Universality for generalized Wigner matrices with Bernoulli distribution

Introduction

The universality of local eigenvalue statistics in the bulk of the spectrum of random matrices has been traditionally considered only for invariant ensembles . For non-invariant ensembles, a new approach to prove the bulk universality was developed in . It consists of the following three steps:

Universality for Gaussian divisible ensembles.

Approximation by Gaussian divisible ensembles.

In Step 2, the universality of the local eigenvalue statistics for a large class of matrices, i.e., Gaussian divisible matrices, was established. Thus in order to prove the universality of a given ensemble, it remains to approximate the matrix elements in this ensemble by Gaussian divisible distribution in such a way that the local eigenvalue statistics are unchanged. This approximation is intrinsically a density theorem and it can be achieved by perturbative expansions in several different ways. In the most recent approach , the universality for Gaussian divisible ensembles was proved via the Dyson Brownian motion and the stability of eigenvalues in Step 3 was provided by the Green function comparison theorem. In Step 2 a technical tool, the logarithmic Sobolev inequality (LSI), was needed to estimate the fluctuations of eigenvalue distribution. This restriction could not be completely removed in Step 3 and thus the Bernoulli measures were excluded in . In this paper, we will improve the local semicircle law so that the LSI is no longer needed. This will enable us to prove the universality for generalized Wigner matrices with Bernoulli distributions. As a byproduct of the new stronger form of local semicircle law, we also obtain much stronger estimates on the eigenvalue density and on the matrix elements of the resolvent.

Recall the Stieltjes transform of the empirical measure of the eigenvalues $\{\lambda_{j}\}_{j=1}^{N}$ is defined by

We have proved in that the difference between $m_{N}(z)$ and $m_{sc}(z)$, the Stieltjes transform of the semicircle law (2.9), is bounded by $(N\eta)^{-1/2}$ where $\eta={\mathfrak{Im}\,}z$. The main result of this paper states that the error can be improved to $(N\eta)^{-1}$. The improvement of a factor $(N\eta)^{-1/2}$ resembles the usual $N^{-1/2}$ factor in the central limit theorem and it results from a new estimate on the correlations of error terms. This estimate also implies that the error between the normalized empirical counting function of the eigenvalues and the one given by the semicircle law is less than $N^{-1+\varepsilon}$ in the bulk of the spectrum for any $\varepsilon>0$. This new input is sufficiently strong to replace the usage of the (LSI) in , see the discussion after Theorem 2.2 for more details.

Notice that this improvement of a factor $(N\eta)^{-1/2}$ and the removal of the LSI need a substantial amount of work. Our motivations to take on this endeavor are for the following two reasons: (1) The distributions of the Bernoulli random matrices are very singular while the Gaussian measures in GOE are very smooth. It is not a priori clear that the universality holds for such singular distributions. (2) The adjacency matrices for random graphs are natural examples of symmetric random matrices. The matrix elements of these matrices take the values or $1$ and thus they form Bernoulli random matrices. Our current results do not cover this case since we require the mean zero condition, but they represent the first step toward the universality of the adjacency matrices of random graphs.

Main results

Matrices with independent, zero mean entries and with the normalization condition (2.1) will be called universal Wigner matrices. The basic parameter of such matrices is the quantity

Note that ${C_{inf}}={C_{sup}}(=1)$ corresponds to the standard Wigner matrices and the conditions $0<{C_{inf}}\leq{C_{sup}}<\infty$ define 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. A special case is the band matrix, where $\sigma_{ij}=0$ for $|i-j|>W$ with some parameter $W$.

Denote by $\Sigma:=\{\sigma^{2}_{ij}\}_{i,j=1}^{N}$ the matrix of variances which is symmetric, doubly stochastic by (2.1), and in particular satisfies $-1\leq\Sigma\leq 1$. Let the spectrum of $\Sigma$ 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 that provided the main motivation for our work.

One important class of universal Wigner matrices is the generalized Wigner ensemble which is defined by the extra condition that

It is easy to check that (2.4) holds with

Another example is the band matrix ensemble whose variances are given by

Define the Stieltjes transform of the empirical eigenvalue distribution of $H$ by

Define $m_{sc}(z)$ as the unique solution of

with positive imaginary part for all $z$ with $\text{Im }z>0$, i.e.,

Here the square root function is chosen with a branch cut in the segment $$so that asymptotically$\sqrt{z^{2}-4}\sim z$at infinity. This guarantees that the imaginary part of$m_{sc}$is non negative for$\text{Im }z>0$ and it is the Wigner semicircle distribution

The Wigner semicircle law states that $m_{N}(z)\to m_{sc}(z)$ for any fixed $z$, i.e., provided that $\eta$ is independent of $N$. We have proved a local version of this result for universal Wigner matrices and the main result can be stated as the following probability estimate:

with some constant $C_{2}$. The accuracy of this estimate can be improved from $(M\eta)^{-1/2}\,\kappa^{-1}$ to $(M\eta)^{-1}\,\kappa^{-1}$, which is the content of the next theorem. It summarizes the results of Theorems 4.1 and 5.1. Prior to our result in , a central limit theorem for the semicircle law on macroscopic scale for band matrices was established by Guionnet and Anderson and Zeitouni ; a semicircle law for Gaussian band matrices was proved by Disertori, Pinson and Spencer . For a review on band matrices, see the recent article by Spencer.

We consider universal Wigner matrices and its special class, the generalized Wigner matrices in parallel. The parameter $A$ will distinguish between the two cases; we set $A=2$ for universal Wigner matrices, and $A=1$ for generalized Wigner matrices, where the results will be stronger.

where \kappa:=\big{|}\,|E|-2\big{|}. 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 $\varepsilon>0$ and $K>0$ the Stieltjes transform of the empirical eigenvalue distribution of $H$ satisfies

for sufficiently large $N$. Furthermore, the diagonal matrix elements of the Green function $G_{ii}(z)=(H-z)^{-1}(i,i)$ satisfy that

and for the off-diagonal elements we have

The subexponential decay condition (2.11) can also be easily 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 $p_{N}(\lambda_{1},\ldots,\lambda_{N})$ 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. The same result was proved in under the additional assumption (2.26).

