Bulk universality for generalized Wigner matrices

Introduction

One key universal quantity for random matrices is the eigenvalue gap distribution. Although the density of eigenvalues may depend on the specific model, the gap distribution or the short distance correlation function are believed to depend only on the symmetry class of the ensembles but are otherwise independent of the details of the distributions. There are two types of universality: the edge universality and the bulk universality. In this paper, we will focus on the bulk universality concerning the interior of the spectrum. The bulk universality was proved for very general classes of invariant ensembles (see, e.g. and references therein). For non-invariant ensembles, in particular for matrices with i.i.d. entries (Wigner matrices), the bulk universality was difficult to establish due to the lack of an explicit expression for the joint distribution of the eigenvalues.

The first rigorous partial result for bulk universality in the non-unitary case was given by Johansson (see also Ben Arous and Péché and the recent improvement ) stating that the bulk universality holds for Gaussian divisible Hermitian ensembles, i.e., Hermitian ensembles of the form

where $\widehat{H}$ is a Wigner matrix, $V$ is an independent standard GUE matrix and $s$ is a positive constant of order one. The restriction on Gaussian divisibility turned out to be very difficult to remove. In a series of papers , we developed a new approach to prove the universality. The first step was to derive the local semicircle law, an estimate of the local eigenvalue density, down to energy scales containing around $\log N$ eigenvalues. Once such a strong form of the local semicircle law was obtained, the result of can be extended to a Gaussian convolution with variance only $s^{2}\asymp N^{-1+\varepsilon}$. This tiny Gaussian component can then be removed via a reverse heat flow argument and this proves the bulk universality for Hermitian ensembles provided that the distributions of the matrix elements are sufficiently differentiable.

The bulk universality for Hermitian ensembles was also proved later on by Tao and Vu under the condition that the first four moments of the matrix elements match those of GUE, but without the differentiability assumption. The condition on the fourth moment was already removed in by using the result for Gaussian divisible ensembles of ; the third moment condition was then removed in by using the result of .

The four moment theorem is also valid for the symmetric ensembles, but the restriction on the matching of the first four moments cannot be weakened for the following reason. The key input to remove the fourth moment matching condition for the Hermitian case, the universality of the Gaussian divisible ensembles , relied entirely on the asymptotic analysis of an explicit formula, closely related to a formula in Brézin-Hikami , for the correlation functions of the eigenvalues for the Hermitian ensembles $\widehat{H}+sV$. Since similar formulas for symmetric matrices are very complicated, the corresponding result is not available and thus the matching of the fourth moment cannot be removed in this way. Although there is a proof of universality for $s^{2}\geq N^{-3/4}$ without using this formula, the main ingredient of that proof, establishing the uniqueness of the local equilibria of the Dyson Brownian motion, still heavily used explicit formulas related to GUE.

In a completely different strategy was introduced based on a local relaxation flow, which locally behaves like a Dyson Brownian motion, but has a faster decay to equilibrium. This approach entirely eliminates explicit formulas and it gives a unified proof for the universality of symmetric and Hermitian Wigner matrices . It was further generalized to quaternion self-dual Wigner matrices and sample covariance matrices. The method not only applies to all these specific ensembles, but it also gives a conceptual interpretation that the occurrence of the universality is due to the relaxation to local equilibrium of the DBM. We remark that very recently the results of were also extended to sample covariance matrices .

The main input of all these methods and is an estimate of the local density of eigenvalues, the local semicircle law. This has been developed in the previous work on Wigner matrices , where the matrix elements were i.i.d. random variables. In this paper, we extend this method to random matrices with independent, but not necessarily identically distributed entries. If we denote the variance of the $(i,j)$ entry of the matrix by $\sigma_{ij}^{2}$, our main interest is the case that $\sigma_{ij}$ are not a constant but they satisfy the normalization condition $\sum_{i}\sigma_{ij}^{2}=1$ for all $j$. We will call such matrix ensembles universal Wigner matrices. For these ensembles Guionnet and Anderson-Zeitouni proved that the density of the eigenvalues converges to the Wigner semi-circle law. The simplest case is that of generalized Wigner matrices, where $N\sigma_{ij}^{2}$ is uniformly bounded from above and below by two fixed positive numbers. In this case, we prove the local semicircle law down to essentially the smallest possible energy scale $N^{-1}$ (modulo $\log N$ factors). A much more difficult case is the Wigner band matrices where, roughly speaking, $\sigma_{ij}^{2}=0$ if $|i-j|>M$ for some $M<N$. In this case, we obtain the local semicircle law to the energy scale $M^{-1}$. We note that a certain three-dimensional version of Gaussian band matrices was considered by Disertori, Pinson and Spencer using the supersymmetric method. They proved that the expectation of the density of eigenvalues is smooth and it coincides with the Wigner semicircle law.

With the local semicircle law proved up to the almost optimal scale, applying the method of leads to the identification of the correlation functions and the gap distribution for generalized Wigner matrices provided that the distribution of the matrix elements is continuous and satisfies the logarithmic Sobolev inequality. These additional assumptions can be removed if one can extend the Tao-Vu theorem to generalized Wigner matrices. In Section 8, we will introduce an approach based on a Green’s function comparison theorem, which states that the joint distributions of Green’s functions of two ensembles at different energies with imaginary parts of order $1/N$ are identical provided that the first three moments of the two ensembles coincide and the fourth moments are close. Since local correlation functions and the gap distribution of the eigenvalues can be identified from Green’s functions, it follows that the local correlation functions of these two ensembles are identical at the scale $1/N$. We can thus use this theorem to remove all continuity and logarithmic Sobolev inequality restrictions in our approach. In particular, this leads to the bulk universality for generalized Wigner matrices with the subexponential decay being essentially the only assumption on the probability law. We note that one major technical difficulty in , the level repulsion estimate, is not needed in the proof of the Green’s function comparison theorem. It will be clear in Section 8 that, once the local semicircle law is established, the Green’s function comparison theorem is a simple consequence of the standard resolvent perturbation theory.

Main results

Matrices with independent, zero mean entries and with the normalization condition (2.1) will be called universal Wigner matrices. For a forthcoming review on this matrix class, see , where the terminology of random band matrices was used.

