The Local Semicircle Law for a General Class of Random Matrices

Introduction

Since the pioneering work W of Wigner in the fifties, random matrices have played a fundamental role in modelling complex systems. The basic example is the Wigner matrix ensemble, consisting of $N\times N$ symmetric or Hermitian matrices $H=(h_{ij})$ whose matrix entries are identically distributed random variables that are independent up to the symmetry constraint $H=H^{*}$. From a physical point of view, these matrices represent Hamilton operators of disordered mean-field quantum systems, where the quantum transition rate from state $i$ to state $j$ is given by the entry $h_{ij}$.

A central problem in the theory or random matrices is to establish the local universality of the spectrum. Wigner observed that the distribution of the distances between consecutive eigenvalues (the gap distribution) in complex physical systems follows a universal pattern. The Wigner-Dyson-Gaudin-Mehta conjecture, formalized in M , states that this gap distribution is universal in the sense that it depends only on the symmetry class of the matrix, but is otherwise independent of the details of the distribution of the matrix entries. This conjecture has recently been established for all symmetry classes in a series of works ESY4 ; EYYrigi ; EKYY2 ; an alternative approach was given in TV for the special Wigner Hermitian case. The general approach of ESY4 ; EYYrigi ; EKYY2 to prove universality consists of three steps: (i) establish a local semicircle law for the density of eigenvalues; (ii) prove universality of Wigner matrices with a small Gaussian component by analysing the convergence of Dyson Brownian motion to local equilibrium; (iii) remove the small Gaussian component by comparing Green functions of Wigner ensembles with a few matching moments. For an overview of recent results and this three-step strategy, see EYBull .

For definiteness, let us focus on the case of a one-dimensional band matrix $H$. A fundamental conjecture, supported by nonrigorous supersymmetric arguments as well as numerics Fy , is that the local spectral statistics of $H$ are governed by random matrix statistics for large $W$ and by Poisson statistics for small $W$. This transition is in the spirit of the Anderson metal-insulator transition Fy ; Spe , and is conjectured to be sharp around the critical value $W=\sqrt{N}$. In other words, if $W\gg\sqrt{N}$, we expect the universality results of EYY ; EYY2 ; EYYrigi to hold. In addition to a transition in the local spectral statistics, an accompanying transition is conjectured to occur in the behaviour localization length of the eigenvectors of $H$, whereby in the large-$W$ regime they are expected to be completely delocalized and in the small-$W$ regime exponentially localized. The localization length for band matrices was recently investigated in great detail in EKYY3 .

Although the Wigner-Dyson-Gaudin-Mehta conjecture was originally stated for Wigner matrices, the methods of ESY4 ; EYYrigi ; EKYY2 also apply to certain ensembles with independent but not identically distributed entries, which however retain the mean-field character of Wigner matrices. More precisely, they yield universality provided the variances

of the matrix entries are only required to be of comparable size (but not necessarily equal):

for some positive constants $c$ and $C$. (Such matrices were called generalized Wigner matrices in EYYrigi .) This condition admits a departure from spatial homogeneity, but still imposes a mean-field behaviour and hence excludes genuinely inhomogeneous models such as band matrices.

In the three-step approach to universality outlined above, the first step is to establish the semicircle law on very short scales. In the scaling of $H$ where its spectrum is asymptotically given by the interval $$, the typical distance between neighbouring eigenvalues is of order$1/N$. The number of eigenvalues in an interval of length$\eta$is typically of order$N\eta$. Thus, the smallest possible scale on which the empirical density may be close to a deterministic density (in our case the semicircle law) is$\eta\gg 1/N$. If we characterize the empirical spectral density around an energy$E$on scale$\eta$by its Stieltjes transform,$m_{N}(z)=N^{-1}\operatorname{Tr}(H-z)^{-1}$for$z=E+i\eta$, then the local semicircle law around the energy$E$and in a spectral window of size$\eta$ is essentially equivalent to

on the variances (instead of (1.1)). Here $M$ is a new parameter that typically satisfies $M\ll N$. (From now on, the relation $A\ll B$ for two $N$-dependent quantities $A$ and $B$ means that $A\leqslant N^{-\varepsilon}B$ for some positive $\varepsilon>0$.) The question of the validity of the local semicircle law under the assumption (1.3) was initiated in EYY , where (1.2) was proved with an error term of order $(M\eta)^{-1/2}$ away from the spectral edges.

The purpose of this paper is twofold. First, we prove a local semicircle law (1.2), under the variance condition (1.3), with a stronger error bound of order $(M\eta)^{-1}$, including energies $E$ near the spectral edge. Away from the spectral edge (and from the origin $E=0$ if the matrix does not have a band structure), the result holds for any $\eta\gg 1/M$. Near the edge there is a restriction on how small $\eta$ can be. This restriction depends explicitly on a norm of the resolvent of the matrix of variances, $S=(s_{ij})$; we give explicit bounds on this norm for various special cases of interest.

As a corollary, we derive bounds on the eigenvalue counting function and rigidity estimates on the locations of the eigenvalues for a general class of matrices. Combined with an analysis of Dyson Brownian motion and the Green function comparison method, this yields bulk universality of the local eigenvalue statistics in a certain range of parameters, which depends on the matrix $S$. In particular, we extend bulk universality, proved for generalized Wigner matrices in EYY , to a large class of matrix ensembles where the upper and lower bounds on the variances (1.1) are relaxed.

The main motivation for the generalizations in this paper is the Anderson transition for band matrices outlined above. While not optimal, our results nevertheless imply that band matrices with a sufficiently broad band plus a negligible mean-field component exhibit bulk universality: their local spectral statistics are governed by random matrix statistics. For example, the local two-point correlation functions coincide if $W\gg N^{33/34}$. Although eigenvector delocalization and random matrix statistics are conjectured to occur in tandem, delocalization was actually proved in EKYY3 under more general conditions than those under which we establish random matrix statistics. In fact, the delocalization results of EKYY3 hold for a mean-field component as small as $(N/W^{2})^{2/3}$, and, provided that $W\gg N^{4/5}$, the mean-field component may even vanish (resulting in a genuine band matrix).

The second purpose of this paper is to provide a coherent, pedagogical, and self-contained proof of the local semicircle law. In recent years, a series of papers ESY1 ; ESY2 ; EYY ; EYY2 ; EYYrigi ; EKYY1 with gradually weaker assumptions, was published on this topic. These papers often cited and relied on the previous ones. This made it difficult for the interested reader to follow all the details of the argument. The basic strategy of our proof (that is, using resolvents and large deviation bounds) was already used in ESY1 ; ESY2 ; EYY ; EYY2 ; EYYrigi ; EKYY1 . In this paper we not only streamline the argument for generalized Wigner matrices (satisfying (1.1)), but we also obtain sharper bounds for random matrices satisfying the much weaker condition (1.3). This allows us to establish universality results for a class of ensembles beyond generalized Wigner matrices.