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

Remark. We can take $b=N^{-c}$ for some small constant $c>0$ so that there is no double limit taken. This is because all our bounds have an effective error estimate $N^{-c}$. In case of hermitian matrices there is no need for averaging in the energy parameter $E^{\prime}$. The limit (2.17) holds even for any fixed energy $E^{\prime}$, with $|E^{\prime}|<2$, since, instead of relying on the local relaxation flow of , we can use the result of for Gaussian divisible ensembles at a fixed energy.

It is well-known that the limiting correlation functions of the GUE ensemble are given by the sine kernel

and a similar universal formula is available for the limiting gap distribution. The formulas for the GOE cases are more complicated and we refer the reader to standard references such as .

We will prove Theorem 2.2 using the approach of . The logarithmic Sobolev inequality was an important tool in these papers and it was the main obstacle why the case of Bernoulli random matrices were not covered. We note that the Bernoulli distribution satisfies the discrete version of the LSI but it would not be sufficient for our purposes. To explain the necessity of LSI, we now review the three basic ingredients of the approach of .

Local semicircle law: It states that the density of eigenvalues is given by the semicircle law down to short scales containing only $N^{\varepsilon}$ eigenvalues for all $\varepsilon>0$, where $N$ is the size of the matrix.

Local ergodicity of the Dyson Brownian motion: The Dyson Brownian motion is given by the flow

be the probability measure of the eigenvalues ${\bf{x}}=(x_{1},x_{2},\ldots,x_{N})$ of the general $\beta$ ensemble, $\beta\geq 1$ ($\beta=2$ for the hermitian case and $\beta=1$ for the symmetric case). Denote the distribution of the eigenvalues of $H_{t}$ at time $t$ by $f_{t}({\bf x})\mu({\rm d}{\bf x})$. Then $f_{t}=f_{t,N}$ satisfies

We now recall the following theorem concerning the universality of the Dyson Brownian motion. Following the convention in , we label the assumptions as Assumptions II–IV since the Assumption I, a convexity property of the Hamiltonian for the invariant measure of the Dyson Brownian motions, is automatically satisfied for any $\beta$ ensembles.

where $\varrho_{sc}$ is the density of the semicircle law (2.10).

Let $\gamma_{j}=\gamma_{j,N}$ denote the location of the $j$-th point under the semicircle law, 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

where $\varepsilon$ is the exponent from Assumption III and $\sigma$ and $\delta$ are arbitrarily small numbers.

We have proved that Assumption IV follows from the local semicircle law and Assumption III also follows from the local semicircle law provided that a uniform LSI for the distributions of the matrix elements is assumed.

Green function comparison theorem: It asserts that the correlation functions of the eigenvalues of two matrix ensembles are identical up to the scale $1/N$ provided that the first four moments of the matrix elements of these two ensembles are almost identical. Given this theorem and the universality for the Dyson Brownian motion for $t\sim N^{-\varepsilon}$, the universality for a matrix ensemble $H$ holds if we can find another matrix ensemble $H_{0}$ such that the first four moments of the matrix elements of $H$ and $H_{t}$ (given by (2.18)) are almost the same. Furthermore, $H_{0}$ is required to satisfy a uniform LSI so that the Assumption III can be verified. This is possible if the first four moments of $H_{0}$ satisfy

where $m_{k}(i,j)$ is the $k$-th moment of the $i,j$ matrix element in the symmetric case. In the hermitian case, the moments of the real and imaginary parts have to satisfy (2.26).

Combining these ingredients, the universality of local eigenvalue statistics in the bulk was proved for all generalized Wigner ensembles (see (2.6) for the definition) satisfying (2.26) and a subexponential decay technical condition. The restriction (2.26) was needed to guarantee the existence of a matching matrix ensemble whose matrix element distributions satisfy the LSI so that the Assumption III can be verified. The local semicircle estimates in Theorem 2.1 imply that the empirical counting function of the eigenvalues is close to the semicircle counting function (Theorem 6.3) and that the location of the eigenvalues are close to their classical location in mean square deviation sense (Theorem 7.1). This provides a direct proof to the Assumption III (2.24) and thus removes the usage of the LSI.

Finally we summarize the recent results related to the bulk universality of local eigenvalue statistics. The local semicircle law for Step 1 was first established for Wigner matrices in a series of papers . The method was based on a self-consistent equation for the Stieltjes transform of the eigenvalues and the continuity of the imaginary part of the spectral parameter in the Stieltjes transform. As a by-product, an eigenvector delocalization estimate was proved.

The universality for Gaussian divisible ensembles was proved by Johansson for hermitian Wigner ensembles. It was extended to complex sample covariance matrices by Ben Arous and Péché . There were two major restrictions of this method: 1. The Gaussian component was fairly large, it was required to be of order one independent of $N$. 2. It relies on explicit formulas for the correlation functions of eigenvalues which are valid only for Gaussian divisible ensembles with unitary invariant Gaussian component. The size of the Gaussian component was reduced to $N^{-1+\varepsilon}$ in by using an improved formula for correlation functions and the local semicircle law from . The Gaussian component was then removed by a perturbation argument using the reverse heat flow. Thus the three step strategy to prove the universality was introduced and it led to the first proof of the bulk universality for hermitian Wigner ensembles. Due to the reverse heat flow argument used in Step 3, the universality class established in was restricted to matrices with smooth distributions for the matrix elements. Shortly after, Tao and Vu proved the four moment theorem which in particular removes the smoothness restriction in Step 3. It thus proved the universality for hermitian Wigner matrices whose matrix element distributions were supported on at least three points. The last condition was removed in by combining the arguments of . The result of also implies that the local statistics of symmetric Wigner matrices and GOE are the same, but under the restriction that the first four moments of the matrix elements match those of GOE. Thus the universality class for the local correlation functions established via the approach of combining and was broader for the hermitian ensembles than for the symmetric ones. This improvement was due to Johansson’s result , which provided the universality for Gaussian divisible ensembles in Step 2, was available only for hermitian ensembles.

