣ Y ] = Σ x y E [ U x 1 U y 1 ] = 4 E [ b x b y Σ x y ] \mathbb{E}[U_x^1U_y^1]=4\mathbb{E}[b_xb_y\Sigma_{xy}] E [ U x 1 U y 1 ] = 4 E [ b x b y Σ x y ] ζ y = Y y ′ − q y v y ∗ \zeta_y=Y'_y-\sqrt{q_y}v_y^* ζ y = Y y ′ − q y v y ∗
U y 2 = Y y ′ b y q y − b y 2 − Σ y y − b y ( v y ∗ − b y ) . U_y^2=\frac{Y'_yb_y}{\sqrt{q_y}}-b_y^2-\Sigma_{yy}-b_y(v_y^*-b_y). U y 2 = q y Y y ′ b y − b y 2 − Σ y y − b y ( v y ∗ − b y ) . and the first three terms are Y \mathcal{Y} Y E [ U x 1 U y 2 ] = − 2 E [ b x b y Σ x y ] \mathbb{E}[U_x^1U_y^2]=-2\mathbb{E}[b_xb_y\Sigma_{xy}] E [ U x 1 U y 2 ] = − 2 E [ b x b y Σ x y ] E [ U x 2 U y 1 ] = − 2 E [ b x b y Σ x y ] \mathbb{E}[U_x^2U_y^1]=-2\mathbb{E}[b_xb_y\Sigma_{xy}] E [ U x 2 U y 1 ] = − 2 E [ b x b y Σ x y ] b x b y Σ x y b_xb_y\Sigma_{xy} b x b y Σ x y E [ U x U y ] = E [ U x 2 U y 2 ] \mathbb{E}[U_xU_y]=\mathbb{E}[U_x^2U_y^2] E [ U x U y ] = E [ U x 2 U y 2 ]
Concentration
We use the bounded-differences inequality of McDiarmid [101 ] : if ζ 1 , … , ζ n \zeta_1,\ldots,\zeta_n ζ 1 , … , ζ n f f f c x c_x c x x x x P ( ∣ f ( ζ ) − E f ( ζ ) ∣ ≥ ϵ ) ≤ 2 exp ( − 2 ϵ 2 / ∑ x c x 2 ) \mathbb{P}(|f(\zeta)-\mathbb{E}f(\zeta)|\ge\epsilon)\le2\exp(-2\epsilon^2/\sum_x c_x^2) P ( ∣ f ( ζ ) − E f ( ζ ) ∣ ≥ ϵ ) ≤ 2 exp ( − 2 ϵ 2 / ∑ x c x 2 ) ϵ ≥ 0 \epsilon\ge0 ϵ ≥ 0
Lemma 7.10. Let β > 0 \beta>0 β > 0 Λ > 0 \Lambda>0 Λ > 0 m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] r ∈ [ 0 , Λ ] V r\in[0,\Lambda]^V r ∈ [ 0 , Λ ] V ϵ > 0 \epsilon>0 ϵ > 0
P ( ∣ F ^ ( t , r ) − F ( t , r ) ∣ > ϵ ) ≤ 4 exp ( − n ϵ 2 C Λ ) , C Λ = 32 ( 4 β 2 + Λ + 1 ) 2 . \mathbb{P}\left(\left|\widehat{F}(t,r)-F(t,r)\right|>\epsilon\right)\le4\exp\left(-\frac{n\epsilon^2}{C_\Lambda}\right),\qquad C_\Lambda=32(4\beta^2+\Lambda+1)^2. P ( F ( t , r ) − F ( t , r ) > ϵ ) ≤ 4 exp ( − C Λ n ϵ 2 ) , C Λ = 32 ( 4 β 2 + Λ + 1 ) 2 . Proof. Fix v ∗ v^* v ∗ ∂ Z e F ^ = n − 1 β ( t S x y ) 1 / 2 ⟨ v x v y ⟩ t , r \partial_{Z_e}\widehat{F}=n^{-1}\beta(tS_{xy})^{1/2}\langle v_xv_y\rangle_{t,r} ∂ Z e F = n − 1 β ( t S x y ) 1/2 ⟨ v x v y ⟩ t , r e = { x , y } e=\{x,y\} e = { x , y } ∂ ζ x F ^ = n − 1 r x 1 / 2 b x \partial_{\zeta_x}\widehat{F}=n^{-1}r_x^{1/2}b_x ∂ ζ x F = n − 1 r x 1/2 b x ∣ ⟨ v x v y ⟩ ∣ ≤ 4 |\langle v_xv_y\rangle|\le4 ∣ ⟨ v x v y ⟩ ∣ ≤ 4 ∣ b x ∣ ≤ 2 |b_x|\le2 ∣ b x ∣ ≤ 2 ∑ e ∈ E S e = n / 2 \sum_{e\in E}S_e=n/2 ∑ e ∈ E S e = n /2 F ^ \widehat{F} F ( Z , ζ ) (Z,\zeta) ( Z , ζ ) n − 2 ( 8 β 2 n + 4 Λ n ) n^{-2}(8\beta^2n+4\Lambda n) n − 2 ( 8 β 2 n + 4Λ n ) F ^ \widehat{F} F ( Z , ζ ) (Z,\zeta) ( Z , ζ ) ( ( 8 β 2 + 4 Λ ) / n ) 1 / 2 ((8\beta^2+4\Lambda)/n)^{1/2} (( 8 β 2 + 4Λ ) / n ) 1/2
P ( ∣ F ^ − Φ 0 ( v ∗ ) ∣ > ϵ / 2 ∣ v ∗ ) ≤ 2 exp ( − n ϵ 2 8 ( 8 β 2 + 4 Λ ) ) , Φ 0 ( v ∗ ) = E [ F ^ ∣ v ∗ ] . \mathbb{P}\left(\left|\widehat{F}-\Phi_0(v^*)\right|>\epsilon/2\mid v^*\right)\le2\exp\left(-\frac{n\epsilon^2}{8(8\beta^2+4\Lambda)}\right),\qquad\Phi_0(v^*)=\mathbb{E}[\widehat{F}\mid v^*]. P ( F − Φ 0 ( v ∗ ) > ϵ /2 ∣ v ∗ ) ≤ 2 exp ( − 8 ( 8 β 2 + 4Λ ) n ϵ 2 ) , Φ 0 ( v ∗ ) = E [ F ∣ v ∗ ] . Next, changing the coordinate v x ∗ v_x^* v x ∗ ν m x \nu_{m_x} ν m x 2 2 2 E t , r ( v ) \mathcal{E}_{t,r}(v) E t , r ( v ) 2 ∣ v x ∣ ( β 2 t ∑ y S x y ∣ v y ∗ v y ∣ + r x ) ≤ 4 ( 4 β 2 + Λ ) 2|v_x|(\beta^2t\sum_yS_{xy}|v_y^*v_y|+r_x)\le4(4\beta^2+\Lambda) 2∣ v x ∣ ( β 2 t ∑ y S x y ∣ v y ∗ v y ∣ + r x ) ≤ 4 ( 4 β 2 + Λ ) v v v F ^ \widehat{F} F 4 ( 4 β 2 + Λ ) / n 4(4\beta^2+\Lambda)/n 4 ( 4 β 2 + Λ ) / n Φ 0 \Phi_0 Φ 0 v ∗ v^* v ∗ E Φ 0 ( v ∗ ) = F ( t , r ) \mathbb{E}\Phi_0(v^*)=F(t,r) E Φ 0 ( v ∗ ) = F ( t , r )
P ( ∣ Φ 0 ( v ∗ ) − F ( t , r ) ∣ > ϵ / 2 ) ≤ 2 exp ( − n ϵ 2 32 ( 4 β 2 + Λ ) 2 ) . \mathbb{P}\left(\left|\Phi_0(v^*)-F(t,r)\right|>\epsilon/2\right)\le2\exp\left(-\frac{n\epsilon^2}{32(4\beta^2+\Lambda)^2}\right). P ( ∣ Φ 0 ( v ∗ ) − F ( t , r ) ∣ > ϵ /2 ) ≤ 2 exp ( − 32 ( 4 β 2 + Λ ) 2 n ϵ 2 ) . Since 8 ( 8 β 2 + 4 Λ ) ≤ C Λ 8(8\beta^2+4\Lambda)\le C_\Lambda 8 ( 8 β 2 + 4Λ ) ≤ C Λ 32 ( 4 β 2 + Λ ) 2 ≤ C Λ 32(4\beta^2+\Lambda)^2\le C_\Lambda 32 ( 4 β 2 + Λ ) 2 ≤ C Λ
The next lemma controls F ^ − F \widehat{F}-F F − F q + η S 2 y q+\eta S^{2}y q + η S 2 y S S S tr S 2 ≤ n / D \operatorname{tr} S^{2}\le n/D tr S 2 ≤ n / D n / ( D α 2 ) n/(D\alpha^{2}) n / ( D α 2 ) S S S α \alpha α
Lemma 7.11. Let β > 0 \beta>0 β > 0 0 < ϑ ≤ Λ 0<\vartheta\le\Lambda 0 < ϑ ≤ Λ C 0 C_{0} C 0 β \beta β ϑ \vartheta ϑ Λ \Lambda Λ D ≥ 1 D\ge1 D ≥ 1 S S S max x , y S x y ≤ 1 / D \max_{x,y}S_{xy}\le1/D max x , y S x y ≤ 1/ D m ∈ [ − 1 , 1 ] V m\in[-1,1]^{V} m ∈ [ − 1 , 1 ] V t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] q ∈ [ ϑ , Λ ] V q\in[\vartheta,\Lambda]^{V} q ∈ [ ϑ , Λ ] V A ≥ 1 A\ge1 A ≥ 1 η ∈ ( 0 , ϑ / ( 2 A ) ] \eta\in(0,\vartheta/(2A)] η ∈ ( 0 , ϑ / ( 2 A )] α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) M = { q + η S 2 y : y ∈ [ − A , A ] V } \mathcal{M}=\{q+\eta S^{2}y:y\in[-A,A]^{V}\} M = { q + η S 2 y : y ∈ [ − A , A ] V }
E sup r ∈ M ∣ F ^ ( t , r ) − F ( t , r ) ∣ ≤ C 0 ( ( log ( 3 / α 2 ) D α 2 ) 1 / 2 + η A α 2 ) . \mathbb{E}\sup_{r\in\mathcal{M}}\left|\widehat{F}(t,r)-F(t,r)\right|\le C_{0}\left(\left(\frac{\log(3/\alpha^{2})}{D\alpha^{2}}\right)^{1/2}+\eta A\alpha^{2}\right). E r ∈ M sup F ( t , r ) − F ( t , r ) ≤ C 0 ( ( D α 2 log ( 3/ α 2 ) ) 1/2 + η A α 2 ) . Proof. Since S 2 S^{2} S 2 S 2 y ∈ [ − A , A ] V S^{2}y\in[-A,A]^{V} S 2 y ∈ [ − A , A ] V y ∈ [ − A , A ] V y\in[-A,A]^{V} y ∈ [ − A , A ] V M \mathcal{M} M [ ϑ / 2 , Λ + ϑ / 2 ] V ⊂ [ ϑ / 2 , 2 Λ ] V [\vartheta/2,\Lambda+\vartheta/2]^{V}\subset[\vartheta/2,2\Lambda]^{V} [ ϑ /2 , Λ + ϑ /2 ] V ⊂ [ ϑ /2 , 2Λ ] V M \mathcal{M} M F ^ ( t , ⋅ ) \widehat{F}(t,\cdot) F ( t , ⋅ ) F ( t , ⋅ ) F(t,\cdot) F ( t , ⋅ )
We first construct a net. Let Π α \Pi_{\alpha} Π α S S S α \alpha α k α k_{\alpha} k α (103) , k α α 2 ≤ tr S 2 ≤ n / D k_{\alpha}\alpha^{2}\le\operatorname{tr}S^{2}\le n/D k α α 2 ≤ tr S 2 ≤ n / D ∣ ( I − Π α ) S 2 w ∣ ≤ α 2 ∣ w ∣ \lvert(I-\Pi_{\alpha})S^{2}w\rvert\le\alpha^{2}\lvert w\rvert ∣( I − Π α ) S 2 w ∣ ≤ α 2 ∣ w ∣ w ∈ R V w\in\mathbb{R}^{V} w ∈ R V ϱ = α 2 A n \varrho=\alpha^{2}A\sqrt{n} ϱ = α 2 A n y ∈ [ − A , A ] V y\in[-A,A]^{V} y ∈ [ − A , A ] V Π α S 2 y \Pi_{\alpha}S^{2}y Π α S 2 y A n A\sqrt{n} A n Π α \Pi_{\alpha} Π α ∥ S ∥ ≤ 1 \lVert S\rVert\le1 ∥ S ∥ ≤ 1 ϱ \varrho ϱ C C C
∣ C ∣ ≤ ( 1 + 2 A n / ϱ ) k α ≤ ( 3 / α 2 ) k α , |C|\le(1+2A\sqrt{n}/\varrho)^{k_{\alpha}}\le(3/\alpha^{2})^{k_{\alpha}}, ∣ C ∣ ≤ ( 1 + 2 A n / ϱ ) k α ≤ ( 3/ α 2 ) k α , by comparing the volumes of the disjoint balls of radius ϱ / 2 \varrho/2 ϱ /2 A n + ϱ / 2 A\sqrt{n}+\varrho/2 A n + ϱ /2 ϱ \varrho ϱ C C C c ∈ C c\in C c ∈ C y ∈ [ − A , A ] V y\in[-A,A]^{V} y ∈ [ − A , A ] V ∣ Π α S 2 y − c ∣ ≤ ϱ \lvert\Pi_{\alpha}S^{2}y-c\rvert\le\varrho ∣ Π α S 2 y − c ∣ ≤ ϱ y c y_{c} y c N \mathcal{N} N q + η S 2 y c ∈ M q+\eta S^{2}y_{c}\in\mathcal{M} q + η S 2 y c ∈ M N \mathcal{N} N ∣ N ∣ ≤ ( 3 / α 2 ) k α |\mathcal{N}|\le(3/\alpha^{2})^{k_{\alpha}} ∣ N ∣ ≤ ( 3/ α 2 ) k α y ∈ [ − A , A ] V y\in[-A,A]^{V} y ∈ [ − A , A ] V c ∈ C c\in C c ∈ C ∣ Π α S 2 y − c ∣ ≤ ϱ \lvert\Pi_{\alpha}S^{2}y-c\rvert\le\varrho ∣ Π α S 2 y − c ∣ ≤ ϱ
∣ S 2 ( y − y c ) ∣ ≤ ∣ Π α S 2 ( y − y c ) ∣ + ∣ ( I − Π α ) S 2 ( y − y c ) ∣ ≤ 2 ϱ + α 2 ∣ y − y c ∣ ≤ 4 ϱ . \lvert S^{2}(y-y_{c})\rvert\le\lvert\Pi_{\alpha}S^{2}(y-y_{c})\rvert+\lvert(I-\Pi_{\alpha})S^{2}(y-y_{c})\rvert\le2\varrho+\alpha^{2}\lvert y-y_{c}\rvert\le4\varrho. ∣ S 2 ( y − y c )∣ ≤ ∣ Π α S 2 ( y − y c )∣ + ∣( I − Π α ) S 2 ( y − y c )∣ ≤ 2 ϱ + α 2 ∣ y − y c ∣ ≤ 4 ϱ . In particular, every r ∈ M r\in\mathcal{M} r ∈ M 4 η α 2 A 4\eta\alpha^{2}A 4 η α 2 A N \mathcal{N} N
Next, by Lemma 7.10 with Λ \Lambda Λ 2 Λ 2\Lambda 2Λ
P ( max r ∈ N ∣ F ^ ( t , r ) − F ( t , r ) ∣ > ϵ ) ≤ 4 ∣ N ∣ e − n ϵ 2 / C , \mathbb{P}\left(\max_{r\in\mathcal{N}}\left|\widehat{F}(t,r)-F(t,r)\right|>\epsilon\right)\le4|\mathcal{N}|e^{-n\epsilon^{2}/C}, P ( r ∈ N max F ( t , r ) − F ( t , r ) > ϵ ) ≤ 4∣ N ∣ e − n ϵ 2 / C , where C = C 2 Λ C=C_{2\Lambda} C = C 2Λ ϵ 0 = ( C log ( 4 ∣ N ∣ ) / n ) 1 / 2 \epsilon_{0}=(C\log(4|\mathcal{N}|)/n)^{1/2} ϵ 0 = ( C log ( 4∣ N ∣ ) / n ) 1/2 ϵ ≥ ϵ 0 \epsilon\ge\epsilon_{0} ϵ ≥ ϵ 0 4 ∣ N ∣ e − n ϵ 2 / C ≤ e − n ( ϵ − ϵ 0 ) 2 / C 4|\mathcal{N}|e^{-n\epsilon^{2}/C}\le e^{-n(\epsilon-\epsilon_{0})^{2}/C} 4∣ N ∣ e − n ϵ 2 / C ≤ e − n ( ϵ − ϵ 0 ) 2 / C ϵ 2 − ϵ 0 2 ≥ ( ϵ − ϵ 0 ) 2 \epsilon^{2}-\epsilon_{0}^{2}\ge(\epsilon-\epsilon_{0})^{2} ϵ 2 − ϵ 0 2 ≥ ( ϵ − ϵ 0 ) 2