Our proof is self-contained and simpler than those of EYY ; EYY2 ; EYYrigi ; EKYY1 . In particular, we give a proof of the Fluctuation Averaging Theorem, Theorems 4.6 and 4.7 below, which is considerably simpler than that of its predecessors in EYY2 ; EYYrigi ; EKYY1 . In addition, we consistently use fluctuation averaging at several key steps of the main argument, which allows us to shorten the proof and relax previous assumptions on the variances $s_{ij}$. The reader who is mainly interested in the pedagogical presentation should focus on the simplest choice of $S$, $s_{ij}=1/N$, which corresponds to the standard Wigner matrix (for which $M=N$), and focus on Sections 2, 4, 5, and 6, as well as Appendix B.

We conclude this section with an outline of the paper. In Section 2 we define the model, introduce basic definitions, and state the local semicircle law in full generality (Theorem 2.3). Section 3 is devoted to some examples of random matrix models that satisfy our assumptions; for each example we give explicit bounds on the spectral domain on which the local semicircle law holds. Sections 4, 5, and 6 are devoted to the proof of the local semicircle law. Section 4 collects the basic tools that will be used throughout the proof. The purpose of Section 5 is mainly pedagogical; in it, we state and prove a weaker form of the local semicircle law, Theorem 5.1. The error bounds in Theorem 5.1 are identical to those of Theorem 2.3, but the spectral domain on which they hold is smaller. Provided one stays away from the spectral edge, Theorems 5.1 and 2.3 are equivalent; near the edge, Theorem 2.3 is stronger. The proof of Theorem 5.1 is very short and contains several key ideas from the proof of Theorem 2.3. The expert reader may therefore want to skip Section 5, but for the reader looking for a pedagogical presentation we recommend first focusing on Sections 4 and 5 (along with Appendix B). The full proof of our main result, Theorem 2.3, is given in Section 6. In Sections 7 and 8 we draw consequences from Theorem 2.3. In Section 7 we derive estimates on the density of states and the rigidity of the eigenvalue locations. In Section 8 we state and prove the universality of the local spectral statistics in the bulk, and give applications to some concrete matrix models. In Appendix A we derive explicit bounds on relevant norms of the resolvent of $S$ (denoted by the abstract control parameters $\widetilde{\Gamma}$ and $\Gamma$), which are used to define the domains of applicability of Theorems 2.3 and 5.1. Finally, Appendix B is devoted to the proof of the fluctuation averaging estimates, Theorems 4.6 and 4.7.

We use $C$ to denote a generic large positive constant, which may depend on some fixed parameters and whose value may change from one expression to the next. Similarly, we use $c$ to denote a generic small positive constant.

Definitions and the main result

for all $i$ and $j$. We regard $N$ as the fundamental parameter of our model, and $M$ as a function of $N$. We introduce the $N\times N$ symmetric matrix $S\equiv S_{N}=(s_{ij})_{i,j=1}^{N}$. We assume that $S$ is (doubly) stochastic:

for all $i$. For simplicity, we assume that $S$ is irreducible, so that 1 is a simple eigenvalue. (The case of non-irreducible $S$ may be trivially dealt with by considering its irreducible components separately.) We shall always assume the bounds

It is sometimes convenient to use the normalized entries

for all $N$, $i$, and $j$. We make this assumption to streamline notation in the statements of results such as Theorem 2.3 and the proofs. In fact, our results (and our proof) also cover the case where (2.6) holds for some finite large $p$; see Remark 2.4.

without further comment, and always assume that $\eta>0$. Wigner semicircle law $\varrho$ and its Stieltjes transform $m$ are defined by

To avoid confusion, we remark that $m$ was denoted by $m_{sc}$ in the papers ESY1 ; ESY2 ; ESY4 ; ESYY ; EYY ; EYY2 ; EYYrigi ; EKYY1 ; EKYY2 , in which $m$ had a different meaning from (2.7). It is well known that the Stieltjes transform $m$ is the unique solution of

satisfying $\operatorname{Im}m(z)>0$ for $\operatorname{Im}z>0$. Thus we have

Some basic estimates on $m$ are collected in Lemma 4.3 below.

the norm of $(1-m(z)^{2}S)^{-1}$ restricted to the subspace orthogonal to the constants. Clearly, $\widetilde{\Gamma}(z)\leqslant\Gamma(z)$. Basic estimates on $\Gamma$ and $\widetilde{\Gamma}$ are collected in Proposition A.2 below. Many estimates in this paper depend critically on $\Gamma$ and $\widetilde{\Gamma}$. Indeed, these parameters quantify the stability of certain self-consistent equations that underlie our proof. However, $\Gamma$ and $\widetilde{\Gamma}$ remain bounded (up to a factor $\log N$) provided $E=\operatorname{Re}z$ is separated from the set $\{-2,0,2\}$; for band matrices (see Example 3.2) it suffices that $E$ be separated from the spectral edges $\{-2,2\}$; see Appendix A. At a first reading, we recommend that the reader neglect $\Gamma$ and $\widetilde{\Gamma}$ (i.e. replace them with a constant). For band matrices, this amounts to focusing on the local semicircle law in the bulk of the spectrum.

We define the resolvent or Green function of $H$ through

and denote its entries by $G_{ij}(z)$. The Stieltjes transform of the empirical spectral measure of $H$ is

The following definition introduces a notion of a high-probability bound that is suited for our purposes. It was introduced (in a slightly different form) in EKYfluc .

be two families of nonnegative random variables, where $U^{(N)}$ is a possibly $N$-dependent parameter set. We say that $X$ is stochastically dominated by $Y$, uniformly in $u$, if for all (small) $\varepsilon>0$ and (large) $D>0$ we have

for large enough $N\geqslant N_{0}(\varepsilon,D)$. Unless stated otherwise, throughout this paper the stochastic domination will always be uniform in all parameters apart from the parameter $\delta$ in (2.4) and the sequence of constants $\mu_{p}$ in (2.6); thus, $N_{0}(\varepsilon,D)$ also depends on $\delta$ and $\mu_{p}$. If $X$ is stochastically dominated by $Y$, uniformly in $u$, we use the notation $X\prec Y$. Moreover, if for some complex family $X$ we have $\lvert X\rvert\prec Y$ we also write $X=O_{\prec}(Y)$.

For example, using Chebyshev’s inequality and (2.6) one easily finds that

The relation $\prec$ is a partial ordering, i.e. it is transitive and it satisfies the familiar arithmetic rules of order relations. For instance if $X_{1}\prec Y_{1}$ and $X_{2}\prec Y_{2}$ then $X_{1}+X_{2}\prec Y_{1}+Y_{2}$ and $X_{1}X_{2}\prec Y_{1}Y_{2}$. More general statements in this spirit are given in Lemma 4.4 below.

a spectral domain. (Recall that $M\equiv M_{N}$ depends on $N$.)