A more general and conceptually very appealing approach for Step 2 is via the local ergodicity of Dyson Brownian motion. This approach, initiated in , was applied to prove the universality for symmetric Wigner matrices with the three point support condition. In , we formulated a general theorem for the bulk universality which applies to all classical ensembles, i.e., real and complex Wigner matrices, real and complex sample covariance matrices and quaternion Wigner matrices. Later on, Tao and Vu also extended their results to the sample covariance matrices with the three point support condition for complex covariance matrices and four moment matching conditions for real ones. Shortly after , Péché also extended the approach to the complex sample covariance matrices and proved the universality in the bulk.

Most recently, we introduced the Green function comparison theorem and extended the local semicircle law to include the matrix elements of the Green functions. This allows us to remove the smoothness restriction from the reverse heat flow argument in Step 3 of our approach. We remark that the comparison theorems in concern individual eigenvalues with a fixed index, while the Green function comparison theorem is at a fixed energy. On the other hand, in the variances of the matrix elements were allowed to vary, i.e., the matrices belonged to generalized Wigner ensembles. The three step strategy can thus be applied and the universality was proved for generalized Wigner ensembles with essentially only one class of measures, the Bernoulli measures, excluded due to the LSI used in verifying Assumption III in Step 2. Finally, in the current paper, Assumption III will be shown to be a consequence of a strong local semicircle law, which will be proved for all ensembles with a subexponential decay property. In particular, Bernoulli measures are now included in the universality class (in the sense of (2.17)) for both hermitian and symmetric generalized Wigner ensembles. We have thus removed all restrictions except the subexponential decay in our approach. A clear picture of the three step strategy emerges: Step 2 and 3 hold under very general conditions and are model independent. The main task of proving the universality is to establish a strong version of the local semicircle law—which can be model dependent. We believe that our method applies to generalized sample covariance matrices as well, but we will not pursue this direction in this paper.

Proof of Universality

We now prove the main universality theorem, Theorem 2.2.

Step 1. Universality for Dyson Brownian Motion: Under the Assumptions II–IV in the introduction, the universality for the Dyson Brownian Motion was proved in . We recall the statement in the following Theorem.

Notice that the assumption on the initial entropy is not needed as was remarked in .

Step 2 Universality for Gaussian divisible ensembles: The Dyson Brownian motion is generated by the matrix flow (2.18). Our task is to determine the initial ensemble $H_{0}$ so that the Assumptions II–IV of Theorem 3.1 can be proved for the flow. The Assumption IV is a direct consequence of the local semicircle law, i.e., Theorem 4.1. The Assumption III will be proved in Proposition 7.1. For the generalized Wigner matrices, the only assumption of Theorem 4.1 and Proposition 7.1 is the subexponential decay property of the distributions of the matrix elements. Since the evolution of the matrix element is given by an Ornstein-Uhlenbeck process, the subexponential property is preserved and we only have to check it for the initial data. We have thus proved the following theorem.

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 $t\geq N^{-\varepsilon_{0}}$, the probability law for the eigenvalues of $H_{t}$ satisfies the universality equation (2.17).

Step 3 Green function comparison theorem: We have proved the universality for all ensembles with the matrix element at $(i,j)$ distributed by $\sigma_{ij}\xi_{t}^{ij}$ with

where $\xi_{G}^{ij}$ are independent Gaussian random variables with mean and variance $1$ and $t\sim N^{-\varepsilon}$. In order to prove Theorem 2.2, it remains to approximate all random variables with the subexponential property by $\xi_{t}$. The only requirement of $\xi_{0}$ is the subexponential decay property and the mean zero and variance one normalization. Our tool is the following Green function comparison theorem from . It implies that the correlation functions of the eigenvalues of two matrix ensembles at a fixed energy are identical up to the scale $1/N$ provided that the first four moments of the matrix elements of these two ensembles are almost identical. Prior to this theorem, it was proved that the joint distribution of individual eigenvalues for Wigner ensembles is the same under the four moment assumption. Tao-Vu’s theorem addresses the distribution of individual eigenvaluesIn a recent preprint (appeared after the current preprint was first posted), it was pointed out that if the four moment condition is violated, then the differences between individual eigenvalues of the two ensembles are bigger than the eigenvalue spacing. Thus the four moment condition is also necessary for locating the individual eigenvalues. This is in contrast with the main theme of this paper that gap distribution and correlation functions are even independent of the second moments as long as they are nonzero. while Theorem 3.3 compares Green functions (and thus eigenvalues) at a fixed energy.

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 (2.11). 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 $z^{m}_{j}=E^{m}_{j}\pm i\eta$, $j=1,\ldots k_{i}$, $m=1,\ldots,n$ with $|E^{m}_{j}|\leq 2-2\kappa$ and with an arbitrary choice of the $\pm$ signs. Let $G^{(v)}(z)=(H^{(v)}-z)^{-1}$ be 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 $\alpha,\beta$, $\sum_{i}k_{i}$ 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 $z^{m}_{j}$

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

Given this theorem, for any matrix ensemble $H$ whose matrix element at $(i,j)$ are distributed according to $\sigma_{ij}\zeta^{ij}$, we need to find $\xi_{0}^{ij}$ such that the first four moments of $\zeta^{ij}$ and $\xi^{ij}_{t}$ are almost the same and $\xi_{0}^{ij}$ has a subexponential decay. Since the real and imaginary parts are i.i.d., it is sufficient to match them individually. This is the content of the following lemma which is stated for real random variables normalized to variance one. With this lemma, we have proved Theorem 2.2. This lemma is essentially the same as Lemma 28 in .

Let $m_{3}$ and $m_{4}$ be two real numbers such that

for some positive constant $C_{2}$. Let $\xi^{G}$ be a real Gaussian random variable with mean and variance $1$. Then for any sufficient small $\gamma>0$ (depending on $C_{2}$), there exists a real random variable $\xi_{\gamma}$ with subexponential decay and independent of $\xi^{G}$, such that the first four moments of

are $m_{1}(\xi^{\prime})=0$, $m_{2}(\xi^{\prime})=1$, $m_{3}(\xi^{\prime})=m_{3}$ and $m_{4}(\xi^{\prime})$, and

for some positive constant $C$ depending on $C_{2}$.