E max r ∈ N ∣ F ^ ( t , r ) − F ( t , r ) ∣ ≤ ϵ 0 + ∫ ϵ 0 ∞ e − n ( ϵ − ϵ 0 ) 2 / C d ϵ ≤ ( C log ( 4 ∣ N ∣ ) n ) 1 / 2 + ( C n ) 1 / 2 . \mathbb{E}\max_{r\in\mathcal{N}}\left|\widehat{F}(t,r)-F(t,r)\right|\le\epsilon_{0}+\int_{\epsilon_{0}}^{\infty}e^{-n(\epsilon-\epsilon_{0})^{2}/C}\,d\epsilon\le\left(\frac{C\log(4|\mathcal{N}|)}{n}\right)^{1/2}+\left(\frac{C}{n}\right)^{1/2}. E r ∈ N max F ( t , r ) − F ( t , r ) ≤ ϵ 0 + ∫ ϵ 0 ∞ e − n ( ϵ − ϵ 0 ) 2 / C d ϵ ≤ ( n C log ( 4∣ N ∣ ) ) 1/2 + ( n C ) 1/2 . Further, log ( 4 ∣ N ∣ ) ≤ log 4 + n log ( 3 / α 2 ) / ( D α 2 ) \log(4|\mathcal{N}|)\le\log4+n\log(3/\alpha^{2})/(D\alpha^{2}) log ( 4∣ N ∣ ) ≤ log 4 + n log ( 3/ α 2 ) / ( D α 2 ) n ≥ D n\ge D n ≥ D α < 1 \alpha<1 α < 1 log ( 3 / α 2 ) ≥ 1 \log(3/\alpha^{2})\ge1 log ( 3/ α 2 ) ≥ 1 log 4 / n \log4/n log 4/ n 1 / n 1/n 1/ n 2 log ( 3 / α 2 ) / ( D α 2 ) 2\log(3/\alpha^{2})/(D\alpha^{2}) 2 log ( 3/ α 2 ) / ( D α 2 ) 4 C 1 / 2 ( log ( 3 / α 2 ) / ( D α 2 ) ) 1 / 2 4C^{1/2}(\log(3/\alpha^{2})/(D\alpha^{2}))^{1/2} 4 C 1/2 ( log ( 3/ α 2 ) / ( D α 2 ) ) 1/2
Finally, we compare M \mathcal{M} M N \mathcal{N} N r ′ ∈ M r' \in\mathcal{M} r ′ ∈ M r ′ r' r ′ ϑ / 2 \vartheta/2 ϑ /2 2 n ∣ ∂ r x F ^ ( t , r ′ ) ∣ ≤ 2 3 / 2 ϑ − 1 / 2 ∣ ζ x ∣ + 12 2n|\partial_{r_x}\widehat{F}(t,r')| \le2^{3/2}\vartheta^{-1/2}|\zeta_x|+12 2 n ∣ ∂ r x F ( t , r ′ ) ∣ ≤ 2 3/2 ϑ − 1/2 ∣ ζ x ∣ + 12 ∣ b x ∣ ≤ 2 |b_x|\le2 ∣ b x ∣ ≤ 2 ∣ v x ∗ ∣ ≤ 2 |v_x^*|\le2 ∣ v x ∗ ∣ ≤ 2 ⟨ ( v x ∗ ) 2 ⟩ ≤ 4 \langle(v_x^*)^2\rangle\le4 ⟨( v x ∗ ) 2 ⟩ ≤ 4 M \mathcal{M} M r , r ′ ∈ M r,r'\in\mathcal{M} r , r ′ ∈ M
∣ F ^ ( t , r ) − F ^ ( t , r ′ ) ∣ ≤ 1 2 n ∑ x ( 2 3 / 2 ϑ − 1 / 2 ∣ ζ x ∣ + 12 ) ∣ r x − r x ′ ∣ ≤ ( 2 1 / 2 ϑ − 1 / 2 ∥ ζ ∥ n + 6 ) ∥ r − r ′ ∥ n . |\widehat{F}(t,r)-\widehat{F}(t,r')|
\le\frac{1}{2n}\sum_x\left(2^{3/2}\vartheta^{-1/2}|\zeta_x|+12\right)|r_x-r'_x|
\le\left(2^{1/2}\vartheta^{-1/2}\|\zeta\|_n+6\right)\|r-r'\|_n. ∣ F ( t , r ) − F ( t , r ′ ) ∣ ≤ 2 n 1 x ∑ ( 2 3/2 ϑ − 1/2 ∣ ζ x ∣ + 12 ) ∣ r x − r x ′ ∣ ≤ ( 2 1/2 ϑ − 1/2 ∥ ζ ∥ n + 6 ) ∥ r − r ′ ∥ n . and ∣ F ( t , r ) − F ( t , r ′ ) ∣ ≤ ∥ r − r ′ ∥ n / 2 |F(t,r)-F(t,r')|\le\|r-r'\|_n/2 ∣ F ( t , r ) − F ( t , r ′ ) ∣ ≤ ∥ r − r ′ ∥ n /2 E ∥ ζ ∥ n ≤ 1 \mathbb{E}\|\zeta\|_n\le1 E ∥ ζ ∥ n ≤ 1
E sup r ∈ M ∣ F ^ ( t , r ) − F ( t , r ) ∣ ≤ 4 C Λ 1 / 2 ( log ( 3 / α 2 ) D α 2 ) 1 / 2 + 4 n α 2 A ( 2 1 / 2 ϑ − 1 / 2 + 7 ) . \mathbb{E}\sup_{r\in\mathcal{M}}|\widehat{F}(t,r)-F(t,r)|
\le4C_{\Lambda}^{1/2}\left(\frac{\log(3/\alpha^2)}{D\alpha^2}\right)^{1/2}
+4n\alpha^2 A(2^{1/2}\vartheta^{-1/2}+7). E r ∈ M sup ∣ F ( t , r ) − F ( t , r ) ∣ ≤ 4 C Λ 1/2 ( D α 2 log ( 3/ α 2 ) ) 1/2 + 4 n α 2 A ( 2 1/2 ϑ − 1/2 + 7 ) . This proves the lemma with C 0 = max ( 4 C Λ 1 / 2 , 4 ( 2 1 / 2 ϑ − 1 / 2 + 7 ) ) C_0=\max(4C_{\Lambda}^{1/2},4(2^{1/2}\vartheta^{-1/2}+7)) C 0 = max ( 4 C Λ 1/2 , 4 ( 2 1/2 ϑ − 1/2 + 7 ))
The time derivative at contact points
We now show that at every point where F ( t , ⋅ ) F(t,\cdot) F ( t , ⋅ ) K / n K/n K / n F F F β 2 ( 4 n ) − 1 p ⋅ S p \beta^2(4n)^{-1}p\cdot Sp β 2 ( 4 n ) − 1 p ⋅ S p p = 2 n ∇ q F p=2n\nabla_qF p = 2 n ∇ q F
Definition 7.12. Let β > 0 \beta>0 β > 0 m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V 0 < ϑ ≤ Λ 0<\vartheta\le\Lambda 0 < ϑ ≤ Λ K > 0 K>0 K > 0 ( t , q ) ∈ ( 0 , 1 ] × [ ϑ , Λ ] V (t,q)\in(0,1]\times[\vartheta,\Lambda]^V ( t , q ) ∈ ( 0 , 1 ] × [ ϑ , Λ ] V ( ϑ , Λ , K ) (\vartheta,\Lambda,K) ( ϑ , Λ , K ) F F F
F ( t , q + h ) ≤ F ( t , q ) + ∇ q F ( t , q ) ⋅ h + K 2 n ∣ h ∣ 2 F(t,q+h)\le F(t,q)+\nabla_qF(t,q)\cdot h+\frac{K}{2n}|h|^2 F ( t , q + h ) ≤ F ( t , q ) + ∇ q F ( t , q ) ⋅ h + 2 n K ∣ h ∣ 2 for every h ∈ [ − ϑ / 2 , ϑ / 2 ] V h\in[-\vartheta/2,\vartheta/2]^V h ∈ [ − ϑ /2 , ϑ /2 ] V
Proposition 7.13. Let β > 0 \beta>0 β > 0 0 < ϑ ≤ Λ 0<\vartheta\le\Lambda 0 < ϑ ≤ Λ K > 0 K>0 K > 0 ϵ β , ϑ , Λ , K : [ 1 , ∞ ) → [ 0 , ∞ ) \epsilon_{\beta,\vartheta,\Lambda,K}:[1,\infty)\to[0,\infty) ϵ β , ϑ , Λ , K : [ 1 , ∞ ) → [ 0 , ∞ ) β \beta β ϑ \vartheta ϑ Λ \Lambda Λ K K K lim D → ∞ ϵ β , ϑ , Λ , K ( D ) = 0 \lim_{D\to\infty}\epsilon_{\beta,\vartheta,\Lambda,K}(D)=0 lim D → ∞ ϵ β , ϑ , Λ , K ( D ) = 0 D ≥ 1 D\ge1 D ≥ 1 S S S max x , y S x y ≤ 1 / D \max_{x,y}S_{xy}\le1/D max x , y S x y ≤ 1/ D m ∈ [ − 1 , 1 ] V m\in[-1,1]^V m ∈ [ − 1 , 1 ] V ( t , q ) (t,q) ( t , q ) ( ϑ , Λ , K ) (\vartheta,\Lambda,K) ( ϑ , Λ , K ) F F F p x = E b x 2 = 2 n ∂ q x F ( t , q ) p_x=\mathbb{E}b_x^2=2n\partial_{q_x}F(t,q) p x = E b x 2 = 2 n ∂ q x F ( t , q )
∂ t F ( t , q ) ≤ β 2 4 n p ⋅ S p + ϵ β , ϑ , Λ , K ( D ) . \partial_tF(t,q)\le\frac{\beta^2}{4n}p\cdot Sp+\epsilon_{\beta,\vartheta,\Lambda,K}(D). ∂ t F ( t , q ) ≤ 4 n β 2 p ⋅ S p + ϵ β , ϑ , Λ , K ( D ) . Proof. All posterior quantities are taken at the deterministic point ( t , q ) (t,q) ( t , q )
Step 1: the Hessian. By Lemma 7.8(c), F ( t , ⋅ ) F(t,\cdot) F ( t , ⋅ ) q q q h ∈ R V h\in\mathbb{R}^V h ∈ R V s > 0 s>0 s > 0 s h sh s h h h h s 2 h ⋅ ∇ q 2 F ( t , q ) h / 2 + o ( s 2 ) ≤ K s 2 ∣ h ∣ 2 / ( 2 n ) s^2h\cdot\nabla_q^2F(t,q)h/2+o(s^2)\le Ks^2|h|^2/(2n) s 2 h ⋅ ∇ q 2 F ( t , q ) h /2 + o ( s 2 ) ≤ K s 2 ∣ h ∣ 2 / ( 2 n ) ∇ q 2 F ( t , q ) ⪯ ( K / n ) I \nabla_q^2F(t,q)\preceq(K/n)I ∇ q 2 F ( t , q ) ⪯ ( K / n ) I Ξ = ( E Σ x y 2 ) x , y ∈ V \Xi=(\mathbb{E}\Sigma_{xy}^2)_{x,y\in V} Ξ = ( E Σ x y 2 ) x , y ∈ V
Ξ ⪯ 2 K I , E ∑ x , y Σ x y 2 = 1 ⋅ Ξ 1 ≤ 2 K n . (120) \Xi\preceq2KI,\qquad\mathbb{E}\sum_{x,y}\Sigma_{xy}^2=\mathbf{1}\cdot\Xi\mathbf{1}\le2Kn. \tag*{(120)} Ξ ⪯ 2 K I , E x , y ∑ Σ x y 2 = 1 ⋅ Ξ 1 ≤ 2 K n . (120) Step 2: reduction to the filtered fluctuation of b ⊙ 2 b^{\odot2} b ⊙ 2 Let w = b ⊙ 2 − p w=b^{\odot2}-p w = b ⊙ 2 − p E w = 0 \mathbb{E}w=0 E w = 0 ∣ w x ∣ ≤ 4 |w_x|\le4 ∣ w x ∣ ≤ 4 ⟨ v x v y ⟩ = Σ x y + b x b y \langle v_xv_y\rangle=\Sigma_{xy}+b_xb_y ⟨ v x v y ⟩ = Σ x y + b x b y
4 n β 2 ∂ t F ( t , q ) − p ⋅ S p = ∑ x , y S x y E Σ x y 2 + 2 ∑ x , y S x y E [ Σ x y b x b y ] + E [ w ⋅ S w ] . \frac{4n}{\beta^2}\partial_tF(t,q)-p\cdot Sp=\sum_{x,y}S_{xy}\mathbb{E}\Sigma_{xy}^2+2\sum_{x,y}S_{xy}\mathbb{E}\left[\Sigma_{xy}b_xb_y\right]+\mathbb{E}\left[w\cdot Sw\right]. β 2 4 n ∂ t F ( t , q ) − p ⋅ S p = x , y ∑ S x y E Σ x y 2 + 2 x , y ∑ S x y E [ Σ x y b x b y ] + E [ w ⋅ S w ] . where we used E [ b ⊙ 2 ⋅ S b ⊙ 2 ] − p ⋅ S p = E [ w ⋅ S w ] \mathbb{E}[b^{\odot2}\cdot Sb^{\odot2}]-p\cdot Sp=\mathbb{E}[w\cdot Sw] E [ b ⊙ 2 ⋅ S b ⊙ 2 ] − p ⋅ S p = E [ w ⋅ S w ] S x y ≤ 1 / D S_{xy}\le1/D S x y ≤ 1/ D 2 K n / D 2Kn/D 2 K n / D ∣ b x ∣ ≤ 2 |b_x|\le2 ∣ b x ∣ ≤ 2
8 ( ∑ x , y S x y 2 ) 1 / 2 ( E ∑ x , y Σ x y 2 ) 1 / 2 ≤ 8 ( n D ) 1 / 2 ( 2 K n ) 1 / 2 . 8\left(\sum_{x,y}S_{xy}^2\right)^{1/2}\left(\mathbb{E}\sum_{x,y}\Sigma_{xy}^2\right)^{1/2}\le8\left(\frac{n}{D}\right)^{1/2}(2Kn)^{1/2}. 8 ( x , y ∑ S x y 2 ) 1/2 ( E x , y ∑ Σ x y 2 ) 1/2 ≤ 8 ( D n ) 1/2 ( 2 K n ) 1/2 . For the third term we only use ∥ S ∥ ≤ 1 \lVert S\rVert\le1 ∥ S ∥ ≤ 1 ∣ w ∣ ≤ 4 n |w|\le4\sqrt{n} ∣ w ∣ ≤ 4 n ∣ w ⋅ S w ∣ ≤ ∣ w ∣ ∣ S w ∣ ≤ 4 n ∣ S w ∣ |w\cdot Sw|\le|w||Sw|\le4\sqrt{n}|Sw| ∣ w ⋅ S w ∣ ≤ ∣ w ∣∣ S w ∣ ≤ 4 n ∣ S w ∣ 4 n ( E ∥ S w ∥ n 2 ) 1 / 2 4n(\mathbb{E}\lVert Sw\rVert_n^2)^{1/2} 4 n ( E ∥ S w ∥ n 2 ) 1/2
∂ t F ( t , q ) − β 2 4 n p ⋅ S p ≤ β 2 4 ( 2 K D + 8 ( 2 K D ) 1 / 2 + 4 ( E ∥ S w ∥ n 2 ) 1 / 2 ) . (121) \partial_tF(t,q)-\frac{\beta^2}{4n}p\cdot Sp\le\frac{\beta^2}{4}\left(\frac{2K}{D}+8\left(\frac{2K}{D}\right)^{1/2}+4\left(\mathbb{E}\lVert Sw\rVert_n^2\right)^{1/2}\right). \tag*{(121)} ∂ t F ( t , q ) − 4 n β 2 p ⋅ S p ≤ 4 β 2 ( D 2 K + 8 ( D 2 K ) 1/2 + 4 ( E ∥ S w ∥ n 2 ) 1/2 ) . (121) Step 3: the term U U U Let ξ = χ − p \xi=\chi-p ξ = χ − p χ \chi χ U = χ − b ⊙ 2 U=\chi-b^{\odot2} U = χ − b ⊙ 2 w = ξ − U w=\xi-U w = ξ − U
E ∣ S U ∣ 2 = tr ( S E [ U U T ] S ) = ∑ x ( S 2 ) x x p x q x + tr ( S Ξ S ) ≤ n D ϑ + 2 K tr S 2 ≤ ( 1 ϑ + 2 K ) n D . \mathbb{E}|SU|^2=\operatorname{tr}\left(S\mathbb{E}[UU^{\mathsf T}]S\right)=\sum_x(S^2)_{xx}\frac{p_x}{q_x}+\operatorname{tr}(S\Xi S)\le\frac{n}{D\vartheta}+2K\operatorname{tr}S^2\le\left(\frac{1}{\vartheta}+2K\right)\frac{n}{D}. E ∣ S U ∣ 2 = tr ( S E [ U U T ] S ) = x ∑ ( S 2 ) xx q x p x + tr ( S Ξ S ) ≤ D ϑ n + 2 K tr S 2 ≤ ( ϑ 1 + 2 K ) D n . where we used p x ≤ 1 p_x\le1 p x ≤ 1 q x ≥ ϑ q_x\ge\vartheta q x ≥ ϑ S Ξ S ⪯ 2 K S 2 S\Xi S\preceq2KS^2 S Ξ S ⪯ 2 K S 2