Note that $\widetilde{\eta}_{E}$ depends on $\gamma$, but we do not explicitly indicate this dependence since we regard $\gamma$ as fixed. At a first reading we advise the reader to think of $\gamma$ as being zero. Note also that the lower bound in (A.3) below implies that $\widetilde{\eta}_{E}\geqslant M^{-1}$. We also define the distance to the spectral edge,

Finally, we introduce the fundamental control parameter

which will be used throughout this paper as a sharp, deterministic upper bound on the entries of $G$. Note that the condition in the definition of $\widetilde{\eta}_{E}$ states that the first term of $\Pi$ is bounded by $M^{-\gamma}\widetilde{\Gamma}^{-2}$ and the second term by $M^{-\gamma}\widetilde{\Gamma}^{-3}$. We may now state our main result.

Fix $\gamma\in(0,1/2)$ and define the spectral domain

We remark that the main estimate for the Stieltjes transform $m_{N}$ is (2.19). The other estimate (2.20) is mainly useful for controlling the norm of $H$, which we do in Section 7. We also recall that uniformity for the spectral parameter $z$ means that the threshold $N_{0}(\varepsilon,D)$ in the definition of $\prec$ is independent of the choice of $z$ within the indicated spectral domain. As stated in Definition 2.1, this uniformity holds for all statements containing $\prec$, and is not explicitly mentioned in the following; all of our arguments are trivially uniform in $z$ and any matrix indices.

Theorem 2.3 has the following variant for matrix entries where the condition (2.6) is only imposed for some large but fixed $p$. More precisely, for any $\varepsilon>0$ and $D>0$ there exists a constant $p(\varepsilon,D)$ such that if (2.6) holds for $p=p(\varepsilon,D)$ then

Most of the previous works ESY1 ; ESY2 ; EYY ; EYY2 ; EYYrigi ; EKYY1 assumed a stronger, subexponential decay condition on $\zeta_{ij}$ instead of (2.6). Under the subexponential decay condition, certain probability estimates in the results were somewhat stronger and precise tolerance thresholds were sharper. Roughly, this corresponds to operating with a modified definition of $\prec$, where the factors $N^{\varepsilon}$ are replaced by high powers of $\log N$ and the polynomial probability bound $N^{-D}$ is replaced with a subexponential one. The proofs of the current paper can be easily adjusted to such a setup, but we shall not pursue this further.

A local semicircle law for Wigner matrices on the optimal scale $\eta\gtrsim 1/N$ was first obtained in ESY2 . The optimal error estimates in the bulk were proved in EYY2 , and extended to the edges in EYYrigi . These estimates underlie the derivation of rigidity estimates for individual eigenvalues, which in turn were used in EYYrigi to prove Dyson’s conjecture on the optimal local relaxation time for the Dyson Brownian motion.

Apart from the somewhat different assumption on the tails of the entries of $H$ (see Remark 2.5), Theorem 2.3, when restricted to generalized Wigner matrices, subsumes all previous local semicircle laws obtained in ESY1 ; ESY2 ; EYY2 ; EYYrigi . For band matrices, a local semicircle law was proved in EYY . (In fact, in EYY the band structure was not required; only the conditions (2.2), (2.3), and the subexponential decay condition for the matrix entries (instead of (2.6)) were used.) Theorem 2.3 improves this result in several ways. First, the error bounds in (2.18) and (2.19) are uniform in $E$, even for $E$ near the spectral edge; the corresponding bounds in Theorem 2.1 of EYY diverged as $\kappa^{-1}$. Second, the bound (2.19) on the Stieltjes transform is better than (2.16) in EYY by a factor $(M\eta)^{-1/2}$. This improvement is due to exploiting the fluctuation averaging mechanism of Theorem 4.6. Third, the domain of $\eta$ for which Theorem 2.3 applies is essentially $\eta\gg\kappa^{-7/2}M^{-1}$, which is somewhat larger than the domain $\eta\gg\kappa^{-4}M^{-1}$ of EYY .

While Theorem 2.3 subsumes several previous local semicircle laws, two previous results are not covered. The local semicircle law for sparse matrices proved in EKYY1 does not follow from Theorem 2.3. However, the argument of this paper may be modified so as to include sparse matrices as well; we do not pursue this issue further. The local semicircle law for one-dimensional band matrices given in Theorem 2.2 of EKYY3 is, however, of a very different nature, and may not be recovered using the methods of the current paper. Under the conditions $W\gg N^{4/5}$ and $\eta\gg N^{2}/W^{3}$, Theorem 2.2 of EKYY3 shows that (focusing for simplicity on the one-dimensional case)

in the bulk spectrum, which is stronger than the bound of order $(W\eta)^{-1/2}$ in (2.18). The proof of (2.21) relies on a very general fluctuation averaging result from EKYfluc , which is considerably stronger than Theorems 4.6 and 4.7; see Remark 4.8 below. The key open problem for band matrices is to establish a local semicircle law on a scale $\eta$ below $W^{-1}$. The estimate (2.21) suggests that the resolvent entries should remain bounded throughout the range $\eta\gtrsim\max\{N^{-1},W^{-2}\}$.

The local semicircle law, Theorem 2.3, has numerous consequences, several of which are formulated in Sections 7 and 8. Here we only sketch them. Theorem 7.5 states that the empirical counting function converges to the counting function of the semicircle law. The precision is of order $M^{-1}$ provided that we have the lower bound $s_{ij}\geqslant c/N$ for some constant $c>0$. As a consequence, Theorem 7.6 states that the bulk eigenvalues are rigid on scales of order $M^{-1}$. Under the same condition, in Theorem 8.2 we prove the universality of the local two-point correlation functions in the bulk provided that $M\gg N^{33/34}$; we obtain similar results for higher order correlation functions, assuming a stronger restriction on $M$. These results generalize the earlier theorems from EYYrigi ; EKYY1 ; EKYY2 , which were valid for generalized Wigner matrices satisfying the condition (1.1), under which $M$ is comparable to $N$. We obtain similar results if the condition $s_{ij}\geqslant c/N$ in (1.1) is relaxed to $s_{ij}\geqslant N^{-1-\xi}$ with some small $\xi$. The exponent $\xi$ can be chosen near 1 for band matrices with a broad band $W\asymp N$. In particular, we prove universality for such band matrices with a rapidly vanishing mean-field component. These applications of the general Theorem 8.2 are listed in Corollary 8.3.

Examples

Let $a\equiv a_{N}$ and $b\equiv b_{N}$ be possibly $N$-dependent positive quantities. We call $H$ an $a$-full Wigner matrix if $S$ satisfies (2.3) and

Similarly, we call $H$ a $b$-flat Wigner matrix if $S$ satisfies (2.3) and

(Note that in this case we have $M\geqslant N/b$.)

If $a$ and $b$ are independent of $N$ we call an $a$-full Wigner matrix simply full and a $b$-flat Wigner matrix simply flat. In particular, generalized Wigner matrices, satisfying (1.1), are full and flat Wigner matrices.

for some fixed $\delta^{\prime}>0$. Define the $d$-dimensional discrete torus

Then $H$ is a $d$-dimensional band matrix with band width $W$ and profile function $f$ if

where $Z_{L}$ is a normalization chosen so that (2.3) holds.

The example of Definition 3.3 is a mixture of the previous two. We are especially interested in the case $\nu\ll 1$, when most of the variance comes from the band matrix, i.e. the profile of $S$ is very close to a sharp band.