Proof. It is easy to see by an explicit construction that the following holds:

For any real random variable $\zeta$, independent of $\xi^{G}$, and with the first 4 moments being , $1$, $m_{3}(\zeta)$ and $m_{4}(\zeta)<\infty$, the first 4 moments of

Using (3), we obtain that for any $\gamma>0$ there exists a real random variable $\xi_{\gamma}$ such that the first four moments are , $1$,

With $m_{4}\leq C_{2}$, we have $m_{3}^{2}\leq C_{2}^{3/2}$, thus

for some positive constant $C$ depending on $C_{2}$. Hence with (3.10) and (3.11), we obtain that $\xi^{\prime}=(1-\gamma)^{1/2}\xi_{\gamma}+\gamma^{1/2}\xi^{G}$ satisfies $m_{3}(\xi^{\prime})=m_{3}$ and (3.8). This completes the proof of Lemma 3.4.

Large Deviation of Local Semicircle Law

We first reprove the large deviation of local semicircle law given in . The result of this section is relevant only for $\eta\geq M^{-1}$.

Let $\kappa\equiv||E|-2|$. Then for all $z=E+i\eta$ with

for sufficiently large N with positive some constants $c$ and $C>0$ that depend only $\alpha$ and $\beta$ in (2.11) and $\delta_{-}$ in (2.4) and (2.5).

The theorem will be proved at the end of the section after collecting several lemmas. The first lemma describes the behavior of $m_{sc}$ in the various regimes, its proof is elementary calculus. We use the notation $f\sim g$ for two positive functions in some domain $D$ if there is a positive universal constant $C$ such that $C^{-1}\leq f(z)/g(z)\leq C$ holds for all $z\in D$.

We have for all $z$ with ${\mathfrak{Im}\,}z>0$ that

From now on, let $z=E+i\eta$ with $|E|\leq 5$ and $\eta>0$. If $\eta\geq 10$, then we have

For the behavior of $|1-{\mathfrak{Re}\,}m_{sc}^{2}(z)|$ and ${\mathfrak{Im}\,}m_{sc}(z)$ we distinguish two cases.

Thus the control function $\theta(z)$ has the following behavior

Note that the precise formula (4.1) for $\theta(z)$ is not important, only its asymptotic behavior for small $\kappa$, $\eta$ and $\delta_{+}$ is relevant. The theorem remains valid if $\theta(z)$ is replaced by $\widetilde{\theta}(z)$ with $\widetilde{\theta}(z)\leq C\theta(z)$. In particular, $\theta(z)$ can be chosen to be order one when $E$ is not near the edges of the spectrum. If we are only concerned with the generalized Wigner ensemble $\eqref{VV}$, then by (2.7) we can choose $\theta(z)=(\kappa+\eta)^{-1/2}$ for any $z=E+i\eta$ ($\eta>0$). For universal Wigner matrices we have $\theta(z)\leq C(\kappa+\eta)^{-1}$ for $|z|\leq 10$, i.e., using the parameter $A$ introduced in Theorem 2.1, we have

Based upon these formulas, we also have, for any $z=E+i\eta$ with $\eta>0$,

First, we introduce some notations. Recall that $G_{ij}=G_{ij}(z)$ denotes the matrix element

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

The following two results were proved in our previous work (Lemma 4.2 and Corollary B.3 of ) and they will be our key inputs. We start with the self-consistent perturbation formulas.

for some constants $C$ depending on $\alpha$ and $\beta$ in (2.11).

We start with determining a system of self-consistent equations for the diagonal matrix elements of the resolvent. We can write $G_{ii}$ as follows,

We will estimate the following key quantities

Both quantities $\Lambda_{d}$ and $\Lambda_{o}$ will be typically small, eventually we will prove that their size is less than $(M\eta)^{-1/2}$, modulo logarithmic corrections and a factor involving the distance to the edge. We thus define the exceptional event

We will always work in $\Omega_{\Lambda}^{c}$, and, in particular, we will have

since $1/\theta(z)\leq C$ by (4.12). Define the set

for any $z\in S$ with some universal constant $c>0$. Here we estimated \big{|}|G_{ii}|-|m_{sc}|\big{|}\leq\Lambda_{d}, and we used from (4.6)–(4.7) that $m_{sc}(z)$ satisfies $|m_{sc}(z)|\sim 1$ for $z\in S$.

Thus, a special case of (4.16) or (4.15),

together with (4.25) implies that for any $i$ and with a sufficiently large constant $C$

with $c$ being the constant in (4.25) and we also used that $\sum_{j}\sigma_{ij}^{2}=1$. Similarly, with one more expansion step, we get

Using these estimates, the following lemma shows that $Z_{i}$ and $Z_{ij}^{(ij)}$ are small assuming $\Lambda_{d}+\Lambda_{o}$ is small and the $h_{ij}$’s are not too large. The control parameter for the $Z$’s is $\Phi=\Phi(z)$, defined below (4.32). These bounds hold uniformly in $S$.

to be the set of all exceptional events. Then we have

Proof: Under the assumption of (2.11), we have

therefore we can work on the complement set $\Omega_{1}^{c}$. Define the event

Notice that the estimates (4.26)–(4.31) also hold on $\widetilde{\Omega}^{c}_{\Lambda}$, maybe with different constants $C$. We now prove that for any fixed $z\in S$, we have

To see (4.36), we apply the estimate (4.18) from the large deviation Lemma 4.4, and we obtain that

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

Denote by $u^{({i})}_{\alpha}$ and $\lambda_{\alpha}^{({i})}$ ($\alpha=1,2,\ldots,N-1$) the eigenvectors and eigenvalues of $H^{({i})}$. Let $u^{({i})}_{\alpha}(l)$ denote the $l$-th coordinate of $u^{({i})}_{\alpha}$. Then, using $\sigma_{il}^{2}\leq 1/M$ and (4.28), we have

Here we defined $|A|^{2}:=A^{*}A$ for any matrix $A$ and we used (4.12) to estimate ${\mathfrak{Im}\,}m_{sc}(z)$. Together with (4.38) we have proved (4.36) for a fixed $z$.