Note that ${C_{inf}}={C_{sup}}$ corresponds to the standard Wigner matrices and the condition $0<{C_{inf}}\leq{C_{sup}}<\infty$ defines more general Wigner matrices with comparable variances.

We will also consider an even more general case when $\sigma_{ij}$ for different $(i,j)$ indices are not comparable. The basic parameter of such matrices is the quantity

A special case is the band matrix, where $\sigma_{ij}=0$ for $|i-j|>W$ with some parameter $W$. In this case, $M$ and $W$ are related by $M\leq CW$.

Denote by $B:=\{\sigma^{2}_{ij}\}_{i,j=1}^{N}$ the matrix of variances which is symmetric and doubly stochastic by (2.1), in particular it satisfies $-1\leq B\leq 1$. Let the spectrum of $B$ be supported in

with some nonnegative constants $\delta_{\pm}$. We will always have the following spectral assumption

The local semicircle law will be proven under this general condition, but the precision of the estimate near the spectral edge will also depend on $\delta_{+}$ in an explicit way. For the orientation of the reader, we mention two special cases of universal Wigner matrices that provided the main motivation for our work.

Example 1. Generalized Wigner matrix. In this case we have

and one can easily prove that $1$ is a simple eigenvalue of $B$ and (2.4) holds with

i.e., both $\delta_{-}$ and $\delta_{+}$ are positive constants independent of $N$.

Example 2. Band matrix. The variances are given by

The Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by

We define the density of the semicircle law

and, for ${\text{Im}}\,z>0$, its Stieltjes transform

The Stieltjes transform $m_{sc}(z)\equiv m_{sc}$ may also be characterized as the unique solution of

satisfying ${\text{Im}}\,m_{sc}(z)>0$ for ${\text{Im}}\,z>0$, i.e.,

Here the square root function is chosen with a branch cut along the positive real axis. This guarantees that the imaginary part of $m_{sc}$ is non-negative. The Wigner semicircle law states that $m_{N}(z)\to m_{sc}(z)$ for any fixed $z$ provided that $\eta={\text{Im}}\,z>0$ is independent of $N$. The local version of this result for universal Wigner matrices is the content of the following Theorem.

Then there exist constants $C_{1}$, $C_{2}$, $C$ and $c>0$, depending only on $\alpha$, $\beta$ and $\delta_{-}$ in (2.5), such that for any $z=E+i\eta$ with $\eta={\mbox{I}m}\,z>0$, $|z|\leq 10$ and

where \kappa:=\big{|}\,|E|-2\big{|}, the Stieltjes transform of the empirical eigenvalue distribution of $H$ satisfies

for sufficiently large $N$. In fact, the same result holds for the individual matrix elements of the Green’s function $G_{ii}(z)=(H-z)^{-1}(i,i)$:

We remark that once a local semicircle law is obtained on a scale essentially $M^{-1}$, it is straightforward to show that eigenvectors are delocalized on a scale at least of order $M$. The precise statement will be formulated in Corollary 3.2. We will prove Theorem 2.1 in Sections 3–5 by extending the approach of . The main ingredients of this approach consist of i) a derivation of a self-consistent equation for the Green’s function and ii) an induction on the scale of the imaginary part of the energy. The key novelty in this paper is that the self-consistent equation is formulated for the array of the diagonal elements of the Green’s function $(G_{11},G_{22},\ldots,G_{NN})$ instead of the Stieltjes transform $m=\frac{1}{N}\mbox{Tr\,}G=\frac{1}{N}\sum_{i}G_{ii}$ itself as in . This yields for the first time a strong pointwise control on the diagonal elements $G_{ii}$, see (2.17).

The subexponential decay condition (2.14) can be weakened if we are not aiming at error estimates faster than any power law of $N$. This can be easily carried out and we will not pursue it in this paper.

Denote the eigenvalues of $H$ by $\lambda_{1},\ldots,\lambda_{N}$ and let $p_{N}(x_{1},\ldots,x_{N})$ be their (symmetric) probability density. For any $k=1,2,\ldots,N$, the $k$-point correlation function of the eigenvalues is defined by

We now state our main result concerning these correlation functions.

We consider a generalized hermitian Wigner matrix such that (2.6) holds. Assume that the distributions $\nu_{ij}$ of the $(i,j)$ matrix elements have a uniformly subexponential decay in the sense of (2.14). Suppose that the real and imaginary parts of $h_{ij}$ are i.i.d., distributed according to ${\omega}_{ij}$, i.e., $\nu_{ij}({\rm d}h)={\omega}_{ij}({\rm d}{\text{Im}}\,h){\omega}_{ij}({\rm d}{\text{Re}}h)$. Let $m_{k}(i,j)=\int x^{k}{\rm d}{\omega}_{ij}(x)$, $1\leq k\leq 4$, denote the $k$-th moment of ${\omega}_{ij}$ ($m_{1}=0$). Suppose that

where $p_{GU\!E,N}^{(k)}$ is the $k$-point correlation function of the GUE ensemble. The same statement holds for generalized symmetric Wigner matrices, with GOE replacing the GUE ensemble.

The limiting correlation functions of the GUE ensemble are given by the sine kernel

and similar universal formula is available for the limiting gap distribution.

Remark: The quantity in the bracket in (2.19) is always greater or equal to 1 for any real distribution with mean zero, which can be obtained by

and it is exactly 1 if the distribution is supported on two points. For example, if ${\omega}_{ij}$ is a rescaling of a fixed distribution $\widetilde{\omega}$ with variance $\frac{1}{2}$, i.e. ${\omega}_{ij}(x){\rm d}x=\sigma_{ij}^{-1}\widetilde{\omega}(x/\sigma_{ij}){\rm d}x$, then condition (2.19) is satisfied under (2.6), as long as the support of $\widetilde{\omega}$ consists of at least three points. The case of a Bernoulli-type distribution supported on two points require a separate argument and it will be treated in the forthcoming paper .