E ∥ S w ∥ n 2 ≤ 2 E ∥ S ξ ∥ n 2 + 2 ( 1 ϑ + 2 K ) 1 D . (122) \mathbb{E}\lVert Sw\rVert_n^2\le2\mathbb{E}\lVert S\xi\rVert_n^2+2\left(\frac{1}{\vartheta}+2K\right)\frac{1}{D}. \tag*{(122)} E ∥ S w ∥ n 2 ≤ 2 E ∥ S ξ ∥ n 2 + 2 ( ϑ 1 + 2 K ) D 1 . (122) Step 4: the term ξ \xi ξ Fix A ≥ 1 A\ge1 A ≥ 1 η ∈ ( 0 , ϑ / ( 2 A ) ] \eta\in(0,\vartheta/(2A)] η ∈ ( 0 , ϑ / ( 2 A )] α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) ξ A \xi^A ξ A max ( − A , min ( A , ξ x ) ) \max(-A,\min(A,\xi_x)) max ( − A , min ( A , ξ x )) h = η S ξ A h=\eta S\xi^A h = η S ξ A h ∈ [ − η A , η A ] V ⊂ [ − ϑ / 2 , ϑ / 2 ] V h\in[-\eta A,\eta A]^V\subset[-\vartheta/2,\vartheta/2]^V h ∈ [ − η A , η A ] V ⊂ [ − ϑ /2 , ϑ /2 ] V q + h q+h q + h M \mathcal{M} M q q q q + h q+h q + h ϑ / 2 \vartheta/2 ϑ /2 ∇ q 2 F ^ ⪰ − n − 1 diag ( 2 1 / 2 ϑ − 3 / 2 ∣ ζ x ∣ ) \nabla_q^2\widehat F\succeq-n^{-1}\operatorname{diag}(2^{1/2}\vartheta^{-3/2}|\zeta_x|) ∇ q 2 F ⪰ − n − 1 diag ( 2 1/2 ϑ − 3/2 ∣ ζ x ∣ ) 2 n ∇ q F ^ ( t , q ) = χ 2n\nabla_q\widehat F(t,q)=\chi 2 n ∇ q F ( t , q ) = χ
F ^ ( t , q + h ) − F ^ ( t , q ) ≥ 1 2 n χ ⋅ h − η 2 A 2 ϑ 3 / 2 ⋅ 1 n ∑ x ∣ ζ x ∣ . \widehat F(t,q+h)-\widehat F(t,q)\ge\frac{1}{2n}\chi\cdot h-\frac{\eta^2A^2}{\vartheta^{3/2}}\cdot\frac{1}{n}\sum_x|\zeta_x|. F ( t , q + h ) − F ( t , q ) ≥ 2 n 1 χ ⋅ h − ϑ 3/2 η 2 A 2 ⋅ n 1 x ∑ ∣ ζ x ∣. On the other hand, (7.18) and 2 n ∇ q F ( t , q ) = p 2n\nabla_qF(t,q)=p 2 n ∇ q F ( t , q ) = p
F ( t , q + h ) − F ( t , q ) ≤ 1 2 n p ⋅ h + K 2 η 2 A 2 . F(t,q+h)-F(t,q)\le\frac{1}{2n}p\cdot h+\frac{K}{2}\eta^2A^2. F ( t , q + h ) − F ( t , q ) ≤ 2 n 1 p ⋅ h + 2 K η 2 A 2 . Subtracting the second inequality from the first, we have
η 2 n ζ ⋅ S 2 ζ A ≤ 2 sup r ∈ M ∣ F ^ ( t , r ) − F ( t , r ) ∣ + η 2 A 2 ( ϑ − 3 / 2 1 n ∑ x ∣ ζ x ∣ + K 2 ) . \frac{\eta}{2n}\zeta\cdot S^{2}\zeta^{A}\le2\sup_{r\in\mathcal{M}}\left|\widehat{F}(t,r)-F(t,r)\right|+\eta^{2}A^{2}\left(\vartheta^{-3/2}\frac{1}{n}\sum_{x}|\zeta_{x}|+\frac{K}{2}\right). 2 n η ζ ⋅ S 2 ζ A ≤ 2 r ∈ M sup F ( t , r ) − F ( t , r ) + η 2 A 2 ( ϑ − 3/2 n 1 x ∑ ∣ ζ x ∣ + 2 K ) . Taking expectations, using E ∣ ζ x ∣ ≤ 1 \mathbb{E}|\zeta_x|\le1 E ∣ ζ x ∣ ≤ 1
E [ 1 n ζ ⋅ S 2 ζ A ] ≤ 4 C 0 η ( log ( 3 / α 2 ) D α 2 ) 1 / 2 + 4 C 0 A α 2 + η A 2 ( 2 ϑ − 3 / 2 + K ) . (123) \mathbb{E}\left[\frac{1}{n}\zeta\cdot S^{2}\zeta^{A}\right]\le\frac{4C_{0}}{\eta}\left(\frac{\log(3/\alpha^{2})}{D\alpha^{2}}\right)^{1/2}+4C_{0}A\alpha^{2}+\eta A^{2}(2\vartheta^{-3/2}+K). \tag*{(123)} E [ n 1 ζ ⋅ S 2 ζ A ] ≤ η 4 C 0 ( D α 2 log ( 3/ α 2 ) ) 1/2 + 4 C 0 A α 2 + η A 2 ( 2 ϑ − 3/2 + K ) . (123) where C 0 C_{0} C 0 (117) and q x ≥ ϑ q_x\ge\vartheta q x ≥ ϑ ∣ ζ x ∣ ≤ φ ( ζ x ) |\zeta_x|\le\varphi(\zeta_x) ∣ ζ x ∣ ≤ φ ( ζ x ) φ ( z ) = 2 ϑ − 1 / 2 ∣ z ∣ + 13 \varphi(z)=2\vartheta^{-1/2}|z|+13 φ ( z ) = 2 ϑ − 1/2 ∣ z ∣ + 13 G ∼ N ( 0 , 1 ) G\sim N(0,1) G ∼ N ( 0 , 1 ) ϖ ( A ) = E [ φ ( G ) 2 1 { φ ( G ) > A } ] \varpi(A)=\mathbb{E}[\varphi(G)^{2}\mathbf{1}_{\{\varphi(G)>A\}}] ϖ ( A ) = E [ φ ( G ) 2 1 { φ ( G ) > A } ] A → ∞ A\to\infty A → ∞ E ∥ ζ ∥ n 2 ≤ E φ ( G ) 2 \mathbb{E}\|\zeta\|_{n}^{2}\le\mathbb{E}\varphi(G)^{2} E ∥ ζ ∥ n 2 ≤ E φ ( G ) 2 E ∥ ζ − ζ A ∥ n 2 ≤ ϖ ( A ) \mathbb{E}\|\zeta-\zeta^{A}\|_{n}^{2}\le\varpi(A) E ∥ ζ − ζ A ∥ n 2 ≤ ϖ ( A ) ∥ S ∥ ≤ 1 \|S\|\le1 ∥ S ∥ ≤ 1
E ∥ S ζ ∥ n 2 = E [ 1 n ζ ⋅ S 2 ζ A ] + E [ 1 n ζ ⋅ S 2 ( ζ − ζ A ) ] ≤ E [ 1 n ζ ⋅ S 2 ζ A ] + ( E φ ( G ) 2 ) 1 / 2 ϖ ( A ) 1 / 2 . \mathbb{E}\|S\zeta\|_{n}^{2}=\mathbb{E}\left[\frac{1}{n}\zeta\cdot S^{2}\zeta^{A}\right]+\mathbb{E}\left[\frac{1}{n}\zeta\cdot S^{2}(\zeta-\zeta^{A})\right]\le\mathbb{E}\left[\frac{1}{n}\zeta\cdot S^{2}\zeta^{A}\right]+\bigl(\mathbb{E}\varphi(G)^{2}\bigr)^{1/2}\varpi(A)^{1/2}. E ∥ S ζ ∥ n 2 = E [ n 1 ζ ⋅ S 2 ζ A ] + E [ n 1 ζ ⋅ S 2 ( ζ − ζ A ) ] ≤ E [ n 1 ζ ⋅ S 2 ζ A ] + ( E φ ( G ) 2 ) 1/2 ϖ ( A ) 1/2 . Let e ( D ) e(D) e ( D ) A ≥ 1 A\ge1 A ≥ 1 η ∈ ( 0 , ϑ / ( 2 A ) ] \eta\in(0,\vartheta/(2A)] η ∈ ( 0 , ϑ / ( 2 A )] α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) (123) and ( E φ ( G ) 2 ) 1 / 2 ϖ ( A ) 1 / 2 \bigl(\mathbb{E}\varphi(G)^{2}\bigr)^{1/2}\varpi(A)^{1/2} ( E φ ( G ) 2 ) 1/2 ϖ ( A ) 1/2 E ∥ S ζ ∥ n 2 ≤ e ( D ) \mathbb{E}\|S\zeta\|_{n}^{2}\le e(D) E ∥ S ζ ∥ n 2 ≤ e ( D ) e e e β \beta β ϑ \vartheta ϑ Λ \Lambda Λ K K K D D D e ( D ) → 0 e(D)\to0 e ( D ) → 0 D → ∞ D\to\infty D → ∞ ι > 0 \iota>0 ι > 0 A A A ( E φ ( G ) 2 ) 1 / 2 ϖ ( A ) 1 / 2 ≤ ι \bigl(\mathbb{E}\varphi(G)^{2}\bigr)^{1/2}\varpi(A)^{1/2}\le\iota ( E φ ( G ) 2 ) 1/2 ϖ ( A ) 1/2 ≤ ι η ≤ ϑ / ( 2 A ) \eta\le\vartheta/(2A) η ≤ ϑ / ( 2 A ) η A 2 ( 2 ϑ − 3 / 2 + K ) ≤ ι \eta A^{2}(2\vartheta^{-3/2}+K)\le\iota η A 2 ( 2 ϑ − 3/2 + K ) ≤ ι α \alpha α 4 C 0 A α 2 ≤ ι 4C_{0}A\alpha^{2}\le\iota 4 C 0 A α 2 ≤ ι D 0 D_{0} D 0 (123) is at most ι \iota ι D ≥ D 0 D\ge D_{0} D ≥ D 0 e ( D ) ≤ 4 ι e(D)\le4\iota e ( D ) ≤ 4 ι D ≥ D 0 D\ge D_{0} D ≥ D 0
Step 5: conclusion. Combining (121) , (122) , and Step 4, we obtain the proposition with
ϵ β , ϑ , Λ , K ( D ) = β 2 4 ( 2 K D + 8 ( 2 K D ) 1 / 2 + 4 ( 2 e ( D ) + 2 ( 1 ϑ + 2 K ) 1 D ) 1 / 2 ) . \epsilon_{\beta,\vartheta,\Lambda,K}(D)=\frac{\beta^{2}}{4}\left(\frac{2K}{D}+8\left(\frac{2K}{D}\right)^{1/2}+4\left(2e(D)+2\left(\frac{1}{\vartheta}+2K\right)\frac{1}{D}\right)^{1/2}\right). ϵ β , ϑ , Λ , K ( D ) = 4 β 2 ( D 2 K + 8 ( D 2 K ) 1/2 + 4 ( 2 e ( D ) + 2 ( ϑ 1 + 2 K ) D 1 ) 1/2 ) . This function is nonincreasing and tends to zero as D → ∞ D\to\infty D → ∞
The scalar Hopf–Lax formula
Lemma 7.14. Let β ≥ 0 \beta\ge0 β ≥ 0 μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] t , r ≥ 0 t,r\ge0 t , r ≥ 0
f μ ( t , r ) = max 0 ≤ s ≤ 1 ( ψ μ ( r + β 2 t s ) − β 2 t s 2 4 ) . f_{\mu}(t,r)=\max_{0\le s\le1}\left(\psi_{\mu}(r+\beta^{2}ts)-\frac{\beta^{2}ts^{2}}{4}\right). f μ ( t , r ) = 0 ≤ s ≤ 1 max ( ψ μ ( r + β 2 t s ) − 4 β 2 t s 2 ) . (a) We have f μ ( 0 , r ) = ψ μ ( r ) f_{\mu}(0,r)=\psi_{\mu}(r) f μ ( 0 , r ) = ψ μ ( r ) f μ ( 1 , 0 ) = max 0 ≤ s ≤ 1 Γ μ ( s ) / 2 f_{\mu}(1,0)=\max_{0\le s\le1}\Gamma_{\mu}(s)/2 f μ ( 1 , 0 ) = max 0 ≤ s ≤ 1 Γ μ ( s ) /2
(b) The function f μ f_{\mu} f μ ∣ f μ ( t , r ) − f μ ( t ′ , r ′ ) ∣ ≤ β 2 ∣ t − t ′ ∣ / 4 + ∣ r − r ′ ∣ / 2 |f_{\mu}(t,r)-f_{\mu}(t',r')|\le\beta^{2}|t-t'|/4+|r-r'|/2 ∣ f μ ( t , r ) − f μ ( t ′ , r ′ ) ∣ ≤ β 2 ∣ t − t ′ ∣/4 + ∣ r − r ′ ∣/2
(c) At every point ( t , r ) ∈ ( 0 , ∞ ) 2 (t,r)\in(0,\infty)^{2} ( t , r ) ∈ ( 0 , ∞ ) 2 f μ f_{\mu} f μ ∂ t f μ = β 2 ( ∂ r f μ ) 2 \partial_t f_{\mu}=\beta^{2}(\partial_r f_{\mu})^{2} ∂ t f μ = β 2 ( ∂ r f μ ) 2
Proof. We write f = f μ f=f_\mu f = f μ ψ μ \psi_\mu ψ μ
(b) For each s s s r ↦ ψ μ ( r + β 2 t s ) − β 2 t s 2 / 4 r\mapsto\psi_\mu(r+\beta^2ts)-\beta^2ts^2/4 r ↦ ψ μ ( r + β 2 t s ) − β 2 t s 2 /4 1 / 2 1/2 1/2 s s s f f f 1 / 2 1/2 1/2 r r r 0 ≤ t < t ′ 0\leq t<t' 0 ≤ t < t ′ s s s ( t , r ) (t,r) ( t , r ) s ′ = s t / t ′ ∈ [ 0 , 1 ] s'=st/t'\in[0,1] s ′ = s t / t ′ ∈ [ 0 , 1 ] r + β 2 t ′ s ′ = r + β 2 t s r+\beta^2t's'=r+\beta^2ts r + β 2 t ′ s ′ = r + β 2 t s β 2 t ′ s ′ 2 / 4 ≤ β 2 t s 2 / 4 \beta^2t's'^2/4\leq\beta^2ts^2/4 β 2 t ′ s ′2 /4 ≤ β 2 t s 2 /4 f ( t ′ , r ) ≥ f ( t , r ) f(t',r)\geq f(t,r) f ( t ′ , r ) ≥ f ( t , r ) s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ]
ψ μ ( r + β 2 t ′ s ) − β 2 t ′ s 2 4 ≤ ψ μ ( r + β 2 t s ) − β 2 t s 2 4 + β 2 ( t ′ − t ) ( s 2 − s 2 4 ) , \psi_\mu(r+\beta^2t's)-\frac{\beta^2t's^2}{4}\leq\psi_\mu(r+\beta^2ts)-\frac{\beta^2ts^2}{4}+\beta^2(t'-t)\left(\frac{s}{2}-\frac{s^2}{4}\right), ψ μ ( r + β 2 t ′ s ) − 4 β 2 t ′ s 2 ≤ ψ μ ( r + β 2 t s ) − 4 β 2 t s 2 + β 2 ( t ′ − t ) ( 2 s − 4 s 2 ) , and s / 2 − s 2 / 4 ≤ 1 / 4 s/2-s^2/4\leq1/4 s /2 − s 2 /4 ≤ 1/4 f ( t ′ , r ) ≤ f ( t , r ) + β 2 ( t ′ − t ) / 4 f(t',r)\leq f(t,r)+\beta^2(t'-t)/4 f ( t ′ , r ) ≤ f ( t , r ) + β 2 ( t ′ − t ) /4
(c) If β = 0 \beta=0 β = 0 f ( t , r ) = ψ μ ( r ) f(t,r)=\psi_\mu(r) f ( t , r ) = ψ μ ( r ) β > 0 \beta>0 β > 0 ( t , r ) ∈ ( 0 , ∞ ) 2 (t,r)\in(0,\infty)^2 ( t , r ) ∈ ( 0 , ∞ ) 2 f f f s ∗ s^* s ∗ ( t , r ) (t,r) ( t , r ) f ~ ( t ′ , r ′ ) = ψ μ ( r ′ + β 2 t ′ s ∗ ) − β 2 t ′ s ∗ 2 / 4 \widetilde f(t',r')=\psi_\mu(r'+\beta^2t's^*)-\beta^2t's^{*2}/4 f ( t ′ , r ′ ) = ψ μ ( r ′ + β 2 t ′ s ∗ ) − β 2 t ′ s ∗ 2 /4 f ~ ≤ f \widetilde f\leq f f ≤ f ( t , r ) (t,r) ( t , r ) ( t , r ) (t,r) ( t , r ) ψ μ \psi_\mu ψ μ r + β 2 t s ∗ > 0 r+\beta^2ts^*>0 r + β 2 t s ∗ > 0 f − f ~ f-\widetilde f f − f ( t , r ) (t,r) ( t , r ) ∇ f ( t , r ) = ∇ f ~ ( t , r ) \nabla f(t,r)=\nabla\widetilde f(t,r) ∇ f ( t , r ) = ∇ f ( t , r )