For the offdiagonal estimate (4.37), for $i\neq j$, we have from (4.19) that

holds with a probability larger than $1-CN^{-c(\log\log N)}$ for sufficiently large $N$. Similarly to (4), by using (4.31), we get

Now we start proving (4.34). First we choose an $N^{-10}$-net $\mathcal{N}$ in the set $S$, i.e., a collection of points, $\{z_{n}\}_{n\in I}\subset S$, such that for any $z\in S$ there is $\widetilde{z}\in\mathcal{N}$ such that $|z-\widetilde{z}|\leq N^{-10}$. The net can be chosen such that $|I|\leq CN^{20}$. Then (4.36) and (4.37) imply that

Now let $z\in S$ be arbitrary and choose $\widetilde{z}\in\mathcal{N}$ such that $|z-\widetilde{z}|\leq N^{-10}$. For any fixed $i\neq j$, we have

By $\partial Z_{ij}^{(ij)}/\partial z=-\sum_{s,k,l\notin{(ij)}}\overline{{\bf{a}}^{\,i}_{k}}G^{(ij)}_{ks}G^{({ij})}_{sl}{\bf{a}}^{j}_{l\,}$ and $\max_{ab}|G^{(ij)}_{ab}|\leq\eta^{-1}$, we have

In the last inequality, we used the assumption $\eta\geq N^{-1}$. Thus

Since $\Phi\geq M^{-1/2}\eta^{-1/4}\geq cN^{-1}$ for $z\in S$, we obtain

Moreover, by estimating $|\partial_{z}G|\leq N^{2}$ in $S$, we see that $\Lambda_{d}(z)$, $\Lambda_{o}(z)$, and $\Phi(z)$ are Lipschitz continuous functions in $S$ with a Lipschitz constant bounded by $CN^{3}$. Therefore $\Phi(\widetilde{z})$ can be replaced with $\Phi(z)$ in the lower bound on $|Z_{ij}^{(ij)}(\widetilde{z})|$ and $|Z_{i}(\widetilde{z})|$ obtained from (4.41), and, furthermore, $\Omega_{\Lambda}^{c}(z)\subset\widetilde{\Omega}_{\Lambda}^{c}(\widetilde{z})$ using a trivial upper bound $\theta(z)\leq N$. Thus we get

Combining this with (4.35), we obtain (4.34) and thus Lemma 4.5.

Our goal is to show that $\Lambda_{o}(z)+\Lambda_{d}(z)$ is smaller than $(M\eta)^{-1/2}$ (modulo edge and logarithmic corrections) for any $z\in S$ in the event $\Omega^{c}(z)$. We will use a continuity argument. In Lemma 4.6 we show for any $z\in S$ that if $\Lambda_{o}(z)+\Lambda_{d}(z)$ is smaller than $(\log N)^{-3/2}$, then it is actually also smaller than $(M\eta)^{-1/2}$. In Lemma 4.9 we show that this input condition holds at least for ${\mathfrak{Im}\,}z=\eta=10$. Then reducing $\eta$, we show by a continuity argument that it holds for each $z\in S$.

Let $z=E+i\eta$ and satisfy (4.2), in particular $z\in S$. Recall $\Lambda_{d}$, $\Lambda_{o}$ and $\Omega$ defined in (4.23) and (4.33). Then we have that, in the event $\Omega^{c}$, if

and we also have a stronger bound for the off-diagonal terms:

Proof of Lemma 4.6. First note that condition (4.43) is equivalent assuming the event $\Omega_{\Lambda}^{c}(z)$ and we have

so the event $\Omega_{d}^{c}(z)\cup\Omega_{o}^{c}(z)$ holds. We recall from (4.12) that

With the assumption (4.43) we have (see (4.25), (4.27))

We first estimate the offdiagonal term $G_{ij}$. From (4.14) we have

where we used (4.50) to show that the first term can be absorbed into the second. From the second inequality in (4.50) we also have

This proves the estimate (4.45). Using (4.47), we also see that (4.44) holds for the summand $\Lambda_{o}$.

Now we estimate the diagonal terms. Recalling $\Upsilon_{i}=A_{i}+h_{ii}-Z_{i}$ from (4.21), with (4.29), (4.50), (4.52) we have,

Again, the first term can be absorbed into the second, so we have proved

Using $(m_{sc}+z)=-m_{sc}^{-1}$, and the fact that $|m_{sc}+z|\geq 1$, so with $\Lambda_{d}+\Upsilon\leq\frac{1}{10}|m_{sc}+z|$ (see in (4.49) and (4.54)), we can expand (4.55) as

Summing up this formula for all $i$ and recalling the definition $\bar{v}\equiv\frac{1}{N}\sum_{i}v_{i}=m-m_{sc}$ yield

Introducing the notations $\zeta:=m_{sc}^{2}(z)$, $\overline{\Upsilon}:=\frac{1}{N}\sum_{i}\Upsilon_{i}$ for simplicity, we have (using $\Lambda_{d}\leq 1$)

and the error terms for each $i$ sums up to zero. Therefore, with $\Upsilon\leq 1$, we have

To estimate the norm of the resolvent, we recall the following elementary lemma (Lemma 5.3 in ).

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

Suppose that $\Sigma$ satisfies (2.4), i.e., $\mbox{Spec}(Q\Sigma)\subset[-1+\delta_{-},1-\delta_{+}]$. Then we have

with some constant $C(\delta_{-})$ depending on $\delta_{-}$ and with $q$ defined in (4.61)

with $\tau$ given in (4.60). By (4.60), we have

Choosing $n_{0}=C\log N/q(z)$ with a large $C$, we have proved the Lemma.