We now state our main comparison theorem for matrix elements of Green’s functions of two Wigner ensembles. As in the paper , we assume conditions on four moments. It will lead quickly to Theorem 6.4 stating that the correlation functions of eigenvalues of two matrix ensembles are identical up to scale $1/N$ provided that the first four moments of all matrix elements of these two ensembles are almost identical. Here we do not assume that the real and imaginary parts are i.i.d., hence the $k$-th moment of $h_{ij}$ is understood as the collection of numbers $\int\bar{h}^{s}h^{k-s}\nu_{ij}({\rm d}h)$, $s=0,1,2,\ldots,k$. The main result in compares the joint distribution of individual eigenvalues — which is not covered by our Theorem 2.3 — but it does not address directly the matrix elements of Green’s functions. The key input for both theorems is the local semicircle law on the almost optimal scale $N^{-1+\varepsilon}$. The eigenvalue perturbation used in requires certain estimates on the eigenvalue level repulsion; the proof of Theorem 2.3 is a straightforward resolvent perturbation theory.

Suppose that we have two generalized $N\times N$ Wigner matrices, $H^{(v)}$ and $H^{(w)}$, with matrix elements $h_{ij}$ given by the random variables $N^{-1/2}v_{ij}$ and $N^{-1/2}w_{ij}$, respectively, with $v_{ij}$ and $w_{ij}$ satisfying the uniform subexponential decay condition

with some $\alpha,\beta>0$. Fix a bijective ordering map on the index set of the independent matrix elements,

and denote by $H_{\gamma}$ the generalized Wigner matrix whose matrix elements $h_{ij}$ follow the $v$-distribution if $\phi(i,j)\leq\gamma$ and they follow the $w$-distribution otherwise; in particular $H^{(v)}=H_{0}$ and $H^{(w)}=H_{\gamma(N)}$. Let $\kappa>0$ be arbitrary and suppose that, for any small parameter $\tau>0$ and for any $y\geq N^{-1+\tau}$, we have the following estimate on the diagonal elements of the resolvent

with some constants $C,c$ depending only on $\tau,\kappa$. Moreover, we assume that the first three moments of $v_{ij}$ and $w_{ij}$ are the same, i.e.

and the difference between the fourth moments of $v_{ij}$ and $w_{ij}$ is much less than 1, say

for some given $\delta>0$. Let $\varepsilon>0$ be arbitrary and choose an $\eta$ with $N^{-1-\varepsilon}\leq\eta\leq N^{-1}$. For any sequence of positive integers $k_{1},\ldots,k_{n}$, set complex parameters $z^{m}_{j}=E^{m}_{j}\pm i\eta$, $j=1,\ldots k_{m}$, $m=1,\ldots,n$, with $|E^{m}_{j}|\leq 2-2\kappa$ and with an arbitrary choice of the $\pm$ signs. Let $G^{(v)}(z)=(H^{(v)}-z)^{-1}$ denote the resolvent and let $F(x_{1},\ldots,x_{n})$ be a function such that for any multi-index $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ with $1\leq|\alpha|\leq 5$ and for any $\varepsilon^{\prime}>0$ sufficiently small, we have

Then, there is a constant $C_{1}$, depending on $\alpha,\beta$, $\sum_{m}k_{m}$ and $C_{0}$ such that for any $\eta$ with $N^{-1-\varepsilon}\leq\eta\leq N^{-1}$ and for any choices of the signs in the imaginary part of $z^{m}_{j}$, we have

where the arguments of $F$ in the second term are changed from the Green’s functions of $H^{(v)}$ to $H^{(w)}$ and all other parameters remain unchanged.

We also remark that Theorem 2.3 holds for generalized Wigner matrices since $C_{sup}<\infty$ in (2.2). The positive lower bound on the variances, $C_{inf}>0$, is not necessary for this theorem.

Remark 2: Although we state Theorem 2.3 for Hermitian and symmetric ensembles, similar results hold for real and complex sample covariance ensembles; the modification of the proof, to be given in Section 8, is obvious and we omit the details.

To summarize, our approach to prove the universality is based on the following three steps; a detailed outline will be given in Section 6. Step 1. Local semicircle law, i.e., Theorem 2.1. This will be proved in Sections 3–5. Step 2. Universality for ensembles with smooth distributions satisfying the logarithmic Sobolev inequality (LSI), Theorem 6.3. The key input is the general theorem, Theorem 6.2, concerning the universality for the local relaxation flow. In Section 7, by using the local semicircle law and the LSI, we verify the assumptions for this theorem. Step 3. Green’s function comparison theorem, Theorem 2.3. This removes the restriction on the smoothness and the LSI, and it will be proved in Section 8.

Convention. We will frequently use the notation $C,c$ for generic positive constants whose exact values are irrelevant and may change from line to line. For two positive quantities $A$, $B$ we also introduce the notation $A\asymp B$ to indicate that there exists a universal constant $C$ such that $C^{-1}\leq A/B\leq C$.

Proof of local semicircle law

Proof of Theorem 2.1 Recall that $G_{ij}=G_{ij}(z)$ denotes the matrix element

We will prove the following more detailed stronger result.

where $\kappa\equiv||E|-2|$ and $\delta_{+}$ is given in (2.4). Then for all $z=E+i\eta$ and

for sufficiently large N, with positive $c$ and $C>0$ depending only $\alpha$ and $\beta$ in (2.14) and $\delta_{-}$ in (2.4) and (2.5).

Remark: The condition (3.3) is effectively a lower bound on $\eta$. The control function $g(z)$ can be estimated by

up to some factor of order one. Note that the precise formula (3.2) for $g(z)$ is not important, only its asymptotic behaviour for small $\kappa$, $\eta$ and $\delta_{+}$ is relevant. The theorem remains valid if $g(z)$ is replaced by $\widetilde{g}(z)$ with $\widetilde{g}(z)\leq Cg(z)$. In particular, $g(z)$ can be chosen to be order one when $E$ is not near the edges of the spectrum. If we are only concerned with the case of generalized Wigner matrices, (2.6), we can choose $g(z)=O(\sqrt{\kappa+\eta})$ for any $z=E+i\eta$ $(\eta>0)$. Note that Theorem 2.1 was obtained by replacing $g(z)$ with the lower bound $\kappa\leq g(z)$ in Theorem 3.1.