∂ r f ( t , r ) = ψ μ ′ ( r + β 2 t s ∗ ) , ∂ t f ( t , r ) = β 2 s ∗ ψ μ ′ ( r + β 2 t s ∗ ) − β 2 s ∗ 2 4 . \partial_r f(t,r)=\psi_\mu'(r+\beta^2ts^*),\qquad\partial_t f(t,r)=\beta^2s^*\psi_\mu'(r+\beta^2ts^*)-\frac{\beta^2s^{*2}}{4}. ∂ r f ( t , r ) = ψ μ ′ ( r + β 2 t s ∗ ) , ∂ t f ( t , r ) = β 2 s ∗ ψ μ ′ ( r + β 2 t s ∗ ) − 4 β 2 s ∗ 2 . The derivative of s ↦ ψ μ ( r + β 2 t s ) − β 2 t s 2 / 4 s\mapsto\psi_\mu(r+\beta^2ts)-\beta^2ts^2/4 s ↦ ψ μ ( r + β 2 t s ) − β 2 t s 2 /4 β 2 t ( ψ μ ′ ( r + β 2 t s ) − s / 2 ) \beta^2t(\psi_\mu'(r+\beta^2ts)-s/2) β 2 t ( ψ μ ′ ( r + β 2 t s ) − s /2 ) s ∗ = 0 s^*=0 s ∗ = 0 s = 0 s=0 s = 0 ψ μ ′ ( r ) = 0 \psi_\mu'(r)=0 ψ μ ′ ( r ) = 0 ∂ r f = ∂ t f = 0 \partial_r f=\partial_t f=0 ∂ r f = ∂ t f = 0 s ∗ ∈ ( 0 , 1 ) s^*\in(0,1) s ∗ ∈ ( 0 , 1 ) s ∗ s^* s ∗ s ∗ = 1 s^*=1 s ∗ = 1 s = 1 s=1 s = 1 ψ μ ′ ≤ 1 / 2 \psi_\mu'\leq1/2 ψ μ ′ ≤ 1/2 ψ μ ′ ( r + β 2 t ) = 1 / 2 \psi_\mu'(r+\beta^2t)=1/2 ψ μ ′ ( r + β 2 t ) = 1/2 ψ μ ′ ( r + β 2 t s ∗ ) = s ∗ / 2 \psi_\mu'(r+\beta^2ts^*)=s^*/2 ψ μ ′ ( r + β 2 t s ∗ ) = s ∗ /2 ∂ r f = s ∗ / 2 \partial_r f=s^*/2 ∂ r f = s ∗ /2 ∂ t f = β 2 s ∗ 2 / 4 = β 2 ( ∂ r f ) 2 \partial_t f=\beta^2s^{*2}/4=\beta^2(\partial_r f)^2 ∂ t f = β 2 s ∗ 2 /4 = β 2 ( ∂ r f ) 2 ∂ r f ∈ [ 0 , 1 / 2 ] \partial_r f\in[0,1/2] ∂ r f ∈ [ 0 , 1/2 ]
The function f μ f_\mu f μ k k k R 2 \mathbb{R}^2 R 2 ( 1 / 2 , 1 ) 2 (1/2,1)^2 ( 1/2 , 1 ) 2 c k = ∫ ∣ ∂ 2 k ( u , v ) ∣ d u d v c_k=\int|\partial_2k(u,v)|\,du\,dv c k = ∫ ∣ ∂ 2 k ( u , v ) ∣ d u d v
Lemma 7.15. Let β ≥ 0 \beta\geq0 β ≥ 0 μ \mu μ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] γ > 0 \gamma>0 γ > 0 ( t , r ) ∈ ( − γ / 2 , ∞ ) 2 (t,r)\in(-\gamma/2,\infty)^2 ( t , r ) ∈ ( − γ /2 , ∞ ) 2
w ( t , r ) = ∫ R 2 f μ ( t + γ u , r + γ v ) k ( u , v ) d u d v . w(t,r)=\int_{\mathbb{R}^2}f_\mu(t+\gamma u,r+\gamma v)k(u,v)\,du\,dv. w ( t , r ) = ∫ R 2 f μ ( t + γ u , r + γ v ) k ( u , v ) d u d v . Then w w w t , r > − γ / 2 t,r>-\gamma/2 t , r > − γ /2
(a) 0 ≤ ∂ r w ( t , r ) ≤ 1 / 2 0\leq\partial_rw(t,r)\leq1/2 0 ≤ ∂ r w ( t , r ) ≤ 1/2 ∣ ∂ r 2 w ( t , r ) ∣ ≤ c k / ( 2 γ ) |\partial_r^2w(t,r)|\leq c_k/(2\gamma) ∣ ∂ r 2 w ( t , r ) ∣ ≤ c k / ( 2 γ )
(b) ∂ t w ( t , r ) ≥ β 2 ( ∂ r w ( t , r ) ) 2 \partial_tw(t,r)\geq\beta^2(\partial_rw(t,r))^2 ∂ t w ( t , r ) ≥ β 2 ( ∂ r w ( t , r ) ) 2
(c) w ( 0 , r ) ≥ ψ μ ( r ) w(0,r)\geq\psi_\mu(r) w ( 0 , r ) ≥ ψ μ ( r ) r ≥ 0 r\geq0 r ≥ 0
(d) w ( 1 , r ) ≤ f μ ( 1 , 0 ) + r / 2 + ( β 2 + 2 ) γ / 4 w(1,r)\leq f_\mu(1,0)+r/2+(\beta^2+2)\gamma/4 w ( 1 , r ) ≤ f μ ( 1 , 0 ) + r /2 + ( β 2 + 2 ) γ /4 r ≥ 0 r\geq0 r ≥ 0
Proof. Since k k k ( 1 / 2 , 1 ) 2 (1/2,1)^2 ( 1/2 , 1 ) 2 f μ f_\mu f μ w w w ( 0 , ∞ ) 2 (0,\infty)^2 ( 0 , ∞ ) 2 w ( t , r ) = γ − 2 ∫ f μ ( t ′ , r ′ ) k ( ( t ′ − t ) / γ , ( r ′ − r ) / γ ) d t ′ d r ′ w(t,r)=\gamma^{-2}\int f_\mu(t',r')k((t'-t)/\gamma,(r'-r)/\gamma)\,\mathrm{d}t'\,\mathrm{d}r' w ( t , r ) = γ − 2 ∫ f μ ( t ′ , r ′ ) k (( t ′ − t ) / γ , ( r ′ − r ) / γ ) d t ′ d r ′ w w w Lemma 7.14 (b) and Rademacher’s theorem, f μ f_\mu f μ ( 0 , ∞ ) 2 (0,\infty)^2 ( 0 , ∞ ) 2
∂ t w ( t , r ) = ∫ ∂ t f μ ( t + γ u , r + γ v ) k ( u , v ) d u d v , ∂ r w ( t , r ) = ∫ ∂ r f μ ( t + γ u , r + γ v ) k ( u , v ) d u d v . \begin{aligned}
\partial_t w(t,r)&=\int\partial_t f_\mu(t+\gamma u,r+\gamma v)k(u,v)\,\mathrm{d}u\,\mathrm{d}v,\\
\partial_r w(t,r)&=\int\partial_r f_\mu(t+\gamma u,r+\gamma v)k(u,v)\,\mathrm{d}u\,\mathrm{d}v.
\end{aligned} ∂ t w ( t , r ) ∂ r w ( t , r ) = ∫ ∂ t f μ ( t + γ u , r + γ v ) k ( u , v ) d u d v , = ∫ ∂ r f μ ( t + γ u , r + γ v ) k ( u , v ) d u d v . The second formula and Lemma 7.14 (c) give 0 ≤ ∂ r w ≤ 1 / 2 0\leq\partial_r w\leq1/2 0 ≤ ∂ r w ≤ 1/2 γ − 2 ∫ ∂ r f μ ( t ′ , r ′ ) k ( ( t ′ − t ) / γ , ( r ′ − r ) / γ ) d t ′ d r ′ \gamma^{-2}\int\partial_r f_\mu(t',r')k((t'-t)/\gamma,(r'-r)/\gamma)\,\mathrm{d}t'\,\mathrm{d}r' γ − 2 ∫ ∂ r f μ ( t ′ , r ′ ) k (( t ′ − t ) / γ , ( r ′ − r ) / γ ) d t ′ d r ′ r r r ∣ ∂ r 2 w ∣ ≤ ( 1 / 2 ) γ − 1 c k |\partial_r^2w|\leq(1/2)\gamma^{-1}c_k ∣ ∂ r 2 w ∣ ≤ ( 1/2 ) γ − 1 c k Lemma 7.14 (c) and Jensen’s inequality,
∂ t w = β 2 ∫ ( ∂ r f μ ) 2 ( t + γ u , r + γ v ) k ( u , v ) d u d v ≥ β 2 ( ∂ r w ) 2 , \partial_t w=\beta^2\int(\partial_r f_\mu)^2(t+\gamma u,r+\gamma v)k(u,v)\,\mathrm{d}u\,\mathrm{d}v\geq\beta^2(\partial_r w)^2, ∂ t w = β 2 ∫ ( ∂ r f μ ) 2 ( t + γ u , r + γ v ) k ( u , v ) d u d v ≥ β 2 ( ∂ r w ) 2 , which proves (b). By Lemma 7.14 (a,b), f μ ( γ u , r + γ v ) ≥ f μ ( 0 , r ) = ψ μ ( r ) f_\mu(\gamma u,r+\gamma v)\geq f_\mu(0,r)=\psi_\mu(r) f μ ( γ u , r + γ v ) ≥ f μ ( 0 , r ) = ψ μ ( r ) u , v ≥ 0 u,v\geq0 u , v ≥ 0 Lemma 7.14 (b), f μ ( 1 + γ u , r + γ v ) ≤ f μ ( 1 + γ , r + γ ) ≤ f μ ( 1 , 0 ) + β 2 γ / 4 + ( r + γ ) / 2 f_\mu(1+\gamma u,r+\gamma v)\leq f_\mu(1+\gamma,r+\gamma)\leq f_\mu(1,0)+\beta^2\gamma/4+(r+\gamma)/2 f μ ( 1 + γ u , r + γ v ) ≤ f μ ( 1 + γ , r + γ ) ≤ f μ ( 1 , 0 ) + β 2 γ /4 + ( r + γ ) /2 u , v ≤ 1 u,v\leq1 u , v ≤ 1
Proof of the relative-entropy bound
The comparison function in the proof of Theorem 7.5 contains a penalty for the distance from a moving convex set of profiles. Since only the initial profiles in this set are used, the initial discrepancy is needed only on the set A S ( a ) \mathcal{A}_S(a) A S ( a )
Lemma 7.16. Let β > 0 \beta>0 β > 0 a > 0 a>0 a > 0 V = β 2 S [ 0 , 1 ] V \mathcal{V}=\beta^2\mathcal{S}[0,1]^\mathcal{V} V = β 2 S [ 0 , 1 ] V t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] A t = a 1 + ( 2 − t ) V \mathcal{A}_t=a\mathbf{1}+(2-t)\mathcal{V} A t = a 1 + ( 2 − t ) V ( t , q ) ∈ [ 0 , 1 ] × R V (t,q)\in[0,1]\times\mathbb{R}^\mathcal{V} ( t , q ) ∈ [ 0 , 1 ] × R V z ( t , q ) z(t,q) z ( t , q ) A t \mathcal{A}_t A t q q q
B ( t , q ) = 1 2 a dist V ( q , A t ) 2 = 1 2 n a ∣ q − z ( t , q ) ∣ 2 . \mathcal{B}(t,q)=\frac{1}{2a}\operatorname{dist}_{\mathcal{V}}(q,\mathcal{A}_t)^2=\frac{1}{2na}|q-z(t,q)|^2. B ( t , q ) = 2 a 1 dist V ( q , A t ) 2 = 2 na 1 ∣ q − z ( t , q ) ∣ 2 . (a) Each A t \mathcal{A}_t A t a 1 ∈ A t a\mathbf{1}\in\mathcal{A}_t a 1 ∈ A t A t ⊂ [ a , a + 2 β 2 ] V \mathcal{A}_t\subset[a,a+2\beta^2]^\mathcal{V} A t ⊂ [ a , a + 2 β 2 ] V A 0 = A S ( a ) \mathcal{A}_0=\mathcal{A}_S(a) A 0 = A S ( a )
(b) For each t t t B ( t , ⋅ ) \mathcal{B}(t,\cdot) B ( t , ⋅ ) ∇ q B ( t , q ) = ( q − z ( t , q ) ) / ( n a ) \nabla_q\mathcal{B}(t,q)=(q-z(t,q))/(na) ∇ q B ( t , q ) = ( q − z ( t , q )) / ( na )
B ( t , q + h ) ≤ B ( t , q ) + ∇ q B ( t , q ) ⋅ h + ∣ h ∣ 2 2 n a ( q , h ∈ R V ) . \mathcal{B}(t,q+h)\leq\mathcal{B}(t,q)+\nabla_q\mathcal{B}(t,q)\cdot h+\frac{|h|^2}{2na}\qquad(q,h\in\mathbb{R}^\mathcal{V}). B ( t , q + h ) ≤ B ( t , q ) + ∇ q B ( t , q ) ⋅ h + 2 na ∣ h ∣ 2 ( q , h ∈ R V ) . (c) For each q q q B ( ⋅ , q ) \mathcal{B}(\cdot,q) B ( ⋅ , q ) [ 0 , 1 ] [0,1] [ 0 , 1 ]
∂ t B ( t , q ) = max v ∈ V ∇ q B ( t , q ) ⋅ v . \partial_t\mathcal{B}(t,q)=\max_{v\in\mathcal{V}}\nabla_q\mathcal{B}(t,q)\cdot v. ∂ t B ( t , q ) = v ∈ V max ∇ q B ( t , q ) ⋅ v . (d) The function B \mathcal{B} B [ 0 , 1 ] × R V [0,1]\times\mathbb{R}^\mathcal{V} [ 0 , 1 ] × R V
Proof. (a) The set V \mathcal{V} V 0 0 0 V ⊂ [ 0 , β 2 ] V \mathcal{V} \subset[0,\beta^2]^{\mathcal{V}} V ⊂ [ 0 , β 2 ] V S S S
(b) Fix t t t z = z ( t , q ) z=z(t,q) z = z ( t , q ) z ′ = z ( t , q + h ) z'=z(t,q+h) z ′ = z ( t , q + h ) z ∈ A t z\in A_t z ∈ A t
2 n a B ( t , q + h ) ≤ ∣ q + h − z ∣ 2 = ∣ q − z ∣ 2 + 2 ( q − z ) ⋅ h + ∣ h ∣ 2 , 2na\mathcal{B}(t,q+h)\leq|q+h-z|^2=|q-z|^2+2(q-z)\cdot h+|h|^2, 2 na B ( t , q + h ) ≤ ∣ q + h − z ∣ 2 = ∣ q − z ∣ 2 + 2 ( q − z ) ⋅ h + ∣ h ∣ 2 , which is the claimed inequality. Since the projection onto a closed convex set is 1-Lipschitz, ∣ z − z ′ ∣ ≤ ∣ h ∣ |z-z'|\leq|h| ∣ z − z ′ ∣ ≤ ∣ h ∣
2 n a B ( t , q + h ) = ∣ q − z ′ ∣ 2 + 2 ( q − z ′ ) ⋅ h + ∣ h ∣ 2 ≥ ∣ q − z ∣ 2 + 2 ( q − z ) ⋅ h − 2 ∣ h ∣ 2 . 2na\mathcal{B}(t,q+h)=|q-z'|^2+2(q-z')\cdot h+|h|^2\geq|q-z|^2+2(q-z)\cdot h-2|h|^2. 2 na B ( t , q + h ) = ∣ q − z ′ ∣ 2 + 2 ( q − z ′ ) ⋅ h + ∣ h ∣ 2 ≥ ∣ q − z ∣ 2 + 2 ( q − z ) ⋅ h − 2∣ h ∣ 2 . From the two bounds, we see that B ( t , ⋅ ) \mathcal{B}(t,\cdot) B ( t , ⋅ ) q q q