We conclude with some explicit bounds for these examples. The behaviour of $\Gamma$ and $\widetilde{\Gamma}$ near the spectral edge is governed by the parameter

For general $H$ and any constant $c>0$, there is a constant $C>0$ such that

provided $\operatorname{dist}(E,\{-2,0,2\})\geqslant c$.

where $C$ depends on the constant $a$ in Definition 3.1 but $c$ does not.

For a band matrix with a mean-field component, as in Definition 3.3, we have

The case $\nu=0$ corresponds to a band matrix from Definition 3.2.

Tools

Let $X\equiv X(H)$ be a random variable. For $i\in\{1,\dots,N\}$ define the operations $P_{i}$ and $Q_{i}$ through

We introduce the random $z$-dependent control parameters

We remark that the letter $\Lambda$ had a different meaning in several earlier papers, such as EYYrigi . The following lemma collects basic bounds on $m$.

There is a constant $c>0$ such that for $E\in$ and $\eta\in(0,10]$ we have

The proof is an elementary exercise using (2.9). ∎

In particular, recalling that $-1\leqslant S\leqslant 1$ and using the upper bound $\lvert m\rvert\leqslant C$ from (4.2), we find that there is a constant $c>0$ such that

The following lemma collects basic algebraic properties of stochastic domination $\prec$. Roughly, it states that $\prec$ satisfies the usual arithmetic properties of order relations. We shall use it tacitly throughout the following.

Suppose that $X(u,v)\prec Y(u,v)$ uniformly in $u\in U$ and $v\in V$. If $\lvert V\rvert\leqslant N^{C}$ for some constant $C$ then

Suppose that $X_{1}(u)\prec Y_{1}(u)$ uniformly in $u$ and $X_{2}(u)\prec Y_{2}(u)$ uniformly in $u$. Then $X_{1}(u)X_{2}(u)\prec Y_{1}(u)Y_{2}(u)$ uniformly in $u$.

If $X\prec Y+N^{-\varepsilon}X$ for some $\varepsilon>0$ then $X\prec Y$.

The claims (i) and (ii) follow from a simple union bound. The claim (iii) is an immediate consequence of the definition of $\prec$. ∎

The following resolvent identities form the backbone of all of our calculations. The idea behind them is that a resolvent matrix element $G_{ij}$ depends strongly on the $i$-th and $j$-th columns of $H$, but weakly on all other columns. The first identity determines how to make a resolvent matrix element $G_{ij}$ independent of an additional index $k\neq i,j$. The second identity expresses the dependence of a resolvent matrix element $G_{ij}$ on the matrix elements in the $i$-th or in the $j$-th column of $H$.

This is an exercise in linear algebra. The first identity (4.6) was proved in Lemma 4.2 of EYY and the second is an immediate consequence of the first. The identity (4.7) is proved in Lemma 6.10 of EKYY2 . ∎

Our final tool consists of the following results on fluctuation averaging. They exploit cancellations in sums of fluctuating quantities involving resolvent matrix entries. A very general result was obtained in EKYfluc ; in this paper we state a special case sufficient for our purposes here, and give a relatively simple proof in Appendix B. We consider weighted averages of diagonal resolvent matrix entries $G_{kk}$. They are weakly dependent, but the correlation between $G_{kk}$ and $G_{mm}$ for $m\neq k$ is not sufficiently small to apply the general theory of sums of weakly dependent random variables; instead, we need to exploit the precise form of the dependence using the resolvent structure.

It turns out that the key quantity that controls the magnitude of the fluctuations is $\Lambda$. However, being a random variable, $\Lambda$ itself is unsuitable as an upper bound. For technical reasons (our proof relies on a high-moment estimate combined with Chebyshev’s inequality), it is essential that $\Lambda$ be estimated by a deterministic control parameter, which we call $\Psi$. The error terms are then estimated in terms of powers of $\Psi$. We shall always assume that $\Psi$ satisfies

Typical example weights are $t_{ik}=s_{ik}$ and $t_{ik}=N^{-1}$. Note that in both of these cases $T$ commutes with $S$. We introduce the average of a vector $(a_{i})_{i=1}^{N}$ through

where we defined $v_{i}\mathrel{\vbox{\hbox{.}\hbox{.}}}=G_{ii}-m$. The estimates (4.11), (4.12), and (4.14) are uniform in the index $i$.

In fact, the first bound of (4.11) can be improved as follows.

The first instance of the fluctuation averaging mechanism appeared in EYY2 for the Wigner case, where $[Z]=N^{-1}\sum_{k}Z_{k}$ was proved to be bounded by $\Lambda_{o}^{2}$. Since $Q_{k}[G_{kk}]^{-1}$ is essentially $Z_{k}$ (see (5.6) below), this corresponds to the first bound in (4.11). A different proof (with a better bound on the constants) was given in EYYrigi . A conceptually streamlined version of the original proof was extended to sparse matrices EKYY1 and to sample covariance matrices PY1 . Finally, an extensive analysis in EKYfluc treated the fluctuation averaging of general polynomials of resolvent entries and identified the order of cancellations depending on the algebraic structure of the polynomial. Moreover, in EKYfluc an additional cancellation effect was found for the quantity $Q_{i}|G_{ij}|^{2}$. These improvements played a key role in obtaining the diffusion profile for the resolvent of band matrices and the estimate (2.21) in EKYY3 .

All proofs of the fluctuation averaging theorems rely on computing expectations of high moments of the averages, and carefully estimating the resulting terms. In EKYfluc , a diagrammatic representation was developed for bookkeeping such terms, but this is necessary only for the case of general polynomials. For the special cases given in Theorem 4.6, the proof is relatively simple and it is presented in Appendix B. Compared with EYY2 ; EYYrigi ; EKYY1 , the algebra of the decoupling of the randomness is greatly simplified in the current paper. Moreover, unlike their counterparts from EYY2 ; EYYrigi ; EKYY1 , the fluctuation averaging results of Theorems 4.6 and 4.7 do not require conditioning on the complement of some “bad” low-probability event, because such events are automatically accounted for by the definition of $\prec$ ; this leads to further simplifications in the proofs of Theorems 4.6 and 4.7.

A simpler proof using ΓΓ\Gamma instead of Γ~~Γ\widetilde{\Gamma}

In this section we prove the following weaker version of Theorem 2.3. In analogy to (2.14), we introduce the lower boundary

Fix $\gamma\in(0,1/2)$ and define the spectral domain

Note that the only difference between Theorems 2.3 and 5.1 is that $\widetilde{\Gamma}$ was replaced with the larger quantity $\Gamma$ in the definition of the threshold $\eta_{E}$ and the spectral domain, so that

Hence Theorem 5.1 is indeed weaker than Theorem 2.3, since it holds on a smaller spectral domain. As outlined after (2.11) and discussed in detail in Appendix A, Theorems 5.1 and 2.3 are equivalent provided $E$ is separated from the set $\{-2,0,2\}$ (for band matrices they are equivalent provided $E$ is separated from the spectral edges $\pm 2$).