We now return to the proof of Lemma 4.6, recall that we are in the set $\Omega^{c}\cap\Omega^{c}_{\Lambda}(z)$. First, inserting (4.5) and (4.62) into (4.59), and using $1/q\leq\theta$, we obtain

By the assumption (4.43), we have $C\theta(z)\Lambda_{d}\log N\leq 1/2$, for large enough $N$, therefore we get

Using the bound on $\Upsilon$ in (4.54) and (4.50), we obtain

which, together with (4.52), completes the proof of (4.44).

recall the definitions of $\Omega_{1}$, $\Omega_{d}$ and $\Omega_{o}$ from (4.33) and define

Furthermore, in the set $\widehat{\Omega}^{c}$ we have

Denote by $u_{\alpha}$ and $\lambda_{\alpha}$ the eigenvectors and eigenvalues of $H$. On the set $\Omega_{H}^{c}$ all eigenvalues are bounded, $|\lambda_{\alpha}|\leq 3$. In this set we have, with $|E|\leq 5$,

with some positive constant $c>0$. We also have the upper bound $|G_{kk}|\leq\eta^{-1}$ and $\Lambda_{o}+\Lambda_{d}\leq C/\eta$. In particular, for $\eta=10$, we have

Similarly, the argument (4.51)–(4.52) shows that in the set $\widehat{\Omega}^{c}$, we have

and the argument (4.53)–(4.54) guarantees that

in $\widehat{\Omega}^{c}$. Finally, to control $\Lambda_{d}$, we use that from the self consistent equation (4.55) and the definition of $m_{sc}$, we have

For $\eta=10$, with (2.9), we have $|z+m_{sc}(z)|>2$. Using $|G_{ii}|\leq\eta^{-1}=\frac{1}{10}$ and $|m_{sc}|\leq\eta^{-1}=\frac{1}{10}$, we obtain

Using (4.69), together with $|z+m_{sc}(z)|>2$ and (4.71), we obtain that the absolute value of the r.h.s of (4.70) is less than

Taking the absolute value of (4.70) and maximizing over $n$, we have

Since the denominator satisfies $|z+m_{sc}(z)|-\sup_{i}|v_{i}|\geq 2-1/5$,

follows from the last equation. Combining it with (4.68) and (4.69), we obtain (4.65), and this completes the proof of Lemma 4.9.

Proof of Theorem 4.1. Lemma 4.6 states that, in the event $\Omega^{c}$, if $\Lambda_{d}(z)+\Lambda_{o}(z)\leq R(z)$ then $\Lambda_{d}(z)+\Lambda_{o}(z)\leq S(z)$ with

By assumption (4.2) of Theorem 4.1, we have $S(z)<R(z)$ for any $z\in S$ and these functions are continuous. Lemma 4.9 states that in the set $\widehat{\Omega}^{c}$ the bound $\Lambda_{d}(z)+\Lambda_{o}(z)\leq R(z)$ holds for $\eta=10$.

Thus by a continuity argument, $\Lambda_{d}(z)+\Lambda_{o}(z)\leq S(z)$ in the set $\Omega^{c}\cap\widehat{\Omega}^{c}$ as long as the condition (4.2) is satisfied. Finally, once $\Lambda_{o}(z)\leq S(z)$ is proven, we can use $S(z)\leq R(z)$ (in the domain $D$) and Lemma 4.6 once more to conclude the stronger bound on $\Lambda_{o}(z)$. This proves Theorem 4.1.

We record that combining the bound on $\Lambda_{d},\Lambda_{o}$ with (4.54), we also proved that under the assumption (4.2) we have

Local semicircle law

In this section we strengthen the estimate of Theorem 4.1 for the Stieltjes transform $m(z)=\frac{1}{N}\sum_{i}G_{ii}$. The key improvement is that $|m-m_{sc}|$ will be estimated with a precision $(M\eta)^{-1}$ while the $|G_{ii}-m_{sc}|$ was controlled by a precision $(M\eta)^{-1/2}$ only (modulo logarithmic terms and terms expressing the deterioration of the estimate near the edge).

Assume the conditions of Theorem 4.1 and recall the notations \kappa=\kappa_{E}:=\big{|}|E|-2\big{|} and $\theta(z)$ from (4.1). Define the domain

Then for any $\varepsilon>0$ and $K>0$ there exists a constant $C=C(\varepsilon,K)$ such that

Proof of Theorem 5.1. We will work in the set $\Omega^{c}\cap\widehat{\Omega}^{c}$, which has almost full probability by (4.34) and (4.64). Note that the set $D^{*}$ is included in the domain defined by (4.2), therefore we can use the estimates from Section 4.

As in (4.57), where $\bar{v}=m(z)-m_{sc}(z)$, we have that

holds with a very high probability. Recall that $\zeta=m_{sc}^{2}(z)$ and we mostly omit the argument $z$ from the notations. The quantities $\Lambda_{d}$, $\Upsilon_{i}$ and $\Upsilon$ were defined in (4.23), (4.21) and (4.53). Then with (4.75) we have

holds with a very high probability for any small $\varepsilon>0$. Recall that $\Upsilon_{i}=A_{i}+h_{ii}-Z_{i}$. We have, from (4.20), (4.25) and $\sigma_{ij}^{2}\leq M^{-1}$,

where we used (4.75) to bound $\Lambda_{o}$ and (4.47) to control the $C/M$ term.

holds with a very high probability. Since $h_{ii}$’s are independent, applying the first estimate in the large deviation Lemma 4.4, we have

On the complement event, the estimate $(\log N)^{3/2+\alpha}(MN)^{-1/2}$ can be included in the last error term in (5.3). It only remains to bound

whose moment is bounded in the next lemma which will be proved in Sections 8 and 9.

For fixed $z$ in domain $D^{*}$ (5.1) and any even number $p$, we have

Using Lemma 5.2, we have that for any $\varepsilon>0$ and $K>0$,

for sufficiently large $N$. Combining this with (5.4) and (5.3) and noting that $|1-\zeta|\sim\sqrt{\kappa+\eta}$, see (4.7), we obtain (5.2) and complete the proof of Theorem 5.1.