(c) Fix q q q s ∈ [ 1 , 2 ] s\in[1,2] s ∈ [ 1 , 2 ] v s ∈ V v_s\in\mathcal{V} v s ∈ V z ( 2 − s , q ) = a 1 + s v s z(2-s,q)=a\mathbf{1}+sv_s z ( 2 − s , q ) = a 1 + s v s s > 0 s>0 s > 0
ϕ ( s ) = ∣ q − a 1 − s v s ∣ 2 = min v ∈ V ∣ q − a 1 − s v ∣ 2 , \phi(s)=|q-a\mathbf{1}-sv_s|^2=\min_{v\in\mathcal{V}}|q-a\mathbf{1}-sv|^2, ϕ ( s ) = ∣ q − a 1 − s v s ∣ 2 = v ∈ V min ∣ q − a 1 − s v ∣ 2 , so that B ( t , q ) = ϕ ( 2 − t ) / ( 2 n a ) \mathcal{B}(t,q)=\phi(2-t)/(2na) B ( t , q ) = ϕ ( 2 − t ) / ( 2 na ) s , s ′ ∈ [ 1 , 2 ] s,s'\in[1,2] s , s ′ ∈ [ 1 , 2 ]
ϕ ( s ′ ) ≤ ∣ q − a 1 − s ′ v s ∣ 2 = ϕ ( s ) − 2 ( s ′ − s ) ( q − a 1 − s v s ) ⋅ v s + ( s ′ − s ) 2 ∣ v s ∣ 2 , \phi(s')\leq|q-a\mathbf{1}-s'v_s|^2=\phi(s)-2(s'-s)(q-a\mathbf{1}-sv_s)\cdot v_s+(s'-s)^2|v_s|^2, ϕ ( s ′ ) ≤ ∣ q − a 1 − s ′ v s ∣ 2 = ϕ ( s ) − 2 ( s ′ − s ) ( q − a 1 − s v s ) ⋅ v s + ( s ′ − s ) 2 ∣ v s ∣ 2 , and the same holds with s s s s ′ s' s ′ s ↦ v s s\mapsto v_s s ↦ v s s j → s s_j\to s s j → s v ˉ \bar v v ˉ ( v s j ) (v_{s_j}) ( v s j ) v ˉ ∈ V \bar v\in\mathcal{V} v ˉ ∈ V ∣ q − a 1 − s v ˉ ∣ 2 = lim j ϕ ( s j ) ≤ lim j ∣ q − a 1 − s j v s j ∣ 2 = ϕ ( s ) |q-a\mathbf{1}-s\bar v|^2=\lim_j\phi(s_j)\leq\lim_j|q-a\mathbf{1}-s_jv_{s_j}|^2=\phi(s) ∣ q − a 1 − s v ˉ ∣ 2 = lim j ϕ ( s j ) ≤ lim j ∣ q − a 1 − s j v s j ∣ 2 = ϕ ( s ) v ˉ = v s \bar v=v_s v ˉ = v s ϕ \phi ϕ [ 1 , 2 ] [1,2] [ 1 , 2 ] ϕ ′ ( s ) = − 2 ( q − z ) ⋅ v s \phi'(s)=-2(q-z)\cdot v_s ϕ ′ ( s ) = − 2 ( q − z ) ⋅ v s z = z ( 2 − s , q ) z=z(2-s,q) z = z ( 2 − s , q ) t = 2 − s t=2-s t = 2 − s ∂ t B ( t , q ) = ∇ q B ( t , q ) ⋅ v s \partial_t\mathcal{B}(t,q)=\nabla_q\mathcal{B}(t,q)\cdot v_s ∂ t B ( t , q ) = ∇ q B ( t , q ) ⋅ v s v ∈ V v\in\mathcal{V} v ∈ V a 1 + s v a\mathbf{1}+sv a 1 + s v A t A_t A t ( q − z ) ⋅ ( a 1 + s v − z ) ≤ 0 (q-z)\cdot(a\mathbf{1}+sv-z)\leq0 ( q − z ) ⋅ ( a 1 + s v − z ) ≤ 0 s ( q − z ) ⋅ ( v − v s ) ≤ 0 s(q-z)\cdot(v-v_s)\leq0 s ( q − z ) ⋅ ( v − v s ) ≤ 0 ∇ q B ( t , q ) ⋅ v ≤ ∇ q B ( t , q ) ⋅ v s \nabla_q\mathcal{B}(t,q)\cdot v\leq\nabla_q\mathcal{B}(t,q)\cdot v_s ∇ q B ( t , q ) ⋅ v ≤ ∇ q B ( t , q ) ⋅ v s v ∈ V v\in\mathcal{V} v ∈ V
(d) Every point a 1 + ( 2 − t ) v a\mathbf{1}+(2-t)v a 1 + ( 2 − t ) v A t A_t A t v ∈ V v\in\mathcal{V} v ∈ V ∣ t − t ′ ∣ ∥ v ∥ n ≤ β 2 ∣ t − t ′ ∣ |t-t'|\|v\|_n\leq\beta^2|t-t'| ∣ t − t ′ ∣∥ v ∥ n ≤ β 2 ∣ t − t ′ ∣ a 1 + ( 2 − t ′ ) v a\mathbf{1}+(2-t')v a 1 + ( 2 − t ′ ) v A t ′ A_{t'} A t ′
∣ dist n ( q , A t ) − dist n ( q ′ , A t ′ ) ∣ ≤ ∥ q − q ′ ∥ n + β 2 ∣ t − t ′ ∣ . \left|\operatorname{dist}_n(q,A_t)-\operatorname{dist}_n(q',A_{t'})\right|\leq\|q-q'\|_n+\beta^2|t-t'|. ∣ dist n ( q , A t ) − dist n ( q ′ , A t ′ ) ∣ ≤ ∥ q − q ′ ∥ n + β 2 ∣ t − t ′ ∣. Proof of Theorem 7.5. (a) If β = 0 \beta=0 β = 0 Q m = P 0 Q_m=P_0 Q m = P 0 D ( m ) = 0 D(m)=0 D ( m ) = 0 Γ μ = 0 \Gamma_\mu=0 Γ μ = 0 A S ( a ) = { a 1 } \mathcal{A}_S(a)=\{a\mathbf{1}\} A S ( a ) = { a 1 } Δ S , a ( m ; μ ) ≥ − ψ μ ( a ) ≥ − a / 2 \Delta_{S,a}(m;\mu)\geq-\psi_\mu(a)\geq-a/2 Δ S , a ( m ; μ ) ≥ − ψ μ ( a ) ≥ − a /2 e 0 , a = 0 e_{0,a}=0 e 0 , a = 0
Let β > 0 \beta>0 β > 0 a > 0 a>0 a > 0 γ = a / ( 1 + β 2 ) \gamma=a/(1+\beta^2) γ = a / ( 1 + β 2 ) ϑ = a / 2 \vartheta=a/2 ϑ = a /2 Λ = 3 a / 2 + 2 β 2 \Lambda=3a/2+2\beta^2 Λ = 3 a /2 + 2 β 2 K = c k / ( 2 γ ) + 1 / a K=c_k/(2\gamma)+1/a K = c k / ( 2 γ ) + 1/ a e β , a = ϵ β , ϑ , Λ , K e_{\beta,a}=\epsilon_{\beta,\vartheta,\Lambda,K} e β , a = ϵ β , ϑ , Λ , K D D D S S S m m m μ \mu μ w w w μ \mu μ γ \gamma γ
Ω ( t , q ) = 1 n ∑ x ∈ V w ( t , q x ) ( t ∈ [ 0 , 1 ] , q ∈ [ 0 , ∞ ) V ) . \Omega(t,q)=\frac{1}{n}\sum_{x\in\mathcal{V}}w(t,q_x)\qquad(t\in[0,1],\ q\in[0,\infty)^{\mathcal{V}}). Ω ( t , q ) = n 1 x ∈ V ∑ w ( t , q x ) ( t ∈ [ 0 , 1 ] , q ∈ [ 0 , ∞ ) V ) . Let V \mathcal{V} V A t A_t A t z ( t , q ) z(t,q) z ( t , q ) B \mathcal{B} B ρ > e β , a ( D ) \rho>e_{\beta,a}(D) ρ > e β , a ( D )
L ( t , q ) = F ( t , q ) − Ω ( t , q ) − B ( t , q ) − ρ t ( ( t , q ) ∈ [ 0 , 1 ] × [ 0 , ∞ ) V ) . \mathcal{L}(t,q)=F(t,q)-\Omega(t,q)-\mathcal{B}(t,q)-\rho t\qquad((t,q)\in[0,1]\times[0,\infty)^{\mathcal{V}}). L ( t , q ) = F ( t , q ) − Ω ( t , q ) − B ( t , q ) − ρt (( t , q ) ∈ [ 0 , 1 ] × [ 0 , ∞ ) V ) . Step 1: L \mathcal{L} L By Lemma 7.8(b), F ( t , q ) ≤ F ( 1 , 0 ) + ∥ q ∥ n / 2 F(t,q)\leq F(1,0)+\|q\|_n/2 F ( t , q ) ≤ F ( 1 , 0 ) + ∥ q ∥ n /2 Ω ≥ 0 \Omega\geq0 Ω ≥ 0 f μ ≥ ψ μ ≥ 0 f_\mu\geq\psi_\mu\geq0 f μ ≥ ψ μ ≥ 0 dist n ( q , A t ) ≥ ∥ q ∥ n − a − 2 β 2 \operatorname{dist}_n(q,A_t)\geq\|q\|_n-a-2\beta^2 dist n ( q , A t ) ≥ ∥ q ∥ n − a − 2 β 2 L ( t , q ) → − ∞ \mathcal{L}(t,q)\to-\infty L ( t , q ) → − ∞ ∥ q ∥ n → ∞ \|q\|_n\to\infty ∥ q ∥ n → ∞ t t t L \mathcal{L} L [ 0 , 1 ] × [ 0 , ∞ ) V [0,1]\times[0,\infty)^{V} [ 0 , 1 ] × [ 0 , ∞ ) V ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 )
Step 2: q 0 ∈ ( 0 , ∞ ) V q_0\in(0,\infty)^{V} q 0 ∈ ( 0 , ∞ ) V Suppose that ( q 0 ) x = 0 (q_0)_x=0 ( q 0 ) x = 0 x x x e x e_x e x h ∈ ( 0 , a ) h\in(0,a) h ∈ ( 0 , a ) F ( t 0 , q 0 + h e x ) ≥ F ( t 0 , q 0 ) F(t_0,q_0+he_x)\ge F(t_0,q_0) F ( t 0 , q 0 + h e x ) ≥ F ( t 0 , q 0 ) Ω ( t 0 , q 0 + h e x ) − Ω ( t 0 , q 0 ) ≤ h / ( 2 n ) \Omega(t_0,q_0+he_x)-\Omega(t_0,q_0)\le h/(2n) Ω ( t 0 , q 0 + h e x ) − Ω ( t 0 , q 0 ) ≤ h / ( 2 n ) z = z ( t 0 , q 0 ) z=z(t_0,q_0) z = z ( t 0 , q 0 ) z x ≥ a z_x\ge a z x ≥ a
B ( t 0 , q 0 + h e x ) − B ( t 0 , q 0 ) ≤ h ( ( q 0 ) x − z x ) n a + h 2 2 n a ≤ − h n + h 2 2 n a . \mathcal{B}(t_0,q_0+he_x)-\mathcal{B}(t_0,q_0)\le\frac{h((q_0)_x-z_x)}{na}+\frac{h^2}{2na}\le-\frac{h}{n}+\frac{h^2}{2na}. B ( t 0 , q 0 + h e x ) − B ( t 0 , q 0 ) ≤ na h (( q 0 ) x − z x ) + 2 na h 2 ≤ − n h + 2 na h 2 . Hence L ( t 0 , q 0 + h e x ) − L ( t 0 , q 0 ) ≥ ( h / n ) ( 1 / 2 − h / ( 2 a ) ) > 0 \mathcal{L}(t_0,q_0+he_x)-\mathcal{L}(t_0,q_0)\ge(h/n)(1/2-h/(2a))>0 L ( t 0 , q 0 + h e x ) − L ( t 0 , q 0 ) ≥ ( h / n ) ( 1/2 − h / ( 2 a )) > 0 ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 )
Step 3: location of q 0 q_0 q 0 By Step 2, Lemma 7.8(c), Lemma 7.15, and Lemma 7.16(b), the function L ( t 0 , ⋅ ) \mathcal{L}(t_0,\cdot) L ( t 0 , ⋅ ) q 0 q_0 q 0 ∇ q L ( t 0 , q 0 ) = 0 \nabla_q\mathcal{L}(t_0,q_0)=0 ∇ q L ( t 0 , q 0 ) = 0
p = 2 n ∇ q F ( t 0 , q 0 ) , p ′ = 2 n ∇ q Ω ( t 0 , q 0 ) = ( 2 ∂ r w ( t 0 , ( q 0 ) x ) ) x ∈ V . p=2n\nabla_qF(t_0,q_0),\qquad p'=2n\nabla_q\Omega(t_0,q_0)=\left(2\partial_rw(t_0,(q_0)_x)\right)_{x\in V}. p = 2 n ∇ q F ( t 0 , q 0 ) , p ′ = 2 n ∇ q Ω ( t 0 , q 0 ) = ( 2 ∂ r w ( t 0 , ( q 0 ) x ) ) x ∈ V . By Lemma 7.8(c) and Lemma 7.15(a), p , p ′ ∈ [ 0 , 1 ] V p,p'\in[0,1]^V p , p ′ ∈ [ 0 , 1 ] V z = z ( t 0 , q 0 ) z=z(t_0,q_0) z = z ( t 0 , q 0 )
q 0 − z a = n ∇ q B ( t 0 , q 0 ) = p − p ′ 2 ∈ [ − 1 2 , 1 2 ] V . \frac{q_0-z}{a}=n\nabla_q\mathcal{B}(t_0,q_0)=\frac{p-p'}{2}\in\left[-\frac{1}{2},\frac{1}{2}\right]^V. a q 0 − z = n ∇ q B ( t 0 , q 0 ) = 2 p − p ′ ∈ [ − 2 1 , 2 1 ] V . Hence ∣ ( q 0 ) x − z x ∣ ≤ a / 2 |(q_0)_x-z_x|\le a/2 ∣ ( q 0 ) x − z x ∣ ≤ a /2 x x x q 0 ∈ [ a / 2 , 3 a / 2 + 2 β 2 ] V = [ ϑ , Λ ] V q_0\in[a/2,3a/2+2\beta^2]^V=[\vartheta,\Lambda]^V q 0 ∈ [ a /2 , 3 a /2 + 2 β 2 ] V = [ ϑ , Λ ] V dist n ( q 0 , A t 0 ) ≤ a / 2 \operatorname{dist}_n(q_0,\mathcal{A}_{t_0})\le a/2 dist n ( q 0 , A t 0 ) ≤ a /2
Step 4: t 0 = 0 t_0=0 t 0 = 0 Suppose that t 0 > 0 t_0>0 t 0 > 0 ( t 0 , q 0 ) (t_0,q_0) ( t 0 , q 0 ) ( ϑ , Λ , K ) (\vartheta,\Lambda,K) ( ϑ , Λ , K ) F F F h ∈ [ − ϑ / 2 , ϑ / 2 ] V h\in[-\vartheta/2,\vartheta/2]^V h ∈ [ − ϑ /2 , ϑ /2 ] V q 0 + h ∈ [ 0 , ∞ ) V q_0+h\in[0,\infty)^V q 0 + h ∈ [ 0 , ∞ ) V L ( t 0 , q 0 + h ) ≤ L ( t 0 , q 0 ) \mathcal{L}(t_0,q_0+h)\le\mathcal{L}(t_0,q_0) L ( t 0 , q 0 + h ) ≤ L ( t 0 , q 0 )
F ( t 0 , q 0 + h ) − F ( t 0 , q 0 ) ≤ Ω ( t 0 , q 0 + h ) − Ω ( t 0 , q 0 ) + B ( t 0 , q 0 + h ) − B ( t 0 , q 0 ) ≤ ∇ q ( Ω + B ) ( t 0 , q 0 ) ⋅ h + ( c k 4 γ + 1 2 a ) ∣ h ∣ 2 n . \begin{aligned}
F(t_0,q_0+h)-F(t_0,q_0)\le\Omega(t_0,q_0+h)-\Omega(t_0,q_0)+\mathcal{B}(t_0,q_0+h)-\mathcal{B}(t_0,q_0) \\
&\le\nabla_q(\Omega+\mathcal{B})(t_0,q_0)\cdot h+\left(\frac{c_k}{4\gamma}+\frac{1}{2a}\right)\frac{|h|^2}{n}.