The rest of this section is devoted to the proof of Theorem 5.1. We give the full proof of Theorem 5.1 for pedagogical reasons, since it is simpler than that of Theorem 2.3 but already contains several of its key ideas. Theorem 2.3 will be proved in Section 6. One big difference between the two proofs is that in Theorem 5.1 the main control parameter is $\Lambda$, while in Theorem 2.3 we have to keep track of two control parameters, $\Lambda$ and the smaller $\Theta$.

The key tool behind the proof is a self-consistent equation for the diagonal entries of $G$. The starting point is Schur’s complement formula, which we write as

The partial expectation with respect to the index $i$ (see Definition 4.2) of the last term on the right-hand side reads

where in the first step we used (2.1) and in the second (4.6). Introducing the notation

and recalling (2.3), we therefore get from (5.6) that

where we introduced the fluctuating error term

Using (2.8), we therefore get the self-consistent equation

Notice that this is an equation for the family $(v_{i})_{i=1}^{N}$, with random error terms $\Upsilon_{i}$.

Self-consistent equations play a crucial role in analysing resolvents of random matrices. The simplest one is the scalar (or first level) self-consistent equation for $m_{N}(z)$, the Stieltjes transform of the empirical density (2.12). By averaging the inverse of (5.7) and neglecting the error terms, one obtains that $m_{N}$ approximately satisfies the equation $m=-(m+z)^{-1}$, which is the defining relation for the Stieltjes transform of the semicircle law (2.8).

The vector (or second level) self-consistent equation, as given in (5.9), allows one to control not only fluctuations of $m_{N}-m$ but also those of $G_{ii}-m$. The equation (5.9) first appeared in EYY , where a systematic study of resolvent entries of random matrices was initiated.

For completeness, we mention that a matrix (or third level) self-consistent equation for local averages of $|G_{ij}|^{2}$, was introduced in EKYY3 . This equation constitutes the backbone of the study of the diffusion profile of the resolvent entries of random band matrices.

We begin with the first statement. We shall often use the fact that, by the lower bound of (4.2) and the assumption $\phi\Lambda\prec M^{-c}$, we have

First we estimate $Z_{i}$, which we split as

We estimate each term using the large deviation estimates from Theorem C.1, by conditioning on $G^{(i)}$ and using the fact that the family $(h_{ik})_{k=1}^{N}$ is independent of $G^{(i)}$. By (C.2), the first term of (5.13) is stochastically dominated by \phi\bigl{(}{\sum_{k}^{(i)}s_{ik}^{2}\bigl{\lvert}G_{kk}^{(i)}\bigr{\rvert}^{2}}\bigr{)}^{1/2}\prec M^{-1/2}, where we used the estimate (2.2) and \phi\bigl{\lvert}G_{kk}^{(i)}\bigr{\rvert}\prec 1, as follows from (4.6), (5.12), and the assumption $\phi\Lambda\prec M^{-c}$. For the second term of (5.13) we apply (C.4) with $a_{kl}=s_{ik}^{1/2}G_{kl}^{(i)}s_{li}^{1/2}$ and $X_{k}=\zeta_{ik}$ (see (2.5)). We find

where the second step follows by spectral decomposition of $G^{(i)}$, and in the last step we used (4.6) and (5.12). Thus we get

where we absorbed the bound $M^{-1/2}$ on the first term of (5.13) into the right-hand side of (5.15), using $\operatorname{Im}m\geqslant\eta$ as follows from (4.4).

Next, we estimate $\Lambda_{o}$. We can iterate (4.7) once to get, for $i\neq j$,

The term $h_{ij}$ is trivially $O_{\prec}(M^{-1/2})$. In order to estimate the other term, we invoke (C.3) with $a_{kl}=s_{ik}^{1/2}G_{kl}^{(ij)}s_{lj}^{1/2}$, $X_{k}=\zeta_{ik}$, and $Y_{l}=\zeta_{lj}$. As in (5.14), we find

where we again absorbed the term $h_{ij}\prec M^{-1/2}$ into the right-hand side.

In order to estimate $A_{i}$ and $h_{ii}$ in the definition of $\Upsilon_{i}$, we use (5.12) to estimate

where the second step follows from $\operatorname{Im}m\geqslant\eta$ (recall (4.4)). This completes the proof of (5.10).

The proof of (5.11) is almost identical to that of (5.10). The quantities \bigl{\lvert}G^{(i)}_{kk}\bigr{\rvert} and \bigl{\lvert}G^{(ij)}_{kk}\bigr{\rvert} are estimated by the trivial deterministic bound $\eta^{-1}$. We omit the details. ∎

3. A rough bound on ΛΛ\Lambda

The next step in the proof of Theorem 5.1 is to establish the following rough bound on $\Lambda$.

Then by definition we have $\phi\Lambda\leqslant M^{-\gamma/4}\Gamma^{-1}\leqslant CM^{-\gamma/4}$, where in the last step we used (4.5). Hence we may invoke (5.10) to estimate $\Lambda_{o}$ and $\Upsilon_{i}$. In order to estimate $\Lambda_{d}$, we expand the right-hand side of (5.9) in $v_{i}$ to get

where we used (4.2) and that $\lvert v_{i}\rvert\leqslant CM^{-\gamma/4}$ on the event $\{\phi=1\}$. Using (5.10) we therefore have

Recalling (4.5) and (5.10), we therefore get

Next, by definition of $\phi$ we may estimate

Plugging this into (5.18) yields $\phi\Lambda\prec M^{-\gamma/2}\Gamma^{-1}$, which is the claim. ∎

In order to start the continuity argument underlying the proof of Proposition 5.3, we need the following bound on $\Lambda$ for large $\eta$.

Moreover, we use (4.6) and (5.16) to estimate

where the last step follows using (C.3), exactly as the estimate of the right-hand side of (5.16) in the proof of Lemma 5.2. We conclude that $\lvert\Upsilon_{i}\rvert\prec M^{-1/2}$.

Using $\lvert m^{-1}\rvert\geqslant 2$ and $\lvert v_{k}\rvert\leqslant 1$ as follows from (5.19), we find

Using $\lvert m\rvert\leqslant 1/2$ we therefore conclude that

from which the claim follows together with the estimate on $\Lambda_{o}$ from (5.20). ∎

We may now conclude the proof of Proposition 5.3 by a continuity argument in $\eta=\operatorname{Im}z$. The gist of the continuity argument is depicted in Figure 5.1.

for $N\geqslant N_{0}$, where $N_{0}\equiv N_{0}(\gamma,D)$ does not depend on $z$.

(Here the bounds (5.23) and (5.24) each hold surely, i.e. for every realization of $\Lambda(z)$.)

4. Iteration step and conclusion of the proof of Theorem 5.1

In the following a key role will be played by deterministic control parameters $\Psi$ satisfying

We now improve the estimate $\Lambda\prec\Psi$ iteratively. The iteration step is the content of the following proposition.

Let $\Psi$ be a control parameter satisfying (5.25) and fix $\varepsilon\in(0,\gamma/3)$. Then

For the proof of Proposition 5.6 we need the following averaging result, which is a simple corollary of Theorem 4.6.