Once the local semicircle law is established on scale $\eta\asymp 1/M$ (modulo logarithmic factors), we obtain the following supremum bound on the eigenvectors that can be interpreted as a lower bound of order $1/M$ on the localization length. The proof of this result now is simpler than in , since we have a pointwise control on the diagonal elements of the Green’s function. Let ${\bf u}_{\alpha}$ denote the normalized eigenvector of $H$ belonging to the eigenvalue $\lambda_{\alpha}$, $\alpha=1,2,\ldots,N$, i.e., $H{\bf u}_{\alpha}=\lambda_{\alpha}{\bf u}_{\alpha}$ and $\|{\bf u}_{\alpha}\|=1$.

Let $H$ be as in Theorem 3.1, for any fixed $\kappa>0$, there exists $C_{\kappa}$ that

For the case of generalized Wigner matrices, (2.6), we have the following more precise bound

Proof of Corollary 3.2. Let $\eta=C_{\kappa}M^{-1}(\log N)^{26+12\alpha}$; $C_{\kappa}$ can be chosen large enough so that (3.3) is satisfied for all $|\kappa^{\prime}|\leq\kappa$, making use of (3.5) . Choose $\{E_{m}\}$ as a grid of points in $[-2+\kappa,2-\kappa]$ such that the distance between any two neighbors is of order $\eta$. Then with (3.4), we have

where we used $g(z)\leq\sqrt{\kappa+\eta}\leq C$ from (3.5). Then, with $|m_{sc}(z)|\leq C$ (see (2.13)) and

where ${\bf u}_{\alpha}=(u_{\alpha}(1),\,\,\,u_{\alpha}(2)\ldots u_{\alpha}(N))$, we have

By the definition of $E_{m}$, for any $\lambda_{\alpha}\in[-2+\kappa,2-\kappa]$, there exists $m^{\prime}$ such that $|E_{m^{\prime}}-\lambda_{\alpha}|$ is of the order of $\eta$. Together with (3.10), we obtain (3.6).

In case of the generalized Wigner matrix (2.6), we have $g(z)=\sqrt{\kappa+\eta}$ and $M\asymp N$. Let $\eta$ be the solution to $\eta=N^{-1}(\log N)^{26+12\alpha}(\kappa+\eta)^{-3/2}$, then $N^{-1}\leq\eta\leq CN^{-1}(\kappa+N^{-1})^{-3/2}(\log N)^{26+12\alpha}$. With this choice of $\eta$, (3.3) is satisfied, and $\max_{i}|G_{ii}-m_{sc}|\leq C(\log N)^{-2}(\kappa+\eta)^{1/2}$ holds with an overwhelming probability by (3.4). Since $|{\text{Im}}\,m_{sc}(z)|\leq C\sqrt{\kappa+\eta}$, so $\max_{i}{\text{Im}}\,G_{ii}\leq C(\kappa+\eta)^{1/2}$. By the argument above, we obtain that $\|{\bf{u}}_{\alpha}\|_{\infty}^{2}\leq C\eta(\kappa+\eta)^{1/2}$ on this event. This proves (3.7).

To prove that $G_{ii}(z)$ is very close to $m_{sc}(z)$ in the sense of (3.4), we will also need to control the off-diagonal elements. In fact we will show that all $G_{ij}$ ($i\neq j$) are bounded by $O((M\eta)^{-1})$ up to some factor $(\log N)^{C}$. To state the result precisely, we first define some events in the probability space.

Recall that $\lambda_{\alpha}$, $\alpha=1,2,\ldots,N$, denote the eigenvalues of $H=(h_{ij})$. Denote by $\Omega^{0}$ the subset of the probability space such that

Finally, denote by $\Omega_{z}^{d}$ the set

and similarly define $\Omega_{z}^{o}$. These sets depend on $N$ but we suppress this from the notations.

Proof of Theorem 3.1. The following proposition immediately implies Theorem 3.1.

Suppose that the assumptions of Theorem 3.1 hold. Then, for sufficiently large N, we have

Following the work of , we will use a continuity argument. In Section 4 we will derive a self-consistent equation of the form

Later we will give an explicit formula for $\Upsilon_{i}(z)$, but for now we take (3.16) as the definition of $\Upsilon_{i}$. Let $\widehat{\Omega}^{\Upsilon}_{z}(N)=\widehat{\Omega}^{\Upsilon}_{z}$ be the subset of $\Omega^{0}$ where the following inequality holds

We will use the following Lemmas that will be proved later in Section 4 and 5.

Let $z=E+i\eta$ be a fixed complex number satisfying (3.3). Then there are constants $C$ and $c$ such that for $N\geq N_{0}$, with $N_{0}$ sufficiently large independent of $E$ and $\eta$, the following estimates hold.

(2) Suppose that $\eta\leq 3$. Setting $z^{\prime}=z+iN^{-5}$, we have

Suppose we are on the event $\widehat{\Omega}^{\Upsilon}_{z}$ for some fixed $z=E+i\eta$ satisfying (3.3). Suppose either $3\leq\eta\leq 10$ or the following inequality hold:

Then, for sufficiently large $N$, we have

Proof of Proposition 3.3. Recall that $\widehat{\Omega}_{z}^{d}$ is the subset of $\Omega^{0}$ where (3.12) holds. Since on $\widehat{\Omega}^{\Upsilon}_{z}$ (3.12) follows from (3.17), the case $3\leq\eta={\text{Im}}\,z\leq 10$ follows from Lemma 3.4 and Lemma 3.5 by taking a union bound for $0\leq k\leq 10N^{5}$.

Now we prove (3.15) for the case $\eta\leq 3$ assuming that $z=E+i\eta$ satisfies (3.3). We have shown that (3.15) holds for $\eta=3$, now we will successively decrease $\eta$ by $N^{-5}$ in each step, and we continue this inductive procedure as long as (3.3) is still satisfied for the reduced $\eta$. More precisely, let $z^{\prime}=z+iN^{-5}$ and assume that (3.15) holds for $z^{\prime}$. Our goal is to prove that

The number of steps we will be taking is of order $N^{5}$. Since $N^{-c\log\log N}\ll N^{-5}$, this proves (3.15) provided that we can establish (3.26).