\end{aligned} F ( t 0 , q 0 + h ) − F ( t 0 , q 0 ) ≤ Ω ( t 0 , q 0 + h ) − Ω ( t 0 , q 0 ) + B ( t 0 , q 0 + h ) − B ( t 0 , q 0 ) ≤ ∇ q ( Ω + B ) ( t 0 , q 0 ) ⋅ h + ( 4 γ c k + 2 a 1 ) n ∣ h ∣ 2 . Since ∇ q ( Ω + B ) ( t 0 , q 0 ) = ∇ q F ( t 0 , q 0 ) \nabla_q(\Omega+\mathcal{B})(t_0,q_0)=\nabla_qF(t_0,q_0) ∇ q ( Ω + B ) ( t 0 , q 0 ) = ∇ q F ( t 0 , q 0 ) c k / ( 4 γ ) + 1 / ( 2 a ) = K / 2 c_k/(4\gamma)+1/(2a)=K/2 c k / ( 4 γ ) + 1/ ( 2 a ) = K /2
∂ t F ( t 0 , q 0 ) ≤ β 2 4 n p ⋅ S p + e β , a ( D ) . \partial_tF(t_0,q_0)\le\frac{\beta^2}{4n}p\cdot Sp+e_{\beta,a}(D). ∂ t F ( t 0 , q 0 ) ≤ 4 n β 2 p ⋅ S p + e β , a ( D ) . Next, by Lemma 7.15(b) and S ⪯ I S\preceq I S ⪯ I
∂ t Ω ( t 0 , q 0 ) = 1 n ∑ x ∂ t w ( t 0 , ( q 0 ) x ) ≥ β 2 n ∑ x ( ∂ r w ( t 0 , ( q 0 ) x ) ) 2 = β 2 4 n ∣ p ′ ∣ 2 ≥ β 2 4 n p ′ ⋅ S p ′ . \partial_t\Omega(t_0,q_0)=\frac{1}{n}\sum_x\partial_tw(t_0,(q_0)_x)\ge\frac{\beta^2}{n}\sum_x\left(\partial_rw(t_0,(q_0)_x)\right)^2=\frac{\beta^2}{4n}|p'|^2\ge\frac{\beta^2}{4n}p'\cdot Sp'. ∂ t Ω ( t 0 , q 0 ) = n 1 x ∑ ∂ t w ( t 0 , ( q 0 ) x ) ≥ n β 2 x ∑ ( ∂ r w ( t 0 , ( q 0 ) x ) ) 2 = 4 n β 2 ∣ p ′ ∣ 2 ≥ 4 n β 2 p ′ ⋅ S p ′ . Further, since S S S p − p ′ = 2 n ∇ q B ( t 0 , q 0 ) p-p'=2n\nabla_q\mathcal{B}(t_0,q_0) p − p ′ = 2 n ∇ q B ( t 0 , q 0 )
β 2 4 n ( p ⋅ S p − p ′ ⋅ S p ′ ) = β 2 4 n S ( p + p ′ ) ⋅ ( p − p ′ ) = v ~ ⋅ ∇ q B ( t 0 , q 0 ) , v ~ = β 2 S ( p + p ′ 2 ) . \frac{\beta^2}{4n}(p\cdot Sp-p'\cdot Sp')=\frac{\beta^2}{4n}S(p+p')\cdot(p-p')=\widetilde v\cdot\nabla_q\mathcal{B}(t_0,q_0),\qquad\widetilde v=\beta^2S\left(\frac{p+p'}{2}\right). 4 n β 2 ( p ⋅ S p − p ′ ⋅ S p ′ ) = 4 n β 2 S ( p + p ′ ) ⋅ ( p − p ′ ) = v ⋅ ∇ q B ( t 0 , q 0 ) , v = β 2 S ( 2 p + p ′ ) . Since ( p + p ′ ) / 2 ∈ [ 0 , 1 ] V (p+p')/2 \in[0,1]^{V} ( p + p ′ ) /2 ∈ [ 0 , 1 ] V v ~ ∈ V \tilde{v} \in V v ~ ∈ V v ~ ⋅ ∇ q B ( t 0 , q 0 ) ≤ ∂ t B ( t 0 , q 0 ) \tilde{v} \cdot\nabla_{q}\mathcal{B}(t_{0},q_{0}) \le\partial_{t}\mathcal{B}(t_{0},q_{0}) v ~ ⋅ ∇ q B ( t 0 , q 0 ) ≤ ∂ t B ( t 0 , q 0 )
∂ t F ( t 0 , q 0 ) ≤ ∂ t Ω ( t 0 , q 0 ) + ∂ t B ( t 0 , q 0 ) + ϵ β , a ( D ) < ∂ t Ω ( t 0 , q 0 ) + ∂ t B ( t 0 , q 0 ) + ρ , \partial_{t}F(t_{0},q_{0}) \le\partial_{t}\Omega(t_{0},q_{0})+\partial_{t}\mathcal{B}(t_{0},q_{0})+\epsilon_{\beta,a}(D)<\partial_{t}\Omega(t_{0},q_{0})+\partial_{t}\mathcal{B}(t_{0},q_{0})+\rho, ∂ t F ( t 0 , q 0 ) ≤ ∂ t Ω ( t 0 , q 0 ) + ∂ t B ( t 0 , q 0 ) + ϵ β , a ( D ) < ∂ t Ω ( t 0 , q 0 ) + ∂ t B ( t 0 , q 0 ) + ρ , that is, ∂ t L ( t 0 , q 0 ) < 0 \partial_{t}\mathcal{L}(t_{0},q_{0})<0 ∂ t L ( t 0 , q 0 ) < 0 t 0 = 1 t_{0}=1 t 0 = 1 t ↦ L ( t , q 0 ) t\mapsto\mathcal{L}(t,q_{0}) t ↦ L ( t , q 0 ) t 0 t_{0} t 0 [ 0 , 1 ] [0,1] [ 0 , 1 ] t 0 ∈ ( 0 , 1 ] t_{0}\in(0,1] t 0 ∈ ( 0 , 1 ] ∂ t L ( t 0 , q 0 ) ≥ 0 \partial_{t}\mathcal{L}(t_{0},q_{0})\ge0 ∂ t L ( t 0 , q 0 ) ≥ 0 t 0 = 0 t_{0}=0 t 0 = 0
Step 5: conclusion. Since a 1 ∈ A 1 a1\in\mathcal{A}_{1} a 1 ∈ A 1 B ( 1 , a 1 ) = 0 \mathcal{B}(1,a1)=0 B ( 1 , a 1 ) = 0 Ω ( 1 , a 1 ) = w ( 1 , a ) \Omega(1,a1)=w(1,a) Ω ( 1 , a 1 ) = w ( 1 , a ) B ≥ 0 \mathcal{B}\ge0 B ≥ 0 t 0 = 0 t_{0}=0 t 0 = 0 ( 0 , q 0 ) (0,q_{0}) ( 0 , q 0 )
F ( 1 , a 1 ) − w ( 1 , a ) − ρ = L ( 1 , a 1 ) ≤ L ( 0 , q 0 ) ≤ F ( 0 , q 0 ) − Ω ( 0 , q 0 ) ≤ 1 n ∑ x ( ψ m x ( ( q 0 ) x ) − ψ μ ( ( q 0 ) x ) ) . \begin{aligned}
F(1,a1)-w(1,a)-\rho=\mathcal{L}(1,a1)\le\mathcal{L}(0,q_{0})\le F(0,q_{0})-\Omega(0,q_{0}) \\
&\le\frac{1}{n}\sum_{x}\left(\psi_{m_x}((q_{0})_x)-\psi_{\mu}((q_{0})_x)\right).
\end{aligned} F ( 1 , a 1 ) − w ( 1 , a ) − ρ = L ( 1 , a 1 ) ≤ L ( 0 , q 0 ) ≤ F ( 0 , q 0 ) − Ω ( 0 , q 0 ) ≤ n 1 x ∑ ( ψ m x (( q 0 ) x ) − ψ μ (( q 0 ) x ) ) . where in the last step we used Lemma 7.8(a) and Lemma 7.15(c). By Step 3 and Lemma 7.16(a), there exists z ∈ A 0 = A S ( a ) z\in\mathcal{A}_{0}=\mathcal{A}_{S}(a) z ∈ A 0 = A S ( a ) ∥ q 0 − z ∥ n ≤ a / 2 \lVert q_{0}-z\rVert_{n}\le a/2 ∥ q 0 − z ∥ n ≤ a /2 ψ m x \psi_{m_x} ψ m x ψ μ \psi_{\mu} ψ μ 1 / 2 1/2 1/2
1 n ∑ x ( ψ m x ( z x ) − ψ μ ( z x ) ) + 1 n ∑ x ∣ ( q 0 ) x − z x ∣ ≤ Δ S , a ( m ; μ ) + ∥ q 0 − z ∥ n ≤ Δ S , a ( m ; μ ) + a 2 . \frac{1}{n}\sum_{x}\left(\psi_{m_x}(z_x)-\psi_{\mu}(z_x)\right)+\frac{1}{n}\sum_{x}|(q_{0})_x-z_x|\le\Delta_{S,a}(m;\mu)+\lVert q_{0}-z\rVert_{n}\le\Delta_{S,a}(m;\mu)+\frac{a}{2}. n 1 x ∑ ( ψ m x ( z x ) − ψ μ ( z x ) ) + n 1 x ∑ ∣ ( q 0 ) x − z x ∣ ≤ Δ S , a ( m ; μ ) + ∥ q 0 − z ∥ n ≤ Δ S , a ( m ; μ ) + 2 a . By Lemma 7.8(a,b), Lemma 7.15(d), and Lemma 7.14(a), and since ( β 2 + 2 ) γ / 4 ≤ a / 2 (\beta^{2}+2)\gamma/4\le a/2 ( β 2 + 2 ) γ /4 ≤ a /2
1 n D ( m ) = F ( 1 , 0 ) ≤ F ( 1 , a 1 ) ≤ w ( 1 , a ) + ρ + Δ S , a ( m ; μ ) + a 2 ≤ 1 2 sup 0 ≤ s ≤ 1 Γ μ ( s ) + 3 a 2 + Δ S , a ( m ; μ ) + ρ . \begin{aligned}
\frac{1}{n}\mathcal{D}(m)=F(1,0)\le F(1,a1)\le w(1,a)+\rho+\Delta_{S,a}(m;\mu)+\frac{a}{2} \\
&\le\frac{1}{2}\sup_{0\le s\le1}\Gamma_{\mu}(s)+\frac{3a}{2}+\Delta_{S,a}(m;\mu)+\rho.
\end{aligned} n 1 D ( m ) = F ( 1 , 0 ) ≤ F ( 1 , a 1 ) ≤ w ( 1 , a ) + ρ + Δ S , a ( m ; μ ) + 2 a ≤ 2 1 0 ≤ s ≤ 1 sup Γ μ ( s ) + 2 3 a + Δ S , a ( m ; μ ) + ρ . Letting ρ ↓ ϵ β , a ( D ) \rho\downarrow\epsilon_{\beta,a}(D) ρ ↓ ϵ β , a ( D )
(b) Fix a > 0 a>0 a > 0 k k k S = S k S=S_k S = S k D = D k D=D_k D = D k m ( k ) m^{(k)} m ( k ) D \mathcal{D} D [ − 1 , 1 ] n k [-1,1]^{n_k} [ − 1 , 1 ] n k Δ S k , a ( ⋅ ; μ ) \Delta_{S_k,a}(\cdot\,;\mu) Δ S k , a ( ⋅ ; μ )
1 n k E D ( m ( k ) ) ≤ 1 2 sup 0 ≤ s ≤ 1 Γ μ ( s ) + 2 a + E sup r ∈ A S k ( a ) ∣ 1 n k ∑ x ( ψ m x ( k ) ( r x ) − ψ μ ( r x ) ) ∣ + ϵ β , a ( D k ) . \frac{1}{n_k}\mathbb{E}\mathcal{D}(m^{(k)})\le\frac{1}{2}\sup_{0\le s\le1}\Gamma_{\mu}(s)+2a+\mathbb{E}\sup_{r\in\mathcal{A}_{S_k}(a)}\left|\frac{1}{n_k}\sum_{x}\left(\psi_{m_x^{(k)}}(r_x)-\psi_{\mu}(r_x)\right)\right|+\epsilon_{\beta,a}(D_k). n k 1 E D ( m ( k ) ) ≤ 2 1 0 ≤ s ≤ 1 sup Γ μ ( s ) + 2 a + E r ∈ A S k ( a ) sup n k 1 x ∑ ( ψ m x ( k ) ( r x ) − ψ μ ( r x ) ) + ϵ β , a ( D k ) . By (7.13) and D k → ∞ D_k\to\infty D k → ∞ k → ∞ k\to\infty k → ∞ sup 0 ≤ s ≤ 1 Γ μ ( s ) / 2 + 2 a \sup_{0\le s\le1}\Gamma_{\mu}(s)/2+2a sup 0 ≤ s ≤ 1 Γ μ ( s ) /2 + 2 a a ↓ 0 a\downarrow0 a ↓ 0
Proof of Corollary 7.6. Let r ≥ 0 r \ge0 r ≥ 0 Lemma 7.2 (d), the function m ↦ ψ m ( r ) m \mapsto\psi_m(r) m ↦ ψ m ( r ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ψ μ k ( r ) → ψ μ + ( r ) \psi_{\mu_k}(r) \to\psi_{\mu_+}(r) ψ μ k ( r ) → ψ μ + ( r ) (109) , the functions ψ μ k \psi_{\mu_k} ψ μ k ψ μ + \psi_{\mu_+} ψ μ + 1 / 2 1/2 1/2 [ 0 , β 2 ] [0,\beta^2] [ 0 , β 2 ] [ 0 , β 2 ] [0,\beta^2] [ 0 , β 2 ] sup 0 ≤ s ≤ 1 ∣ Γ μ k ( s ) − Γ μ + ( s ) ∣ = 2 sup 0 ≤ r ≤ β 2 ∣ ψ μ k ( r ) − ψ μ + ( r ) ∣ → 0 \sup_{0 \le s \le1}|\Gamma_{\mu_k}(s)-\Gamma_{\mu_+}(s)| = 2\sup_{0 \le r \le\beta^2}|\psi_{\mu_k}(r)-\psi_{\mu_+}(r)| \to0 sup 0 ≤ s ≤ 1 ∣ Γ μ k ( s ) − Γ μ + ( s ) ∣ = 2 sup 0 ≤ r ≤ β 2 ∣ ψ μ k ( r ) − ψ μ + ( r ) ∣ → 0
Proofs of the main theorems
This section proves Theorems 1.1 and 1.2 and Corollary 1.3. § 9.1 combines the results of Sections 5–§ 8 into a lower bound for tori of a fixed side length and for hypercubes. § 9.2 shows that the pressure of the torus T L d \mathbb{T}^{d}_{L} T L d L L L β 2 / ( 4 L ) \beta^{2}/(4L) β 2 / ( 4 L ) d d d § 9.3 completes the proofs, using Proposition 4.1 for the upper bound.