Empirical counting function

In this section we translate the information on the Stieltjes transform obtained in Theorem 5.1 to an asymptotic on the empirical counting function. The main ingredient for the first step is the following lemma based upon the Helffer-Sjöstrand formula. We will formulate this lemma for general signed measures, but we will apply it to the Stieltjes transform $m^{\Delta}=m-m_{sc}$ of the difference between the empirical density and the semicircle law. A similar statement was already proven in Lemma B.1 in and Lemma 7.7 in .

and in case of $A>0$ we additionally assume $\eta\leq\frac{1}{2}\min\{\kappa_{E_{1}},\kappa_{E_{2}}\}$. Then

with some constant $C$ depending on $K$ and $A$.

where $\chi(y)$ is a smooth cutoff function with support in $$, with$\chi(y)=1$for$|y|\leq 1/2$ and with bounded derivatives. The first term is estimated by, with (6.1),

For the second term in r.h.s of (6) we use that from (6.1) it follows for any $1\geq y>0$ that

With $|f^{\prime\prime}|\leq C\eta^{-2}$ and

As in (B.17) and (B.19) in , we integrate the third term in (6) by parts first in $x$, then in $y$. Then we bound it with an absolute value by

The second term is bounded in (6.4). By using (6.1) and (6.6) in the first term and (6.1) in the third, we have

Let $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{N}$ be the ordered eigenvalues of a universal Wigner matrix. We define the normalized empirical counting function by

be the distribution function of the semicircle law which is very close to the counting function of $\gamma$’s, $n^{\gamma}(E):=\frac{1}{N}\#[{\gamma}_{j}\leq E]$.

We will need some control on the spectral edge, we recall the Lemma 7.2 from .

(1) Let the universal Wigner matrix $H$ satisfy (2.1), (2.2) and (2.11) with $M\geq(\log N)^{9}$. Then we have

(2) Let $H$ be a generalized Wigner matrix with subexponential decay, i.e., (2.1), (2.2), (2.6) and (2.11) hold. Then

for any small $\varepsilon>0$ with an $\varepsilon^{\prime}>0$ depending on $\varepsilon$. Furthermore, for $K\geq 3$,

With these preliminary lemmas, we have the following theorem that we state for universal Wigner matrices and for their subclass, the generalized Wigner matrices in parallel.

Let $A=2$ for universal Wigner matrices and $A=1$ for generalized Wigner matrices. Suppose that the universal Wigner matrix ensemble satisfies (2.1), (2.2) and (2.11) with $M\geq(\log N)^{24+6\alpha}$ and the generalized Wigner matrix ensemble satisfies (2.1), (2.2), (2.6) and (2.11). We recall $M=N$ in the latter case. Then for any $\varepsilon>0$ and $K\geq 1$ there exists a constant $C(\varepsilon,K)$ such that

where the ${\mathfrak{n}}(E)$ and $n_{sc}(E)$ were defined in (6.8) and (6.10) and \kappa_{E}=\big{|}|E|-2\big{|}.

Proof. For definiteness, we will consider the case of generalized Wigner matrices, i.e., $A=1$. In this case $M=N$, $\delta_{+}\geq C_{inf}>0$ (see (2.7)) and thus $\theta(z)\leq C(\kappa+\eta)^{-1/2}$ for $|z|\leq 10$, see (4.10). For simplicity of the presentation, we assume that $\theta(z)=(\kappa+\eta)^{-1/2}$ as overall constant factors do not matter (see the remark after (4.10)). We set $\eta=1/N$, $U=N^{\varepsilon-1}$ and apply Lemma 6.1 to the difference $m^{\Delta}=m-m_{sc}$. Let $\varrho^{\Delta}=\varrho-\varrho_{sc}$, where $\varrho(x)=\frac{1}{N}\sum_{j}\delta(x-\lambda_{j})$ is the normalized empirical counting measure of eigenvalues. First we check the conditions of Lemma 6.1. To check that (6.1) holds, set $L=(\log N)^{24+6\alpha}$ and for a fixed $x$, let $y_{x}$ satisfy $Ny_{x}(\kappa_{x}+y_{x})^{3/2}=L$, so that $x+iy_{x}\in D^{*}$. Clearly (6.1) holds for any $y\geq y_{x}$ with a very high probability by (5.2). In particular, we know that

Consider $y<y_{x}$, set $z=x+iy$, $z_{x}=x+iy_{x}$ and estimate

Now we use the fact that the functions $y\to y{\mathfrak{Im}\,}m(x+iy)$ and $y\to y{\mathfrak{Im}\,}m_{sc}(x+iy)$ are monotone increasing for any $y>0$ since both are Stieltjes transforms of a positive measure. Therefore the integral in (6.15) can be bounded by

By the choice of $y_{x}$ and using that ${\mathfrak{Im}\,}\,m_{sc}(z_{x})\leq C\sqrt{\kappa_{x}+y_{x}}$, we have

with a possible larger $C$ in the r.h.s. Thus (6.1) holds for the difference $m^{\Delta}=m-m_{sc}$.

The application of Lemma 6.1 shows that for $\eta=1/N$

Recall that $f_{E_{1},E_{2},\eta}$ the characteristic function of the interval $[E_{1},E_{2}]$, smoothed on scale $\eta$ at the edges. The additional $1$ in the denominator in the r.h.s. of (6.20) comes from the case when $\kappa_{E_{1}}$, $\kappa_{E_{2}}$ are very small and the trivial estimate $f\leq 1$ with $\int\varrho=\int\varrho_{sc}=1$ gives a better bound than Lemma 6.1.

With the fact $y\to y{\mathfrak{Im}\,}m(x+iy)$ is monotone increasing for any $y>0$, (6.19) implies a crude upper bound on the empirical density. Indeed, for any interval $I:=[x-\eta,x+\eta]$, with $\eta=1/N$, we have

Choose arbitrary $E_{1},E_{2}\in$, then we have

from (6.21). Since $\varrho_{sc}$ is bounded, we also have

Subtracting (6.22) and (6.23) and using (6.20), we obtain that for any $E_{1},E_{2}\in$

with a very high probability, i.e., apart from a set of probability smaller than $C(\varepsilon,K)N^{-K}$ for any $K$. The estimate (6.13) from Lemma 6.2 on the extreme eigenvalues shows that $\varrho$ is supported in $$with very high probability, i.e.,${\mathfrak{n}}(-3)=n_{sc}(-3)=0$,${\mathfrak{n}}(3)=n_{sc}(3)=1$. Thus we obtain that

holds for any fixed $E\in$ with an overwhelming probability.

We now choose a fine grid of equidistant points $E_{j}\in$ with $|E_{j}-E_{j+1}|\leq N^{-1}$, then (6.24) holds simultaneously for every $E=E_{j}$ with an overwhelming probability. For any $E\in$ we can find an $E_{j}$ with $|E-E_{j}|\leq N^{-1}$ and by (6.21) we obtain

This guarantees that (6.24) holds simultaneously for all $E$. Since $\varepsilon>0$ was arbitrary, this proves Theorem 6.3 for generalized Wigner matrices.