From (3.21), the difference between the probabilities of the sets $\widehat{\Omega}_{z}^{o}\cap\Omega_{z^{\prime}}^{d}\cap\Omega_{z^{\prime}}^{o}$ and $\Omega_{z^{\prime}}^{d}\cap\Omega_{z^{\prime}}^{o}$ is negligible. With the definition of $\Omega_{z}^{d}$ and $\Omega_{z}^{o}$ in (3.14), we have

Then, to prove (3.26), it remains to prove

i.e., we need to estimate the probability of the complement of $\widehat{\Omega}_{z}^{d}$ on the set $\widehat{\Omega}_{z}^{o}\cap\Omega_{z^{\prime}}^{d}\cap\Omega_{z^{\prime}}^{o}$. On this set, using (3.22), we can assume that the estimate (3.17) holds with a very high probability. We will show below that (3.24) holds on $\Omega_{z^{\prime}}^{d}$. Then (3.25) together with (3.17) imply (3.12), the defining relation of $\widehat{\Omega}_{z}^{d}$. This will conclude (3.28) and complete the proof of Proposition 3.3. Therefore, we only have to verify (3.24).

Now we show that (3.24) holds on $\Omega^{d}_{z^{\prime}}$. Recall $z^{\prime}=z+iN^{-5}$ and we have the trivial estimate

In the set $\Omega^{d}_{z^{\prime}}$, we have

where in the second inequality we used (3.3). By the definition of $g(z)$ from (3.2), we have $g(z)\leq\sqrt{\kappa+\eta}$. Thus, if (3.3) holds, then, in particular,

This sets a lower bound on $\eta$. Together with $|z-z^{\prime}|=1/N^{5}$, we have the trivial continuity bound

using $|\partial_{z}m_{sc}(z)|\leq|{\text{Im}}\,z|^{-2}$, $|\partial_{z}G_{ii}(z)|=|[(H-z)^{-2}]_{ii}|\leq\|(H-z)^{-2}\|\leq|{\text{Im}}\,\,z|^{-2}$ and $\eta>N^{-1}$ from (3.31). Thus

Using $|g(z)|\geq C\eta\geq CN^{-1}$ and $|g^{\prime}(z)|\leq C\eta^{-1}\leq CN$ for $\eta\leq 3$, we have the following estimate

in the set $\Omega^{d}_{z^{\prime}}$. Thus the assumption (3.24) holds in the set $\Omega^{d}_{z^{\prime}}$.

Under the assumptions of Theorem 3.1, with (3.15), (3.20), (3.23) and the definitions in (3.14), all these $\Omega$’s are sets of almost full probability, i.e.,

Self-consistent equation for Green’s function

For ${\bf T}=\{k_{1},k_{2},\ldots k_{s}\}$, we define

These quantities depend on $z$, but we mostly neglect this dependence in the notation.

We start the proof with deriving some identities between the matrix elements of $G=(H-z)^{-1}$ and $G^{(k_{1},k_{2},\ldots,k_{s})}$ using the following well known result in linear algebra that we quote without proof.

Let $A$ , $B$, $C$ be $n\times n$, $m\times n$ and $m\times m$ matrices. We define $(m+n)\times(m+n)$ matrix $D$ as

Furthermore, let ${\bf T}$ denote the unordered set $\{k_{1},k_{2},\ldots k_{s}\}$ and $1\leq k_{i}\leq n$, $1\leq i\leq s$. We define $D^{({\bf T})}$ to be the $n+m-s$ by $n+m-s$ submatrix of $D$ after removing the $k_{i}$-th $(1\leq i\leq s)$ rows and columns and define $\widehat{D}^{({\bf T})}$ to be the $n-s$ by $n-s$ submatrix of $\widehat{D}$ after removing the $k_{i}$-th $(1\leq i\leq s)$ rows and columns. Then for any $1\leq i,j\leq n$ and $i,j\notin{\bf T}$, we have

Using Lemma 4.1 and Definition 4.1, for $1\leq i\not=j\leq N$, we have

For the off diagonal matrix elements $G_{ij}$, $(i\neq j)$, we have

Let ${\bf T}$ be an unordered set $\{k_{1}$, $k_{2}$, $\ldots$, $k_{s}\}$ with $1\leq k_{t}\leq N$ for $(1\leq t\leq s)$ or $\bf T=\emptyset$. For simplicity, we use the notation $(i\,{\bf T})$ for $\{i\}\cup{\bf T}$ and $(ij\,{\bf T})$ for $\{i,j\}\cup{\bf T}$. Then we have the following identities:

For any indices $i$, $j$ and $k$ that are different and $i,j,k\notin{\bf T}$

Proof of Lemma 4.2. The first two identities (4.5) and (4.6) are obvious extensions of (4.3) and (4.4). To prove (4.7), without loss of generality, we may assume that $i=1$, $j=2$ and ${\bf T}=\emptyset$. Let $D=H-z$ and $\widehat{D}$ defined as in (4.2) with $n=2$ and $m=N-2$. With Lemma 4.1, we have that for $i,j=1$ or $2$.

Since $\widehat{D}$ is just a $2\times 2$ matrix, one can easily check that (4.7) holds. With the same method, one can obtain (4.8) .

The diagonal matrix elements of the resolvent satisfy the following self-consistent equation.

hold with a probability larger than $1-CN^{-c(\log\log N)}$ for sufficiently large $N$.

Proof of Lemma 4.3. We can write $G_{11}$ as follows,

Combining this identity with (4.7), we have

Clearly $G_{11}$ can be replaced with any $G_{ii}$ and this proves (4.11) with the definition (4.12).

We note $h_{ii}$, $v_{ij}$ and $G^{(i{\bf T})}_{kl}$ are independent for $k,l\notin(i{\bf T})$. With the sub-exponential decay (2.14) and $\sigma_{ij}^{2}\leq 1/M$, we have for any $i,j$

In Corollary B.3 of Appendix B we will prove a general large deviations result. Applying (B.15) to the last term in (4.19), with the choice

and with $\sum_{j}\sigma_{ij}^{2}=1$ and $\sigma_{ii}^{2}\leq 1/M$, we obtain that

holds with a probability larger than $1-CN^{-c(\log\log N)}$. Together with (4.21), we obtain that (4.13) holds with a probability larger than $1-CN^{-c(\log\log N)}$ for sufficiently large $N$.