The lower bound at a fixed side length
Proposition 8.1. Let β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R L ≥ 3 L \ge3 L ≥ 3
lim inf d → ∞ p d , L ( β , h ) ≥ P ( β , h ) , and lim inf d → ∞ p Q d ( β , h ) ≥ P ( β , h ) . \liminf_{d\to\infty}p_{d,L}(\beta,h)\ge\mathcal{P}(\beta,h),\qquad\text{and}\qquad\liminf_{d\to\infty}p_{\mathbb{Q}^{d}}(\beta,h)\ge\mathcal{P}(\beta,h). d → ∞ lim inf p d , L ( β , h ) ≥ P ( β , h ) , and d → ∞ lim inf p Q d ( β , h ) ≥ P ( β , h ) . Proof. At β = 0 \beta=0 β = 0 log 2 cosh h \log2\cosh h log 2 cosh h Lemma 2.1 (a) for p G p_G p G p N S K p_{N}^{\mathrm{SK}} p N SK p G p_G p G P \mathcal{P} P h h h β > 0 \beta>0 β > 0 h ≥ 0 h\ge0 h ≥ 0 ( G d ) d ≥ 1 (G_d)_{d\ge1} ( G d ) d ≥ 1 T L d \mathbb{T}^{d}_{L} T L d D d = 2 d D_d=2d D d = 2 d Q d Q_d Q d D d = d D_d=d D d = d G d ∈ G D d G_d\in\mathcal{G}_{D_d} G d ∈ G D d D d → ∞ D_d\to\infty D d → ∞ n = ∣ V ( G d ) ∣ → ∞ n=|V(G_d)|\to\infty n = ∣ V ( G d ) ∣ → ∞
We first reduce to h > 0 h>0 h > 0 Lemma 2.1 (d), which also holds for P \mathcal{P} P ∣ p G d ( β , 0 ) − p G d ( β , η ) ∣ ≤ η |p_{G_d}(\beta,0)-p_{G_d}(\beta,\eta)|\le\eta ∣ p G d ( β , 0 ) − p G d ( β , η ) ∣ ≤ η ∣ P ( β , 0 ) − P ( β , η ) ∣ ≤ η |\mathcal{P}(\beta,0)-\mathcal{P}(\beta,\eta)|\le\eta ∣ P ( β , 0 ) − P ( β , η ) ∣ ≤ η η > 0 \eta>0 η > 0 ( β , η ) (\beta,\eta) ( β , η )
lim inf d → ∞ p G d ( β , 0 ) ≥ lim inf d → ∞ p G d ( β , η ) − η ≥ P ( β , η ) − η ≥ P ( β , 0 ) − 2 η , \liminf_{d\to\infty}p_{G_d}(\beta,0)\ge\liminf_{d\to\infty}p_{G_d}(\beta,\eta)-\eta\ge\mathcal{P}(\beta,\eta)-\eta\ge\mathcal{P}(\beta,0)-2\eta, d → ∞ lim inf p G d ( β , 0 ) ≥ d → ∞ lim inf p G d ( β , η ) − η ≥ P ( β , η ) − η ≥ P ( β , 0 ) − 2 η , and letting η → 0 \eta\to0 η → 0 ( β , 0 ) (\beta,0) ( β , 0 ) β > 0 \beta>0 β > 0 h > 0 h>0 h > 0 S = A G d / D d S=A_{G_d}/D_d S = A G d / D d Definition 6.1 and max x , y S x y = 1 / D d \max_{x,y}S_{xy}=1/D_d max x , y S x y = 1/ D d
Fix ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) T T T μ ε \mu_\varepsilon μ ε T T T G d G_d G d m = m ( d ) m=m(d) m = m ( d ) Proposition 5.2 . We first bound the relative entropy of the planted product model at m m m m ′ ∈ [ − 1 , 1 ] m'\in[-1,1] m ′ ∈ [ − 1 , 1 ] r ≥ 0 r\ge0 r ≥ 0 Υ ( m ′ , r ) = ψ m ′ ( r ) \Upsilon(m',r)=\psi_{m'}(r) Υ ( m ′ , r ) = ψ m ′ ( r ) Lemma 7.2 , Υ \Upsilon Υ [ − 1 , 1 ] × [ 0 , ∞ ) [-1,1]\times[0,\infty) [ − 1 , 1 ] × [ 0 , ∞ ) 1 / 2 1/2 1/2 ψ μ \psi_\mu ψ μ ∫ Υ ( m ′ , r ) μ ε ( d m ′ ) = ψ μ ε ( r ) \int\Upsilon(m',r)\mu_\varepsilon(dm')=\psi_{\mu_\varepsilon}(r) ∫ Υ ( m ′ , r ) μ ε ( d m ′ ) = ψ μ ε ( r ) a > 0 a>0 a > 0 Proposition 5.2 (c), applied with c = 2 β 2 c=2\beta^{2} c = 2 β 2 B = 1 B=1 B = 1 (115) ) of Theorem 7.5 (b) with μ = μ ε \mu=\mu_\varepsilon μ = μ ε Theorem 7.5 (b) implies that
lim sup d → ∞ 1 n E D ( m ) ≤ ω ( ε ) , ω ( ε ) = 1 2 sup 0 ≤ s ≤ 1 Γ μ ε ( s ) . (124) \limsup_{d\to\infty}\frac{1}{n}\mathbb{E}\mathcal{D}(m)\le\omega(\varepsilon),\qquad\omega(\varepsilon)=\frac{1}{2}\sup_{0\le s\le1}\Gamma_{\mu_\varepsilon}(s). \tag*{(124)} d → ∞ lim sup n 1 E D ( m ) ≤ ω ( ε ) , ω ( ε ) = 2 1 0 ≤ s ≤ 1 sup Γ μ ε ( s ) . (124) We have ω ( ε ) ≥ 0 \omega(\varepsilon) \ge0 ω ( ε ) ≥ 0 Γ μ ε ( 0 ) = 0 \Gamma_{\mu_\varepsilon}(0) = 0 Γ μ ε ( 0 ) = 0 μ ε → μ + \mu_\varepsilon\to\mu_+ μ ε → μ + ε → 0 \varepsilon\to0 ε → 0 W 1 ( μ ε , μ + ) ≤ ε W_1(\mu_\varepsilon,\mu_+) \le\varepsilon W 1 ( μ ε , μ + ) ≤ ε Proposition 5.2 (b), and Corollary 7.6 gives lim sup ε → 0 ω ( ε ) ≤ 0 \limsup_{\varepsilon\to0} \omega(\varepsilon) \le0 lim sup ε → 0 ω ( ε ) ≤ 0 Theorem 6.2 applies with δ = ω ( ε ) \delta= \omega(\varepsilon) δ = ω ( ε ) Proposition 5.2 (a), we obtain
lim inf d → ∞ p G d ( β , h ) ≥ lim inf d → ∞ 1 n E [ β H G d ( m ) + h ∑ x m x + ∑ x ent ( m x ) + β 2 4 ⟨ κ , S κ ⟩ ] − 4 β ω ( ε ) ≥ P ( β , h ) − ε − 4 β ω ( ε ) . \begin{aligned}
\liminf_{d\to\infty} p_{G_d}(\beta,h) \ge\liminf_{d\to\infty} \frac{1}{n}\mathbb{E}\left[\beta H_{G_d}(m) + h\sum_x m_x + \sum_x \operatorname{ent}(m_x) + \frac{\beta^2}{4}\langle\kappa,S\kappa\rangle\right] - 4\beta\sqrt{\omega(\varepsilon)} \\
&\ge\mathcal{P}(\beta,h) - \varepsilon- 4\beta\sqrt{\omega(\varepsilon)}.
\end{aligned} d → ∞ lim inf p G d ( β , h ) ≥ d → ∞ lim inf n 1 E [ β H G d ( m ) + h x ∑ m x + x ∑ ent ( m x ) + 4 β 2 ⟨ κ , S κ ⟩ ] − 4 β ω ( ε ) ≥ P ( β , h ) − ε − 4 β ω ( ε ) . The left side does not depend on ε \varepsilon ε ε → 0 \varepsilon\to0 ε → 0
The side length
The proof of Theorem 1.1 reduces the uniformity in L L L Γ = ( V Γ , E Γ ) \Gamma= (V_\Gamma,E_\Gamma) Γ = ( V Γ , E Γ ) D ≥ 1 D \ge1 D ≥ 1
p Γ D ( β , h ) = 1 ∣ V Γ ∣ E log ∑ σ ∈ { − 1 , 1 } V Γ exp ( β D ∑ x y ∈ E Γ g x y σ x σ y + h ∑ x ∈ V Γ σ x ) . p^D_\Gamma(\beta,h) = \frac{1}{|V_\Gamma|}\mathbb{E}\log\sum_{\sigma\in\{-1,1\}^{V_\Gamma}}\exp\left(\frac{\beta}{\sqrt{D}}\sum_{xy\in E_\Gamma}g_{xy}\sigma_x\sigma_y+h\sum_{x\in V_\Gamma}\sigma_x\right). p Γ D ( β , h ) = ∣ V Γ ∣ 1 E log σ ∈ { − 1 , 1 } V Γ ∑ exp D β x y ∈ E Γ ∑ g x y σ x σ y + h x ∈ V Γ ∑ σ x . where ( g e ) e ∈ E Γ (g_e)_{e\in E_\Gamma} ( g e ) e ∈ E Γ Γ ∈ G D \Gamma\in\mathcal{G}_D Γ ∈ G D p Γ D = p Γ p^D_\Gamma= p_\Gamma p Γ D = p Γ
The next lemma states that the pressure increases when Gaussian couplings are added, and bounds the increase. The monotonicity was proved by Contucci and Graffi for short-range Gaussian spin glasses, in the form of the superadditivity of the pressure [44 Theorem 2] , using the interpolation of Guerra and Toninelli [75 ] . Contucci and Lebowitz extended it to couplings with symmetric distributions [46 Theorem 1] . We include the short proof, since we also need the upper bound.
Lemma 8.2. Let D ≥ 1 D \ge1 D ≥ 1 β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h \in\mathbb{R} h ∈ R Γ \Gamma Γ Γ ′ \Gamma' Γ ′ n n n E Γ ⊂ E Γ ′ E_\Gamma\subset E_{\Gamma'} E Γ ⊂ E Γ ′ E Γ ′ ∖ E Γ E_{\Gamma'} \setminus E_\Gamma E Γ ′ ∖ E Γ k k k
0 ≤ p Γ ′ D ( β , h ) − p Γ D ( β , h ) ≤ β 2 k 2 D n . 0 \le p^D_{\Gamma'}(\beta,h) - p^D_\Gamma(\beta,h) \le\frac{\beta^2 k}{2Dn}. 0 ≤ p Γ ′ D ( β , h ) − p Γ D ( β , h ) ≤ 2 D n β 2 k . Further, log 2 cosh h ≤ p Γ D ( β , h ) ≤ log 2 cosh h + β 2 ∣ E Γ ∣ 2 D n \log2\cosh h \le p^D_\Gamma(\beta,h) \le\log2\cosh h + \frac{\beta^2|E_\Gamma|}{2Dn} log 2 cosh h ≤ p Γ D ( β , h ) ≤ log 2 cosh h + 2 D n β 2 ∣ E Γ ∣ Γ \Gamma Γ Γ 1 , … , Γ r \Gamma_1,\ldots,\Gamma_r Γ 1 , … , Γ r p Γ D = ∑ j = 1 r ( ∣ V Γ j ∣ / n ) p Γ j D p^D_\Gamma= \sum_{j=1}^r (|V_{\Gamma_j}|/n)p^D_{\Gamma_j} p Γ D = ∑ j = 1 r ( ∣ V Γ j ∣/ n ) p Γ j D
Proof. Let F = E Γ ′ ∖ E Γ F = E_{\Gamma'} \setminus E_\Gamma F = E Γ ′ ∖ E Γ s ∈ [ 0 , 1 ] s \in[0,1] s ∈ [ 0 , 1 ] ⟨ ⋅ ⟩ s \langle\cdot\rangle_s ⟨ ⋅ ⟩ s
β D ∑ x y ∈ E Γ g x y σ x σ y + β s D ∑ x y ∈ F g x y σ x σ y + h ∑ x σ x \frac{\beta}{\sqrt{D}}\sum_{xy\in E_\Gamma}g_{xy}\sigma_x\sigma_y+\frac{\beta\sqrt{s}}{\sqrt{D}}\sum_{xy\in F}g_{xy}\sigma_x\sigma_y+h\sum_x\sigma_x D β x y ∈ E Γ ∑ g x y σ x σ y + D β s x y ∈ F ∑ g x y σ x σ y + h x ∑ σ x and let φ ( s ) \varphi(s) φ ( s ) n − 1 n^{-1} n − 1 φ ( 0 ) = p Γ D \varphi(0)=p_{\Gamma}^{D} φ ( 0 ) = p Γ D φ ( 1 ) = p Γ ′ D \varphi(1)=p_{\Gamma'}^{D} φ ( 1 ) = p Γ ′ D s ∈ ( 0 , 1 ] s \in(0,1] s ∈ ( 0 , 1 ] g x y g_{xy} g x y x y ∈ F xy \in F x y ∈ F
φ ′ ( s ) = β 2 n s D ∑ x y ∈ F E [ g x y ⟨ σ x σ y ⟩ s ] = β 2 2 n D ∑ x y ∈ F ( 1 − E ⟨ σ x σ y ⟩ s 2 ) ∈ [ 0 , β 2 k 2 n D ] . \varphi'(s)=\frac{\beta}{2n\sqrt{sD}}\sum_{xy\in F}\mathbb{E}\left[g_{xy}\langle\sigma_x\sigma_y\rangle_s\right]=\frac{\beta^2}{2nD}\sum_{xy\in F}\left(1-\mathbb{E}\langle\sigma_x\sigma_y\rangle_s^2\right)\in\left[0,\frac{\beta^2 k}{2nD}\right]. φ ′ ( s ) = 2 n sD β x y ∈ F ∑ E [ g x y ⟨ σ x σ y ⟩ s ] = 2 n D β 2 x y ∈ F ∑ ( 1 − E ⟨ σ x σ y ⟩ s 2 ) ∈ [ 0 , 2 n D β 2 k ] . Since φ \varphi φ [ 0 , 1 ] [0,1] [ 0 , 1 ] p Γ D p_{\Gamma}^{D} p Γ D V Γ V_{\Gamma} V Γ log 2 cosh h \log2\cosh h log 2 cosh h Γ \Gamma Γ
For integers d ≥ 1 d\ge1 d ≥ 1 N ≥ 1 N\ge1 N ≥ 1 Λ N d = { 0 , … , N − 1 } d \Lambda_N^d=\{0,\ldots,N-1\}^d Λ N d = { 0 , … , N − 1 } d 1 1 1 b d , N = p Λ N d 2 d b_{d,N}=p_{\Lambda_N^d}^{2d} b d , N = p Λ N d 2 d 1 / ( 2 d ) 1/(2d) 1/ ( 2 d ) d d d
The existence of the thermodynamic limit of short-range spin glasses is classical [93 , 137 , 138 ] ; see also [44 , 46 , 47 ] . Contucci and Graffi compared free and periodic boundary conditions for the EA model with the interpolation of Lemma 8.2 , and they bounded the difference of the pressures by a multiple of the size of the boundary [45 Theorems 1 and 2] . For the torus, the existence of the limit and a bound of order 1 / L 1/L 1/ L [139 ] . The next lemma is a form of these bounds that is uniform in d d d
Lemma 8.3. Let d ≥ 1 d\ge1 d ≥ 1 β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R p d ∞ ( β , h ) = lim N → ∞ b d , N ( β , h ) p_d^\infty(\beta,h)=\lim_{N\to\infty}b_{d,N}(\beta,h) p d ∞ ( β , h ) = lim N → ∞ b d , N ( β , h ) L ≥ 3 L\ge3 L ≥ 3
∣ p d , L ( β , h ) − p d ∞ ( β , h ) ∣ ≤ β 2 4 L . \left|p_{d,L}(\beta,h)-p_d^\infty(\beta,h)\right|\le\frac{\beta^2}{4L}. ∣ p d , L ( β , h ) − p d ∞ ( β , h ) ∣ ≤ 4 L β 2 . Proof. We first compare the torus with the box. Let L ≥ 3 L\ge3 L ≥ 3 T L d T_L^d T L d Λ L d \Lambda_L^d Λ L d j ≤ d j\le d j ≤ d d − 1 d-1 d − 1 j j j L − 1 L-1 L − 1 0 0 0 L ≥ 3 L\ge3 L ≥ 3 d L d − 1 dL^{d-1} d L d − 1 Λ L d \Lambda_L^d Λ L d T L d ∈ G 2 d T_L^d\in\mathcal{G}_{2d} T L d ∈ G 2 d L d L^d L d Lemma 8.2 with D = 2 d D=2d D = 2 d