Suppose that $\Lambda\prec\Psi$ for some deterministic control parameter $\Psi$ satisfying (4.8). Then $[\Upsilon]=O_{\prec}(\Psi^{2})$ (recall the definition of the average $[\cdot]$ from (4.10)).

The claim easily follows from Schur’s complement formula (5.6) written in the form

We may therefore estimate $[\Upsilon]$ using the trivial bound $\lvert A_{i}\rvert\prec\Psi^{2}$ as well as the fluctuation averaging bound from the first estimate of (4.11) with $t_{ik}=1/N$. ∎

Suppose that $\Lambda\prec\Psi$ for some deterministic control parameter $\Psi$ satisfying (5.25). We invoke Lemma 5.2 with $\phi=1$ (recall the bound (4.5)) to get

Next, we estimate $\Lambda_{d}$. Define the $z$-dependent indicator function

By (5.25), (4.5), and the assumption $\Lambda\prec\Psi$, we have $1-\psi\prec 0$. On the event $\{\psi=1\}$, we expand the right-hand side of (5.9) to get the bound

Using the fluctuation averaging estimate (4.12) as well as (5.27), we find

where we again used the lower bound from (4.5). Using $1-\psi\prec 0$ we conclude

Using Young’s inequality and the assumption $\Psi\leqslant M^{-\gamma/3}\Gamma^{-1}$ we conclude the proof. ∎

We may therefore iterate (5.26). This yields a bound on $\Lambda$ that is essentially the fixed point of the map $\Psi\mapsto F(\Psi)$, which is $\Pi$ (up to the factor $M^{\varepsilon}$). More precisely, the iteration is started with $\Psi_{0}\mathrel{\vbox{\hbox{.}\hbox{.}}}=M^{-\gamma/3}\Gamma^{-1}$; the initial hypothesis $\Lambda\prec\Psi_{0}$ is provided by the rough bound from Proposition 5.3. For $k\geqslant 1$ we set $\Psi_{k+1}\mathrel{\vbox{\hbox{.}\hbox{.}}}=F(\Psi_{k})$. Hence from (5.26) we conclude that $\Lambda\prec\Psi_{k}$ for all $k$. Choosing $k\mathrel{\vbox{\hbox{.}\hbox{.}}}=\lceil{\varepsilon^{-1}}\rceil$ yields

Since $\varepsilon$ was arbitrary, we have proved that

What remains is to prove (5.4), i.e. to estimate $\Theta$. We expand (5.9) on $\{\psi=1\}$ to get

By (5.31) and (5.27) with $\Psi=\Pi$, we have $\Lambda+\lvert\Upsilon_{i}\rvert\prec\Pi$. Moreover, by Lemma 5.7 we have $\lvert[\Upsilon]\rvert\prec\Pi^{2}$. Thus we get

Since $1-\psi\prec 0$, we conclude that $[v]=m^{2}[v]+O_{\prec}(\Pi^{2})$. Therefore

Proof of Theorem 2.3

In Section 6.2, we control $\Lambda$ in terms of $\Theta$, which allows us to obtain a self-consistent equation involving only $\Theta$. In this step we use the Fluctuation Averaging Theorem to obtain a quadratic estimate which, very roughly, states that $\Lambda\lesssim\Theta+\Lambda^{2}$ (see (6.20) below for the precise statement). This implies $\Lambda\lesssim\Theta$ in the regime $\Lambda\ll 1$.

Finally, in Section 6.3, we solve the quadratic iteration for $\Theta$. Since the corresponding quadratic equation has a dichotomy and for large $\eta=\operatorname{Im}z$ we know that $\Theta$ is small by direct expansion, a continuity argument similar to the proof of Proposition 5.3 will complete the proof.

In this section we prove the following apriori bounds on both control parameters, $\Lambda$ and $\Theta$.

Before embarking on the proof of Proposition 6.1, we state some preparatory lemmas. First, we derive the key equation for $[v]=N^{-1}\sum_{i}v_{i}$, the average of $v_{i}$.

Define the $z$-dependent indicator function

For the whole proof we work on the event $\{\phi=1\}$, i.e. every quantity is multiplied by $\phi$. We consistently drop these factors $\phi$ from our notation in order to avoid cluttered expressions. In particular, $\Lambda\leqslant CM^{-\gamma/4}$ throughout the proof.

We begin by estimating $\Lambda_{o}$ and $\Lambda_{d}$ in terms of $\Theta$. Recalling (4.5), we find that $\phi$ satisfies the hypotheses of Lemma 5.2, from which we get

In order to estimate $\Lambda_{d}$, we expand the self-consistent equation (5.9) (on the event $\{\phi=1\}$) to get

here we used the bound (6.4) on $|\Upsilon_{i}|$. Next, we subtract the average $N^{-1}\sum_{i}$ from each side to get

Combining with the bound $\Lambda_{o}\prec r(\Lambda)$ from (6.4), we therefore get

By definition of $\phi$ we have $\widetilde{\Gamma}\Lambda^{2}\leqslant M^{-\gamma/4}\Lambda$, so that by Lemma 4.4 (iii) the second term on the right-hand side of (6.7) may be absorbed into the left-hand side:

If (6.9) is proved, clearly (6.3) follows from (6.8). In order to prove (6.9), we use (6.8) and the Cauchy-Schwarz inequality to get

for any $\varepsilon>0$. We conclude that

Since $\varepsilon>0$ was arbitrary, (6.9) follows.

Next, we estimate $\Theta$. We expand (5.9) to second order:

In order to take the average and get a closed equation for $[v]$, we write, using (6.6),

Plugging this back into (6.10) and taking the average over $i$ gives

Estimating $[\Upsilon]$ by $\max\lvert\Upsilon_{i}\rvert\prec r(\Lambda)$ (recall (6.4)) yields

In order to conclude the proof of (6.2), we observe that, by the estimates $\Theta\leqslant\Lambda\leqslant CM^{-\gamma/4}$, $\widetilde{\Gamma}^{2}r(\Lambda)\leqslant 1$, and $\Lambda\leqslant M^{-\gamma/4}\widetilde{\Gamma}^{-1}$, we have

Let $\varepsilon\in(0,\gamma/12)$. Define the $z$-dependent indicator function

Therefore, on the event $\{\phi\chi=1\}$, in (6.2) we may absorb the second term on the left-hand side and the second term on the right-hand side into the first term on the left-hand side:

Solving this quadratic relation for $[v]$, we get

We now have all of the ingredients to complete the proof of Proposition 6.1.