Next we prove (4.14). By the definition of $K^{(ij{\bf T})}_{ij}$, $i\neq j$, we can write

Applying (B.16), (4.21) and $\sigma^{2}_{ij}\leq 1/M$, we obtain that

holds with a probability larger than $1-CN^{-c(\log\log N)}$ for sufficiently large $N$. With Schwarz’s inequality, for any $i,j$,

Denote $u^{({ij\bf T})}_{\alpha}$ and $\lambda_{\alpha}^{({ij\bf T})}$ ($\alpha=1,2,\ldots,N-|{\bf T}|-2$) the $l^{2}$-normalized eigenvectors and eigenvalues of $H^{({ij\bf T})}$. Let $u^{({ij\bf T})}_{\alpha}(l)$ denote the $l$-th coordinate of $u^{({ij\bf T})}_{\alpha}$, then for any $l$

Here we defined $|A|^{2}:=A^{*}A$ for any matrix $A$. Inserting (4.27) into (4.25) and using the definition of $M$ in (2.3), we obtain that (4.14) holds with a probability larger than $1-CN^{-c(\log\log N)}$ for sufficiently large $N$.

and $\max_{\alpha}|\lambda_{\alpha}|\leq 3$, we have that

holds in $\Omega^{0}$. Together with $3\leq\eta\leq 10$ and $|E|\leq 10$, we obtain

with some positive constants. From the interlacing property of the eigenvalues of the matrix and its submatrices, we find that not only $\|H\|\leq 3$ but also $\|H^{(\bf T)}\|\leq 3$ holds on the set $\Omega^{0}$. Thus for any $j,k$ such that $i$, $j$ and $k$ are all different, the bounds

hold in $\Omega^{0}$ by a similar argument that led to (4.30). Thus (4.6) implies

and (3.18) follows make use of (4.14) and $\eta>3$.

Now we prove (3.19). Recall that the self consistent equation (3.16) with the error term $\Upsilon_{i}(z)$ is given by (4.12), i.e.,

Now we bound $\Upsilon_{i}(z)$ in $\widehat{\Omega}^{o}_{z}$. Since $\sigma^{2}_{ii}\leq M^{-1}$, with (4.31), the first term of the r.h.s. of (4.33) is less than $O(M^{-1})$. Then with (2.1), and using the bound on $G_{ij}$ ($i\neq j$) from (3.13) and the one on $G_{ii}$ from (4.31), we obtain that the second term of the r.h.s. of (4.33) is less than $C(\log N)^{10+8\alpha}(M\eta)^{-1}$ (and with (3.31), we know it is much less than 1), i.e., in $\widehat{\Omega}^{o}_{z}$

The last term of the r.h.s. of (4.33) can be bounded, using (4.13) with ${\bf T}=\emptyset$, with a very large probability. Using (4.8) and (4.31), the $G^{(i)}_{kl}$’s in (4.13) can be bounded as

Therefore, again with the bound on $G_{ij}$ ($i\neq j$) in (3.13) and the one on $G_{ii}$ from (4.31), we see that

Now we prove (3.20) for $\eta\geq 3$. By the definition of $\Upsilon_{i}$ in (4.33), we have

We now claim that for some large enough $C>0$ there exists $c>0$ such that

The first estimate follows from the definition of $Z_{ii}^{(i)}$ given in Definition 4.1 by using the sub-exponential decay of the matrix elements and by using the trivial bound $|G_{kl}^{(i)}|\leq\eta^{-1}\neq N$. The second estimate is a trivial consequence of the first one and the definition of $K_{ii}^{(i)}$. Together with (4.38), we obtain (3.20) in the case that $3\leq\eta\leq 10$.

We now prove (3.21) and (3.22) for the case $\eta\leq 3$ satisfying (3.3). We will work in the event $\Omega_{z^{\prime}}^{d}\cap\Omega_{z^{\prime}}^{d}$ where $z^{\prime}=z+iN^{-5}$. Similarly as we proved (3.33), from the bound below (3.31) and the Lipschitz continuity of $g(z)$ , we obtain that

hold in $\Omega_{z^{\prime}}^{d}\cap\Omega_{z^{\prime}}^{d}$. We note the r.h.s of these inequalities are much less than $(\log N)^{-1}$ by (3.3). From the explicit formula (2.13) we obtain that $c\leq|m_{sc}(z)|\leq C$ for any $|z|\leq 10$ with some positive constants. Using this observation and the fact that the r.h.s. of (4.40) is much less than $(\log N)^{-1}$, we have

Hence, using (3.31), (4.7), (4.8) and the lower bound of $|G_{ii}|$, one can easily obtain that

hold in $\Omega_{z^{\prime}}^{o}\cap\Omega_{z^{\prime}}^{d}$, (for the third term in l.h.s., we have also used the lower bounds of $G^{(j)}_{ii}$’s as above). Then we also have

The definition of $m_{sc}(z)$ implies ${\text{Im}}\,\,\,m_{sc}(z)\leq C\sqrt{\kappa+\eta}$. Then with (4.42), (3.3) and $g(z)\leq\sqrt{\kappa+\eta}$, we have that

holds in $\Omega_{z^{\prime}}^{o}\cap\Omega_{z^{\prime}}^{d}$ for some constant $C>0$. Inserting it into (4.14), we obtain that

Then, as we proved in (4.34) and (4.36), we get that

Finally, similarly as using (4.37)- (4.38) to prove (3.20), we can obtain (3.23) in the case that $\eta<3$.

Stability of the self-consistent equation: proof of Lemma 3.5

In this section, we prove Lemma 3.5, i.e., we will prove the stability of the self-consistent equation with a precise error estimate given in (3.25). We set $m_{sc}=m_{sc}(z)$ and $\Upsilon=\max_{i}|\Upsilon_{i}(z)|$ for simplicity of notation and we will omit all $z$ dependences in all the symbols. With the definition of $m_{sc}(z)$ in (2.12) and (2.13), the following properties of $m_{sc}(z)$ can be easily established:

Let $z=E+i\eta$ with $\eta>0$ and $|z|\leq 20$. Then we have

for some constant $C$. Furthermore, suppose that either $2\leq|E|\leq 10$ or $\kappa\leq\eta$. Then

For small values of $|z^{2}-4|\asymp\kappa+\eta$, $m_{sc}(z)$ has the asymptotic expansion

We first prove (3.25) for the case that $3\leq\eta\leq 10$. In this case, we can easily check that $g(z)=\sqrt{\kappa+\eta}$. Denote the difference between $G_{ii}$ and $m_{sc}$ by