The proof for universal Wigner matrices is very similar, just $M$ replaces $N$ in the estimates, $U=N^{\varepsilon}M^{-1}$ and instead of $\theta(z)\leq C(\kappa+\eta)^{-1/2}$ one uses $\theta(z)\leq C(\kappa+\eta)^{-1}$ which follows from (4.10). The main technical estimate (6.20) is modified to

Location of eigenvalues

In this section we estimate the mean square deviation of the eigenvalues from their classical location. The main input is Theorem 6.3, the estimate on the counting function. For simplicity, we consider only the case of generalized Wigner matrices. Similar, but weaker results can be obtained along the same lines for universal Wigner matrices.

Let $H$ be a generalized Wigner matrix with subexponential decay, i.e., assume that (2.1), (2.2), (2.6) and (2.11) hold. Let $\lambda_{j}$ denote the eigenvalues of $H$ and $\gamma_{j}$ be their classical location, defined by (2.23). Then for any $\varepsilon_{0}<1/7$ and for any $K>1$ there exists a constant $C$, depending on $K$ and $\varepsilon_{0}$, such that

Proof. The proof of (7.2) directly follows from (7.1) by using the estimates on the extreme eigenvalue (6.13) from Lemma 6.2. For the proof of (7.1), we can assume that $\max_{j}|\lambda_{j}|\leq 2+N^{-1/7}$ since the complement event has a negligible probability by (6.12) and (6.13) of Lemma 6.2. From Theorem 6.3 we can also assume that

From the definition of $\gamma_{j}$ it follows that for $j\leq N/2$, i.e., $\gamma_{j}\leq 0$,

with some positive constants $C_{1},C_{2}$.

Choose $\beta=\frac{2}{5}-\varepsilon$. Consider first those $j$-indices for which $C_{0}N^{1-3\beta/2}\leq j\leq N-C_{0}N^{1-3\beta/2}$ with a sufficiently large constant. We choose $C_{0}$ so that (7.4) would imply $-2+2N^{-\beta}\leq\gamma_{j}\leq 2-2N^{-\beta}$. We then claim that

We will show that $\lambda_{j}\geq-2+N^{-\beta}$, the upper bound is analogous. Suppose that $\lambda_{j}$ were smaller than $-2+N^{-\beta}$, then ${\mathfrak{n}}(-2+N^{-\beta})\geq j$. On the other hand, $n_{sc}(-2+2N^{-\beta})\leq j$ and thus

with some positive constant $c$. Therefore

where the second inequality follows from (7.3), but this contradicts to the choice $\beta=\frac{2}{5}-\varepsilon$.

Let $j$ satisfy $C_{0}N^{1-3\beta/2}\leq j\leq N/2$; the indices $N/2\leq j\leq N-C_{0}N^{1-3\beta/2}$ can be treated analogously. Note that $\lambda_{N/2}\leq CN^{-1+\varepsilon}$ by (7.3). Define $c(j)$ to be index of the $\gamma$-point right below $\lambda_{j}$, i.e.,

By (7.5) we see that $-2+\frac{1}{2}N^{-\beta}\leq\gamma_{c(j)}\leq CN^{-1+\varepsilon}$ and from (7.3) and (7.4) it follows that

By the choice of $\beta$ we have $\varepsilon+\beta<1-\frac{3}{2}\beta$, i.e., (7.6) implies $|c(j)-j|\ll j$. Using now (7.4), we have

using $|c(j)-j|\ll j$ and hence $(2+\gamma_{j})$ and $(2+\gamma_{c(j)})$ are comparable. In the last step we also used (7.4). Combining this with (7.7), we have

and the same estimate holds for $|\gamma_{c(j)+1}-\gamma_{j}|$ and thus

by the choice of $\beta$ and similar estimate holds for the sum over the indices $N/2\leq j\leq N-C_{0}N^{1-3\beta/2}$ as well.

Now we consider the indices $j\leq C_{0}N^{1-3\beta/2}$ and $\lambda_{j}\geq-2-N^{-\beta}$. By a similar argument that proved (7.5), we can see that there is a constant $C_{3}$ such that $\lambda_{j}\leq-2+C_{3}N^{-\beta}$, otherwise ${\mathfrak{n}}(-2+C_{3}N^{-\beta})\leq j$, but $n_{sc}(-2+C_{3}N^{-\beta})\geq j+cN^{-3\beta/2}$, which would contradict (7.3). It is easy to see that $\gamma_{j}\leq-2+CN^{-\beta}$ for all $j\leq C_{0}N^{1-3\beta/2}$, therefore in this regime we estimate $|\lambda_{j}-\gamma_{j}|\leq CN^{-\beta}$ and thus

The indices $j\geq N-C_{0}N^{1-3\beta/2}$ and $\lambda_{j}\leq 2+N^{-\beta}$ can be treated similarly.

Finally we deal with the extreme eigenvalues $\lambda_{j}\leq-2-N^{-\beta}$ with index $j\leq C_{0}N^{1-3\beta/2}$ and we can assume that $\lambda_{j}\geq-2-N^{-1/7}$. For these indices $-2\leq\gamma_{j}\leq-2+CN^{-\beta}$ and we can estimate

For any $a$ with $N^{-\beta}\leq a\leq N^{-1/7}$, we have $n_{sc}(-2-a)=0$, thus we obtain from (7.3) that

The other extreme eigenvalues, $\lambda_{j}\geq 2+N^{-\beta}$, are treated analogously.

Combining (7.8), (7.9) and (7) and choosing $\varepsilon$ sufficiently small in the definition of $\beta$, we proved (7.1) with any $\varepsilon_{0}<1/7$.

Moment Estimates of Error Terms

In this section we prove the second and fourth moment estimates of Lemma 5.2; the general cases will be proved in Section 9.

Recall the definition of $Z_{i}$, which we rewrite as

We first prove a bound on the Green function $G_{k\,l}^{(i)}$.

and for some constant $c$, $C$ independent of $t$,