The proof is a continuity argument similar to the proof of Proposition 5.3. In a first step, we prove that

The claim now follows since we may choose $\varepsilon\in(0,\gamma/12)$ to be arbitrarily small. This concludes the proof of Proposition 6.1. ∎

2. An improved bound on ΛΛ\Lambda in terms of ΘΘ\Theta

In (6.3) we already estimated $\Lambda$ in terms of $\Theta$; the goal of this section is to improve this bound by removing the factor $\widetilde{\Gamma}$ from that estimate. We do this using the Fluctuation Averaging Theorem, but we stress that the removal of a factor $\widetilde{\Gamma}$ is not the main rationale for using the fluctuation averaging mechanism. Its fundamental use will take place in Lemma 6.6 below. A technical consequence of invoking fluctuation averaging is that we have to use deterministic control parameters instead of random ones. Thus, we introduce a deterministic control parameter $\Phi$ that captures the size of the random control parameter $\Theta$ through the relation $\Theta\prec\Phi$. Throughout the following we shall make use of the control parameter

which differs from $q(\Phi)$ only by a factor $\widetilde{\Gamma}$ in the second term.

We remark that, by Proposition 6.1, the choice $\Psi=M^{-\gamma/4}\widetilde{\Gamma}^{-1}$ and $\Phi=(M\eta)^{-1/3}\leqslant M^{-\gamma/4}\widetilde{\Gamma}^{-1}$ satisfies the assumptions of Lemma 6.5.

Choosing $\phi=1$ in Lemma 5.2 and recalling (4.5), we get

In order to estimate $\Lambda_{d}$, as in (5.32), we expand (5.9) to get

As in the proof of (5.32) and (6.5), the expansion of (5.9) is only possible on the event $\{\Lambda\leqslant M^{-\delta}\}$ for some $\delta>0$. By $\Lambda\prec\Psi$ and (6.15), the indicator function of this event is $1+O_{\prec}(0)$; the contribution $O_{\prec}(0)$ of the complementary event can be absorbed in the error term $O_{\prec}(\Psi^{2})$.

Subtracting the average $N^{-1}\sum_{i}$ from both sides of (6.18) and estimating $m^{2}$ by a constant (see (4.2)) yields

where in the last step we used the fluctuation averaging estimate (4.14) and $\lvert\Upsilon_{i}\rvert\prec r(\Psi)$ from (6.17). Together with $\lvert[v]\rvert=\Theta\prec\Phi$ , this gives the estimate $\Lambda_{d}\prec\Gamma\Psi^{2}+\Phi+r(\Psi)$. Combining it with the bound (6.17), we conclude that

Now fix $\varepsilon\in(0,\gamma/4)$. Using the assumption $\widetilde{\Gamma}\Psi\leqslant CM^{-\gamma/4}\leqslant M^{-\varepsilon}$, we conclude: if $\Psi$ and $\Phi$ satisfy (6.15) then

This implies the claimed bound (6.16) on $\Lambda$. Calling the right-hand side of (6.22) $\Psi$, we find

Hence the claimed bound (6.16) on $\Lambda_{o}$ and $Z_{i}$ follows from (6.17). ∎

3. Iteration for ΘΘ\Theta and conclusion of the proof of Theorem 2.3

Next, we prove the following version of (5.9), which is the key tool for estimating $\Theta$.

Notice that this bound is stronger than the previous formula (6.2), as the power of $p(\Phi)$ is two instead of one. The improvement is due to using fluctuation averaging in $[\Upsilon]$. Otherwise the proof is very similar to that of (6.2).

since $\Theta\leqslant\Lambda\prec M^{-\gamma/4}\widetilde{\Gamma}^{-1}$. From Lemma 6.5 we get $\Lambda_{o}+\lvert Z_{i}\rvert\prec p(\Phi)$ and $\Lambda\prec\Psi$, where

Now we expand the right-hand side of (5.9) exactly as in (6.10) to get

Using Theorem 4.7 and the bound $\Lambda_{o}\prec p(\Phi)$ from Lemma 6.5, we may prove, exactly as in Lemma 5.7, that $\lvert[\Upsilon]\rvert\prec p(\Phi)^{2}$. Taking the average over $i$ in (6.27) therefore yields

Using the estimates (6.19) and (6.23), we write the quadratic term on the left-hand side as

where we also used $\widetilde{\Gamma}\Psi\leqslant 2M^{-\gamma/4}$, as observed after (6.26). From (6.28) we therefore get

The bound on $\Theta$ will follow by iterating the following estimate.

If $\Phi\geqslant M^{3\varepsilon}(M\eta)^{-1}$ then

If $\lvert E\rvert\geqslant 2$, $\frac{M^{3\varepsilon}}{M(\kappa+\eta)}\leqslant\Phi\leqslant M^{\varepsilon}\sqrt{\kappa+\eta}$, and $M\eta\sqrt{\kappa+\eta}\geqslant M^{2\varepsilon}$, then

such constant exists by (4.2) and (4.3). Define the indicator function

Hence on the event $\{\psi=1\}$ we may absorb the quadratic term on the left-hand side of (6.24) into the linear term, to get the bound

which replaces (6.32) and (6.33). The rest of the argument is unchanged. ∎

Choosing $k=\lceil{\varepsilon^{-1}}\rceil$ yields $\Theta\prec M^{3\varepsilon}(M\eta)^{-1}$. Since $\varepsilon$ can be made as small as desired, we therefore obtain $\Theta\prec(M\eta)^{-1}$. This is (2.19).

In the regime $\lvert E\rvert\geqslant 2$, the same argument with the better iteration bound (6.30) yields (2.20). The iteration can be started with $\Phi_{0}=M^{3\varepsilon}(M\eta)^{-1}$ from (2.19).

Finally, the bound $\Lambda\prec\Pi$ in (2.18) follows from (2.19) and Lemma 6.5. This concludes the proof of Theorem 2.3.

Density of states and eigenvalue locations

In this section we apply the local semicircle law to obtain information on the density of states and on the location of eigenvalues. The techniques used here have been developed in a series of papers ESY2 ; ESYY ; EYYrigi ; EKYY1 .

The first result is to translate the local semicircle law, Theorem 2.3, into a statement on the counting function of the eigenvalues. Let $\lambda_{1}\leqslant\lambda_{2}\leqslant\cdots\leqslant\lambda_{N}$ denote the ordered eigenvalues of $H$, and recall the semicircle density $\varrho$ defined in (2.7). We define the distribution functions

For given $E_{1}<E_{2}$ in $$ we abbreviate

Then, for $-10\leqslant E_{1}<E_{2}\leqslant 10$, we have

The accuracy of the estimate (7.4) depends on $\widetilde{\Gamma}$ (see (A.3) for explicit bounds on $\widetilde{\Gamma}$), since $\widetilde{\Gamma}$ determines $\widetilde{\eta}_{E}$, the smallest scale on which the local semicircle law (Theorem 2.3) holds around the energy $E$. In the regime away from the spectral edges $E=\pm 2$ and away from $E=0$, the parameter $\widetilde{\Gamma}$ is essentially bounded (see the example (i) from Section 3); in this case $\widetilde{\eta}_{E}\asymp M^{-1}$ (up to an irrelevant logarithmic factor). For $E$ near , the parameter $\widetilde{\Gamma}$ blows up as $E^{-2}$, so that $\widetilde{\eta}_{E}\sim E^{-12}M^{-1}$; however, if $S$ has a positive gap $\delta_{-}$ at the bottom of its spectrum, $\widetilde{\Gamma}$ remains bounded in the vicinity of $E=0$ (see (A.3)). See Definition A.1 in Appendix A for the definition of the spectral gaps $\delta_{\pm}$.

where $A$ is an $L\times(N-L)$ rectangular matrix with independent centred entries. The eigenvalues of $H$ are the square roots (with both signs) of the eigenvalues of the random covariance matrices $AA^{*}$ and $A^{*}A$, whose spectral density is asymptotically given by the Marchenko-Pastur law MP . The instability near $E=0$ arises from the fact that $H$ has a macroscopically large kernel unless $L/N\to 1/2$. In the latter case the support of the Marchenko-Pastur law extends to zero and in fact the density diverges as $E^{-1/2}$. We remark that a local version of the Marchenko-Pastur law was given in ESYY for the case when the limit of $L/N$ differs from 0, 1/2 and $\infty$; the “hard edge” case, $L/N\to 1/2$, in which the density near the lower spectral edge is singular, was treated in CMS .