By the self consistent equation (3.16), (2.1) and (2.12), we have

For $\eta\geq 3$, $|z+m_{sc}(z)|>2$ by (2.13). Using $|G_{ii}|\leq\eta^{-1}$ and $|m_{sc}|\leq\eta^{-1}$, we obtain

From the assumption (3.17) and (3.3), we have $\Upsilon=\max_{i}|\Upsilon_{i}|\ll 1$ in this region. Together with $|z+m_{sc}(z)|>2$ and (5.6), we obtain that the absolute value of the r.h.s. of (5.5) is less than

Taking the absolute value of (5.5) and maximizing over $n$, we have

The denominator satisfies $|z+m_{sc}(z)|-\sup_{i}|v_{i}|\geq 2-2/3=4/3$, therefore we obtain $\sup|v_{i}|=$ $\sup_{i}|G_{ii}-m_{sc}(z)|\leq O(\Upsilon)$, which shows (3.25) for $3\leq\eta\leq 10$.

Next, we prove (3.25) in the case that $\eta\leq 3$ with $\eta$ satisfying (3.3) and under the condition (3.24). Define

Combining (3.17), (3.3), (3.24) with the fact that $g(z)\leq C$, we can see that

for some $C>0$. Furthermore (3.24) implies

Therefore, expanding the self consistent equation (3.16) around $z+m(z)$, we obtain that

where $\Omega_{i}$ is defined by the second equality and it satisfies

with error bounds uniform in $i$. Here $\|{\bf u}\|_{\infty}=\max_{i}|u_{i}|$. Taking the average of the r.h.s of (5.13) with respect to $i$, we obtain that

Here we used $\sum_{i}\sum_{j}\sigma_{ij}^{2}u_{j}=\sum_{j}u_{j}=0$. The bound (3.24) , (5.11) and $g(z)\leq\sqrt{\kappa+\eta}$ (from (3.2)) implies that

Together with (5.10) and (5.17), we obtain

To bound $m(z)$, we use the following lemma.

with ${\text{Im}}\,s_{z}(t)>0$ and the estimate

Proof of Lemma 5.2. It follows from (5.20) that

We denote by $s^{1}_{z}(t)$ and $s^{2}_{z}(t)$ the two solutions of this equation, which are continuous with respect to $t$ locally in the neighborhood (5.19). When $t=0$, one of them is equal to $m_{sc}(z)$, we choose $s^{1}_{z}(0)=m_{sc}(z)$. From (5.23), we have

Then, for small enough $\delta$, if $|t|\leq\delta(\kappa+\eta)$, then $|s^{1}-s^{2}|\geq\frac{1}{2}\min\{|z-2|,|z+2|\}$ by using (5.24) and that $\kappa+\eta\asymp\min\{|z-2|,|z+2|\}$. We thus see that only one out of $s^{1}$ and $s^{2}$ can satisfy (5.21). With the assumption that $s^{1}_{z}(0)=m_{sc}(z)$, it is $s^{1}$ that satisfies (5.21). Then

where for the second inequality, we used $|t|\leq\delta(\kappa+\eta)$.

Using Lemma 5.2, for $s_{z}(t)=m(z)$ and $t=-\Omega$, we have

where in the second inequality we used (5.17) and (5.18). Subtracting (5.17) from (5.13), we have the equation for $u_{i}$

where $w_{i}$ is defined as $u_{i}-(\sum_{j}\sigma_{ij}^{2}u_{j})(z+m_{sc})^{-2}$. By (5.17), it is bounded by

Then, using (5.11) and (5.1), we obtain that

Inserting this into (5.29), using the bounds on $\|{\bf u}\|_{\infty}$ in (5.18) and (5.27), we have

From (5.3) in Lemma 5.1, whenever $|E|\geq 2$ or $\kappa\leq\eta$, in which case $g(z)\asymp\sqrt{\kappa+\eta}$, we have

for some $C>0$. Therefore (5.28) imply in this region that

and using $\|{\bf u}\|_{\infty}\ll\sqrt{\kappa+\eta}$ from (5.18), we conclude that

Combining this with the bound on $m-m_{sc}$ (5.27) and $G_{ii}-m_{sc}=m-m_{sc}+u_{i}$, we obtain (3.25).

Finally, we consider the main interesting regime: $|E|\leq 2$ and $\kappa\geq\eta$. We claim that the following inequality about $m_{sc}(z)$ holds.

Let $1>\delta_{-}>0$ be a given constant. Then there exist small real numbers $\tau\geq 0$ and $c_{1}>0$, depending only on $\delta_{-}$, such that we have

for any positive number $\delta_{+}$ such that $-1+\delta_{-}\leq 1-\delta_{+}$.

We postpone the proof of this lemma to the end of this subsection and we first complete the main argument. Recall that $B=\{\sigma_{ij}^{2}\}_{i,j=1}^{N}$ is the matrix of variances which is symmetric. We also recall $\delta_{\pm}$ from (2.4) and we will apply Lemma 5.3 with these $\delta_{-}$ and $\delta_{+}$. Fix $z$, set $\zeta:=m_{sc}^{2}(z)=(m_{sc}(z)+z)^{-2}$ and rewrite (5.28) as

by the Lemma 5.3 and ${\bf{w}}\perp(1,...,1)$, the Neumann expansion of (5.38) converges on span $((1,...,1))^{\perp}$ and

since $|\zeta|=|m_{sc}|^{2}\leq 1$ and $\sum_{j}|B_{ij}|=\sum_{j}B_{ij}=\sum_{j}\sigma_{ij}^{2}=1$. Then we have

by Lemma 5.3. Since for any $N\times N$ matrix we have

Thus, estimating the first $n\leq n_{0}:=(\log N)(c_{1}\widehat{g}(z))^{-1}$ terms in (5.39) by (5.40), and the rest by (5.41), we get

Using the bound (5.31) on $\|{\bf w}\|_{\infty}$ and the bound (5.18) on $\|{\bf u}\|_{\infty}$, we have

for some $C>0$. Combining this with (5.27), we find

which implies (3.25), since $g(z)=\min\{\sqrt{\kappa+\eta},\widehat{g}(z)\}$.