0 ≤ p d , L − b d , L ≤ β 2 d L d − 1 2 ⋅ 2 d ⋅ L d = β 2 4 L . (125) 0\le p_{d,L}-b_{d,L}\le\frac{\beta^2dL^{d-1}}{2\cdot2d\cdot L^d}=\frac{\beta^2}{4L}. \tag*{(125)} 0 ≤ p d , L − b d , L ≤ 2 ⋅ 2 d ⋅ L d β 2 d L d − 1 = 4 L β 2 . (125) Next, we compare boxes of different sizes. Fix integers m ≥ 1 m\ge1 m ≥ 1 N ≥ m N\ge m N ≥ m N = k m + r N=km+r N = k m + r k = ⌊ N / m ⌋ k=\lfloor N/m\rfloor k = ⌊ N / m ⌋ { 0 , … , N − 1 } \{0,\ldots,N-1\} { 0 , … , N − 1 } m m m r > 0 r > 0 r > 0 r r r Λ N d \Lambda_N^d Λ N d k d k^d k d Λ m d \Lambda_m^d Λ m d Γ \Gamma Γ Λ N d \Lambda_N^d Λ N d Λ N d \Lambda_N^d Λ N d d d d N d − 1 N^{d-1} N d − 1 Λ N d \Lambda_N^d Λ N d k k k d k N d − 1 ≤ d N d / m dkN^{d-1} \le dN^d/m d k N d − 1 ≤ d N d / m Λ N d \Lambda_N^d Λ N d Γ \Gamma Γ D = 2 d D = 2d D = 2 d 0 ≤ b d , N − p Γ 2 d ≤ β 2 / ( 4 m ) 0 \le b_{d,N} - p_{\Gamma}^{2d} \le\beta^2/(4m) 0 ≤ b d , N − p Γ 2 d ≤ β 2 / ( 4 m ) p Γ 2 d p_{\Gamma}^{2d} p Γ 2 d Λ m d \Lambda_m^d Λ m d b d , m b_{d,m} b d , m n ′ n' n ′ d n ′ dn' d n ′ b d , m b_{d,m} b d , m [ log 2 cosh h , log 2 cosh h + β 2 / 4 ] [\log2\cosh h,\log2\cosh h+\beta^2/4] [ log 2 cosh h , log 2 cosh h + β 2 /4 ] f N = ( k m / N ) d f_N=(km/N)^d f N = ( k m / N ) d Λ N d \Lambda_N^d Λ N d Λ m d \Lambda_m^d Λ m d
b d , m − ( 1 − f N ) β 2 4 ≤ b d , N ≤ b d , m + ( 1 − f N ) β 2 4 + β 2 4 m . b_{d,m}-(1-f_N)\frac{\beta^2}{4}\le b_{d,N}\le b_{d,m}+(1-f_N)\frac{\beta^2}{4}+\frac{\beta^2}{4m}. b d , m − ( 1 − f N ) 4 β 2 ≤ b d , N ≤ b d , m + ( 1 − f N ) 4 β 2 + 4 m β 2 . Since f N = ( 1 − r / N ) d → 1 f_N=(1-r/N)^d\to1 f N = ( 1 − r / N ) d → 1 N → ∞ N\to\infty N → ∞ d d d m m m
b d , m ≤ lim inf N → ∞ b d , N ≤ lim sup N → ∞ b d , N ≤ b d , m + β 2 4 m . (126) b_{d,m}\le\liminf_{N\to\infty}b_{d,N}\le\limsup_{N\to\infty}b_{d,N}\le b_{d,m}+\frac{\beta^2}{4m}. \tag*{(126)} b d , m ≤ N → ∞ lim inf b d , N ≤ N → ∞ lim sup b d , N ≤ b d , m + 4 m β 2 . (126) Taking the limit inferior in m m m lim sup N b d , N ≤ lim inf m b d , m \limsup_{N}b_{d,N}\le\liminf_{m}b_{d,m} lim sup N b d , N ≤ lim inf m b d , m p d ∞ p_d^\infty p d ∞ b d , m ≤ p d ∞ ≤ b d , m + β 2 / ( 4 m ) b_{d,m}\le p_d^\infty\le b_{d,m}+\beta^2/(4m) b d , m ≤ p d ∞ ≤ b d , m + β 2 / ( 4 m ) m ≥ 1 m\ge1 m ≥ 1 m = L m=L m = L p d , L p_{d,L} p d , L p d ∞ p_d^\infty p d ∞ [ b d , L , b d , L + β 2 / ( 4 L ) ] [b_{d,L},b_{d,L}+\beta^2/(4L)] [ b d , L , b d , L + β 2 / ( 4 L )]
Remark 8.4. Lemma 8.3 is the only place where the geometry of the torus enters the passage from fixed to arbitrary side lengths. It shows that the limits N → ∞ N\to\infty N → ∞ L → ∞ L\to\infty L → ∞ lim d → ∞ p d ∞ ( β , h ) = P ( β , h ) \lim_{d\to\infty}p_d^\infty(\beta,h)=P(\beta,h) lim d → ∞ p d ∞ ( β , h ) = P ( β , h )
Proofs of the main results
Proof of Theorem 1.1. Fix β ≥ 0 \beta\ge0 β ≥ 0 h ∈ R h\in\mathbb{R} h ∈ R T L d ∈ G 2 d T_L^d\in\mathcal{G}_{2d} T L d ∈ G 2 d L ≥ 3 L\ge3 L ≥ 3
lim sup d → ∞ sup L ≥ 3 p d , L ( β , h ) ≤ lim sup D → ∞ sup G ∈ G D p G ( β , h ) ≤ P ( β , h ) . \limsup_{d\to\infty}\sup_{L\ge3}p_{d,L}(\beta,h)\le\limsup_{D\to\infty}\sup_{G\in\mathcal{G}_D}p_G(\beta,h)\le P(\beta,h). d → ∞ lim sup L ≥ 3 sup p d , L ( β , h ) ≤ D → ∞ lim sup G ∈ G D sup p G ( β , h ) ≤ P ( β , h ) . For the lower bound, fix an integer m ≥ 3 m\ge3 m ≥ 3 L ≥ m L\ge m L ≥ m L L L m m m
p d , L ≥ p d ∞ − β 2 4 L ≥ p d , m − β 2 4 m − β 2 4 L ≥ p d , m − β 2 2 m . p_{d,L}\ge p_d^\infty-\frac{\beta^2}{4L}\ge p_{d,m}-\frac{\beta^2}{4m}-\frac{\beta^2}{4L}\ge p_{d,m}-\frac{\beta^2}{2m}. p d , L ≥ p d ∞ − 4 L β 2 ≥ p d , m − 4 m β 2 − 4 L β 2 ≥ p d , m − 2 m β 2 . Hence inf L ≥ 3 p d , L ≥ min 3 ≤ L ≤ m p d , L − β 2 2 m \inf_{L\ge3} p_{d,L} \ge\min_{3\le L\le m} p_{d,L} - \frac{\beta^2}{2m} inf L ≥ 3 p d , L ≥ min 3 ≤ L ≤ m p d , L − 2 m β 2
lim inf d → ∞ inf L ≥ 3 p d , L ( β , h ) ≥ P ( β , h ) − β 2 2 m . \liminf_{d\to\infty}\inf_{L\ge3} p_{d,L}(\beta,h) \ge\mathcal{P}(\beta,h) - \frac{\beta^2}{2m}. d → ∞ lim inf L ≥ 3 inf p d , L ( β , h ) ≥ P ( β , h ) − 2 m β 2 . Letting m → ∞ m \to\infty m → ∞ Theorem 1.1 .
Proof of Theorem 1.2. Since Q d ∈ G d Q_d \in\mathcal{G}_d Q d ∈ G d lim sup d p Q d ( β , h ) ≤ P ( β , h ) \limsup_d p_{Q_d}(\beta,h) \le\mathcal{P}(\beta,h) lim sup d p Q d ( β , h ) ≤ P ( β , h )
Proof of Corollary 1.3. Let G ∈ G D G \in\mathcal{G}_D G ∈ G D h ∈ R h \in\mathbb{R} h ∈ R β > 0 \beta> 0 β > 0 (7) , e ( G ; h ) e(G;h) e ( G ; h ) [ β − 1 p G ( β , β h ) − β − 1 log 2 , β − 1 p G ( β , β h ) ] [\beta^{-1}p_G(\beta,\beta h)-\beta^{-1}\log2,\ \beta^{-1}p_G(\beta,\beta h)] [ β − 1 p G ( β , β h ) − β − 1 log 2 , β − 1 p G ( β , β h )] (8) , P ∗ ( h ) P_*(h) P ∗ ( h ) [ β − 1 P ( β , β h ) − β − 1 log 2 , β − 1 P ( β , β h ) ] [\beta^{-1}\mathcal{P}(\beta,\beta h)-\beta^{-1}\log2,\ \beta^{-1}\mathcal{P}(\beta,\beta h)] [ β − 1 P ( β , β h ) − β − 1 log 2 , β − 1 P ( β , β h )]
∣ e ( G ; h ) − P ∗ ( h ) ∣ ≤ ∣ p G ( β , β h ) − P ( β , β h ) ∣ β + log 2 β . |e(G;h)-P_*(h)| \le\frac{|p_G(\beta,\beta h)-\mathcal{P}(\beta,\beta h)|}{\beta}+\frac{\log2}{\beta}. ∣ e ( G ; h ) − P ∗ ( h ) ∣ ≤ β ∣ p G ( β , β h ) − P ( β , β h ) ∣ + β log 2 . Taking the supremum over L ≥ 3 L\ge3 L ≥ 3 G = T L d G=T_L^d G = T L d d → ∞ d\to\infty d → ∞ Theorem 1.1 at ( β , β h ) (\beta,\beta h) ( β , β h ) lim sup d sup L ≥ 3 ∣ e ( T L d ; h ) − P ∗ ( h ) ∣ ≤ ( log 2 ) / β \limsup_d\sup_{L\ge3}|e(T_L^d;h)-P_*(h)|\le(\log2)/\beta lim sup d sup L ≥ 3 ∣ e ( T L d ; h ) − P ∗ ( h ) ∣ ≤ ( log 2 ) / β β → ∞ \beta\to\infty β → ∞ Theorem 1.2 proves it for hypercubes.
Finally, we justify Remark 1.4. By Lemma 2.1(d), for each B ≥ 0 B\ge0 B ≥ 0 p G p_G p G G ∈ ⋃ D G D G\in\bigcup_D\mathcal{G}_D G ∈ ⋃ D G D p N S K p_N^{\mathrm{SK}} p N SK N ≥ 1 N\ge1 N ≥ 1 ∣ f ( β , h ) − f ( β ′ , h ′ ) ∣ ≤ B ∣ β − β ′ ∣ / 2 + ∣ h − h ′ ∣ |f(\beta,h)-f(\beta',h')|\le B|\beta-\beta'|/2+|h-h'| ∣ f ( β , h ) − f ( β ′ , h ′ ) ∣ ≤ B ∣ β − β ′ ∣/2 + ∣ h − h ′ ∣ β , β ′ ∈ [ 0 , B ] \beta,\beta'\in[0,B] β , β ′ ∈ [ 0 , B ] h , h ′ ∈ R h,h'\in\mathbb{R} h , h ′ ∈ R P \mathcal{P} P K ⊂ [ 0 , B ] × R K\subset[0,B]\times\mathbb{R} K ⊂ [ 0 , B ] × R η > 0 \eta>0 η > 0 K ′ K' K ′ K K K K K K η \eta η K ′ K' K ′
sup ( β , h ) ∈ K sup L ≥ 3 ∣ p d , L ( β , h ) − P ( β , h ) ∣ ≤ max ( β , h ) ∈ K ′ sup L ≥ 3 ∣ p d , L ( β , h ) − P ( β , h ) ∣ + ( B + 2 ) η . \sup_{(\beta,h)\in K}\sup_{L\ge3}|p_{d,L}(\beta,h)-\mathcal{P}(\beta,h)|\le\max_{(\beta,h)\in K'}\sup_{L\ge3}|p_{d,L}(\beta,h)-\mathcal{P}(\beta,h)|+(B+2)\eta. ( β , h ) ∈ K sup L ≥ 3 sup ∣ p d , L ( β , h ) − P ( β , h ) ∣ ≤ ( β , h ) ∈ K ′ max L ≥ 3 sup ∣ p d , L ( β , h ) − P ( β , h ) ∣ + ( B + 2 ) η . and the maximum tends to zero as d → ∞ d\to\infty d → ∞ Theorem 1.1 . Since η \eta η Theorem 1.1 is uniform on K K K h h h P ∗ ( h ) P_*(h) P ∗ ( h ) m N ( h ) m_N(h) m N ( h )
We also prove the limit in Remark 1.8. A similar interpolation, with independent SK models on the two parts, is in [16 Section 3.1] . Let D ≥ 1 D\ge1 D ≥ 1 n = 2 D n=2D n = 2 D G = K D , D G=K_{D,D} G = K D , D V 1 V_1 V 1 V 2 V_2 V 2 D D D v ∈ { − 1 , 1 } V v\in\{-1,1\}^V v ∈ { − 1 , 1 } V V 1 V_1 V 1 − 1 -1 − 1 V 2 V_2 V 2 σ 1 , σ 2 ∈ { − 1 , 1 } V \sigma^1,\sigma^2\in\{-1,1\}^V σ 1 , σ 2 ∈ { − 1 , 1 } V u = σ 1 ⊙ σ 2 u=\sigma^1\odot\sigma^2 u = σ 1 ⊙ σ 2 ∑ x ∈ V 1 u x \sum_{x\in V_1}u_x ∑ x ∈ V 1 u x ∑ y ∈ V 2 u y \sum_{y\in V_2}u_y ∑ y ∈ V 2 u y ( ⟨ u , 1 ⟩ ± ⟨ u , v ⟩ ) / 2 (\langle u,\mathbf{1}\rangle\pm\langle u,v\rangle)/2 (⟨ u , 1 ⟩ ± ⟨ u , v ⟩) /2 V 1 V_1 V 1 V 2 V_2 V 2
E H G ( σ 1 ) H G ( σ 2 ) = 1 D ∑ x ∈ V 1 ∑ y ∈ V 2 u x u y = 1 2 n ( ⟨ u , 1 ⟩ 2 − ⟨ u , v ⟩ 2 ) . \mathbb{E}H_G(\sigma^1)H_G(\sigma^2)=\frac{1}{D}\sum_{x\in V_1}\sum_{y\in V_2}u_xu_y=\frac{1}{2n}\left(\langle u,\mathbf{1}\rangle^2-\langle u,v\rangle^2\right). E H G ( σ 1 ) H G ( σ 2 ) = D 1 x ∈ V 1 ∑ y ∈ V 2 ∑ u x u y = 2 n 1 ( ⟨ u , 1 ⟩ 2 − ⟨ u , v ⟩ 2 ) . We identify V V V { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } H n S K H_n^{\mathrm{SK}} H n SK { − 1 , 1 } V \{-1,1\}^{V} { − 1 , 1 } V ( g e ) e ∈ E (g_e)_{e\in E} ( g e ) e ∈ E E H n S K ( σ 1 ) H n S K ( σ 2 ) = ( ⟨ u , 1 ⟩ 2 − n ) / ( 2 n ) \mathbb{E}H_n^{\mathrm{SK}}(\sigma^1)H_n^{\mathrm{SK}}(\sigma^2)=(\langle u,\mathbf{1}\rangle^2-n)/(2n) E H n SK ( σ 1 ) H n SK ( σ 2 ) = (⟨ u , 1 ⟩ 2 − n ) / ( 2 n )
C ( σ 1 , σ 2 ) = E H n S K ( σ 1 ) H n S K ( σ 2 ) − E H G ( σ 1 ) H G ( σ 2 ) = ⟨ u , v ⟩ 2 − n 2 n . C(\sigma^1,\sigma^2)=\mathbb{E}H_n^{\mathrm{SK}}(\sigma^1)H_n^{\mathrm{SK}}(\sigma^2)-\mathbb{E}H_G(\sigma^1)H_G(\sigma^2)=\frac{\langle u,v\rangle^2-n}{2n}. C ( σ 1 , σ 2 ) = E H n SK ( σ 1 ) H n SK ( σ 2 ) − E H G ( σ 1 ) H G ( σ 2 ) = 2 n ⟨ u , v ⟩ 2 − n . In particular, C ( σ , σ ) = − 1 / 2 C(\sigma,\sigma)=-1/2 C ( σ , σ ) = − 1/2 σ \sigma σ ⟨ 1 , v ⟩ = 0 \langle\mathbf{1},v\rangle=0 ⟨ 1 , v ⟩ = 0 θ ∈ [ 0 , 1 ] \theta\in[0,1] θ ∈ [ 0 , 1 ] ⟨ ⋅ ⟩ θ \langle\cdot\rangle_\theta ⟨ ⋅ ⟩ θ β ( 1 − θ H G ( σ ) + θ H n S K ( σ ) ) + h ∑ x σ x \beta(\sqrt{1-\theta}H_G(\sigma)+\sqrt{\theta}H_n^{\mathrm{SK}}(\sigma))+h\sum_x\sigma_x β ( 1 − θ H G ( σ ) + θ H n SK ( σ )) + h ∑ x σ x φ ( θ ) \varphi(\theta) φ ( θ ) n − 1 n^{-1} n − 1 φ ( 0 ) = p G ( β , h ) \varphi(0)=p_G(\beta,h) φ ( 0 ) = p G ( β , h ) φ ( 1 ) = p n S K ( β , h ) \varphi(1)=p_n^{\mathrm{SK}}(\beta,h) φ ( 1 ) = p n SK ( β , h ) θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) [75 ] ,
φ ′ ( θ ) = β 2 2 n E ⟨ C ( σ 1 , σ 1 ) − C ( σ 1 , σ 2 ) ⟩ θ = − β 2 4 n 2 E ⟨ ⟨ σ 1 ⊙ σ 2 , v ⟩ 2 ⟩ θ ≤ 0. \varphi'(\theta)=\frac{\beta^2}{2n}\mathbb{E}\langle C(\sigma^1,\sigma^1)-C(\sigma^1,\sigma^2)\rangle_\theta=-\frac{\beta^2}{4n^2}\mathbb{E}\langle\langle\sigma^1\odot\sigma^2,v\rangle^2\rangle_\theta\leq0. φ ′ ( θ ) = 2 n β 2 E ⟨ C ( σ 1 , σ 1 ) − C ( σ 1 , σ 2 ) ⟩ θ = − 4 n 2 β 2 E ⟨⟨ σ 1 ⊙ σ 2 , v ⟩ 2 ⟩ θ ≤ 0. where σ 1 \sigma^1 σ 1 σ 2 \sigma^2 σ 2 φ \varphi φ [ 0 , 1 ] [0,1] [ 0 , 1 ] p K D , D ( β , h ) ≥ p 2 D S K ( β , h ) p_{K_{D,D}}(\beta,h)\geq p_{2D}^{\mathrm{SK}}(\beta,h) p K D , D ( β , h ) ≥ p 2 D SK ( β , h ) P ( β , h ) \mathcal{P}(\beta,h) P ( β , h ) D → ∞ D\to\infty D → ∞ K D , D ∈ G D K_{D,D}\in\mathcal{G}_D K D , D ∈ G D Proposition 4.1 gives the matching upper bound.
References 