Proof of Lemma 5.3. First, if $\widehat{g}(z)=\delta_{+}$, then we choose $\tau=0$. With $|m_{sc}|\leq 1$ in (5.1), one can see that (5.36) holds.

In the case of $\widehat{g}(z)=|1-{\rm Re}\,m_{sc}^{2}|$, we have $\widehat{g}\leq 2$ by using $|m_{sc}|\leq 1$. We choose $\tau=\delta_{-}/10$, then

For the other term in r.h.s. of (5.44), we have

With $|m_{sc}|\leq 1$ in (5.1) and $\widehat{g}(z)=|1-{\rm Re}\,(m_{sc}^{2})|$ in this case, (5.46) is bounded as

for some $C$ depending on $\delta_{-}$. At last, we complete the proof by combining (5.47) and (5.45).

Proof of the universality of local statistics

We now outline the main steps to prove Theorem 2.2.

Step 1. Local relaxation flow. Following , we first prove that the local eigenvalue statistics of Dyson Brownian motion (DBM) at a fixed time $t$ are the same as those of GUE if $t\asymp N^{-\varepsilon_{0}}$ for some $\varepsilon_{0}$. The DBM is generated by the flow

($\beta=2$ for GUE) be the probability measure of the eigenvalues of the general $\beta$ ensemble, $\beta\geq 1$ (in this section, we often use the notation $x_{j}$ for the eigenvalues to follow the notations of ). In this paper we consider the $\beta=2$ case for simplicity, but we stress that our proof applies to the case of symmetric matrices as well. Denote the distribution of the eigenvalues at time $t$ by $f_{t}({\bf x})\mu({\rm d}{\bf x})$. Then $f_{t}$ satisfies

Suppose that the probability law for the initial matrix $H_{0}$ satisfies the assumptions of Theorem 2.2. Then there exists $\varepsilon_{0}>0$ such that for any

where $p_{GU\!E,N}^{(k)}$ is the $k$-point correlation function of the GUE ensemble.

Proof of Theorem 6.1. We first recall the following general theorem concerning the Dyson Brownian motion from that asserts that under four general assumptions, the local eigenvalue statistics of the time evolved matrix $H_{t}$ coincide with GUE. The first assumption (called Assumption I in ) is a convexity bound on ${\mathcal{H}}$ which is automatically satisfied in our case and we only have to verify the following three assumptions.

For the next assumption, we introduce a notation. Let $\gamma_{j}=\gamma_{j,N}$ denote the location of the $j$-th point under the limiting density, i.e., $\gamma_{j}$ is defined by

We will call $\gamma_{j}$ the classical location of the $j$-th point.

Assumption III. There exists an $\varepsilon>0$ such that

Assumption IV. For any compact subinterval $I_{0}\subset\{E\;:\;\varrho(E)>0\}$ independent of $N$, and for any $\delta>0$, $\sigma>0$ and $r>0$, there are constants $c$ depending on $I_{0}$, $\delta$, $\sigma$ and $r$ such that for any interval $I\subset I_{0}$ with $|I|\geq N^{-1+\sigma}$, we have

where $\varepsilon$ is the exponent from Assumption III.

[19, Theorem 2.1] Let $\varepsilon>0$ be the exponent from Assumption III. Suppose that there is a time $\tau<N^{-2\varepsilon}$ such that the following entropy bound holds

Theorem 6.2 was exactly Theorem 2.1 of except that the assumption (6.10) on the entropy in was stated for the initial probability density $f_{0}$. Clearly, we can start the flow (6.4) from a fixed time $\tau\ll N^{-2\varepsilon+\delta}$ since the statement of Theorem 6.2 concerns only the time $t\geq N^{-2\varepsilon+\delta}$. In the case that the flow (6.4) is generated from the matrix evolution (6.1), the entropy assumption (6.10) is satisfied automatically. To see this, let $\nu_{t}^{ij}$ denote the probability measure of the $ij$-th element of the matrix $H_{t}$, $i\leq j$, and $\bar{\nu}_{t}$ the probability measure of the matrix $H_{t}$. Let $\bar{\mu}$ denote the probability measure of the GUE and $\mu^{ij}$ the probability measure of its $ij$-th element which is a Gaussian measure with mean zero and variance $1/N$. Since the dynamics of matrix elements are independent (subject to the Hermitian condition), we have the identity

The process $t\to\nu^{ij}_{t}$ is an Ornstein-Uhlenbeck process and each entropy term on the right hand side of the last equation is bounded by $CN$ provided that $t\geq 1/N$ and $\nu^{ij}_{0}$ has a subexponential decay. It is easy to check from the explicit OU kernel. Since the entropy of the marginal distribution on the eigenvalues is bounded by the entropy of the total measure on the matrix, we have proved that

and this verifies (6.10). Therefore, in order to apply Theorem 6.2, we only have to verify the Assumptions II, III and IV. Clearly, Assumption II follows from Theorem 2.1 (note that in the case of generalized Wigner matrix, $M\asymp N$ and $g(z)\asymp\sqrt{\kappa+\eta}$). Assumption IV also follows from Theorem 2.1 by noting that ${\mathcal{N}}_{I}\leq C\,{\text{Im}}(E+i\eta)$ if $I$ is an interval of length $\eta$ about $E$. We also note that Assumption IV in was stated in a slightly stronger form, requiring a large deviation bound (6.9) for all $K\geq 1$, but inspecting the proof of Theorem 2.1 of reveals that Assumption IV is used only for $K$ larger than some positive power of $N$ and smaller than $N$ (the main observation is that the upper limit of the summation in (7.16) of is effectively $N$ and not $\infty$).

Having verified all other assumptions, it remains to prove (6.8), which we state as the next theorem.

Suppose $H$ satisfies the assumptions of Theorem 2.2, in particular, it is a generalized Wigner matrix with positive constants $C_{inf}$, $C_{sup}$ in (2.6). Let $\widetilde{\nu}_{ij}(x){\rm d}x:=\sigma_{ij}\nu_{ij}(\sigma_{ij}x){\rm d}x$ be the rescaling of the distributions $\nu_{ij}$ of the matrix elements and suppose that they satisfy the logarithmic Sobolev inequality (LSI) with a constant $C_{S}$ independent of $N,i,j$, i.e.,