This example shows that the vanishing of $\delta_{-}$ may lead to a very different behaviour of the spectral statistics. Although our technique is also applicable to random covariance matrices, for simplicity in this section we assume that $\delta_{-}\geqslant c$ for some positive constant $c$. By Proposition A.3, this holds for random band matrices, for full Wigner matrices (see Definition 3.1), and for their combinations; these examples are our main interest in this paper.

Under the condition $\delta_{-}\geqslant c$, the upper bound of (A.3) yields

where $\theta$ was defined in (3.2) and $\delta_{+}$ is the upper gap of the spectrum of $S$ given in Definition A.1. Notice that $\theta$ vanishes near the spectral edge $E=\pm 2$ as $\eta\to 0$. For the purpose of estimating $\widetilde{\Gamma}$, this deterioration is mitigated if the upper gap $\delta_{+}$ is non-vanishing. While full Wigner matrices satisfy $\delta_{+}\geqslant c$, the lower bound on $\delta_{+}$ for band matrices is weaker; see Proposition A.3 for a precise statement.

We first give an estimate on $\widetilde{\eta}_{x}$ using the explicit bound (7.5). While not fully optimal, this estimate is sufficient for our purposes and in particular reproduces the correct behaviour when $\delta_{+}\geqslant c$.

Suppose that $\delta_{-}\geqslant c$ (so that (7.5) holds). Then we have for any $\lvert x\rvert\leqslant 2$

In the regime $2\leqslant\lvert x\rvert\leqslant 10$ we have the improved bound

For any $\lvert x\rvert\leqslant 2$ define $\eta_{x}^{\prime}$ as the solution of the equation

This solution is unique since the left-hand side is decreasing in $\eta$. An elementary but tedious analysis of (7.8) yields

(The calculation is based on the observation that if $\eta(a+\eta^{\alpha})\leqslant b$ for some $a,b>0$ and $\alpha\geqslant 0$, then $\eta\leqslant 2b(b^{\frac{\alpha}{1+\alpha}}+a)^{-1}$.) From (7.5), $\operatorname{Im}m(x+i\eta)\leqslant C\sqrt{\kappa_{x}+\eta}$ (see (4.4)) and the simple bound $\theta(x+i\eta)\geqslant c(\kappa_{x}+\eta^{2/3})$, we get for $\eta\geqslant\eta^{\prime}_{x}$

The proof of (7.7) is similar, but we use $\theta=\sqrt{\kappa+\eta}$ and the stronger bound $\operatorname{Im}m\leqslant\eta/\sqrt{\kappa+\eta}$ available in the regime $\lvert x\rvert\geqslant 2$. For $2\leqslant\lvert x\rvert\leqslant 10$, define $\eta_{x}^{\prime}$ to be the solution of the equation

As for (7.9), a tedious calculation yields

Next, we obtain an estimate on the extreme eigenvalues.

Suppose that $\delta_{-}\geqslant c$ (so that (7.5) holds) and that $N^{3/4}\leqslant M\leqslant N$. Then we have

where we introduced the control parameter

In particular, if $\delta_{+}\geqslant c$ then

Note that (7.13) yields the optimal error bound $O_{\prec}(N^{-2/3})$ in the case of a full and flat Wigner matrix (see Definition 3.1). Under stronger assumptions on the law of the entries of $H$, Theorem 7.3 can be improved as follows.

Suppose that the matrix elements $h_{ij}$ have a uniform subexponential decay, i.e. that there exist positive constants $C$ and $\vartheta$ such that

If in addition the law of each matrix entry is symmetric (i.e. $h_{ij}$ and $-h_{ij}$ have the same law), then (7.11) holds with

We remark that (7.15) can obtained via a relatively standard moment method argument combined with refined combinatorics. Obtaining the bound (7.16) is fairly involved; it makes use of the Chebyshev polynomial representation first used by Feldheim and Sodin FSo ; So1 in this context for a special distribution of $h_{ij}$, and extended in EK2 to general symmetric entries.

We shall prove a lower bound on the smallest eigenvalue $\lambda_{1}$ of $H$; the largest eigenvalue $\lambda_{N}$ may be estimated similarly from above. Fix a small $\gamma>0$ and set

We distinguish two regimes depending on the location of $\lambda_{1}$, i.e. we decompose

In the first regime we further decompose the probability space by estimating

Clearly, $\eta_{k}\leqslant\kappa_{k}$ since $M\leqslant N$. On the support of $\phi_{1,k}$ we have $\lvert\lambda_{1}-E_{k}\rvert\leqslant C/N\leqslant\eta_{k}$, so that we get the lower bound

for some positive constant $c$. On the other hand, by (4.4), we have

for some positive constant $c^{\prime}$. Here in the second step we used that $\eta_{k}/\sqrt{\kappa_{k}}\leqslant M^{-\gamma}(N\eta_{k})^{-1}$.

Comparing this bound with (7.18) we conclude that $\phi_{1,k}\prec 0$ (i.e. the event $\{\phi_{1,k}=1\}$ has very small probability). Summing over $k$ yields $\phi_{1}\prec 0$. Note that in this proof the stronger bound (2.20) outside of the spectrum was essential; the general bound of order $(M\eta_{k})^{-1}$ from (2.19) is not smaller than the right-hand side of (7.18).

The preceding proof of $\phi_{1}\prec 0$ assumed the existence of a spectral gap $\delta_{+}\geqslant c$. The above argument easily carries over to the case without a gap of constant size, in which case we choose

All that remains to complete the proof of (7.11) and (7.13) is the estimate $\phi_{2}\prec 0$. Clearly

In part (2) of Lemma 7.2 in EYY it was shown, using the moment method, that the right-hand side is bounded by $CN^{-c\log\log N}$ provided the matrix entries $h_{ij}$ have subexponential decay, i.e.

for some constants $\alpha,\beta$ (recall the notation (2.5)). In this paper we only assume polynomial decay, (2.6). However, the subexponential decay assumption of EYY was only used in the first truncation step, Equations (7.28)–(7.29) in EYY , where a new set of independent random variables $\widehat{h}_{ij}$ was constructed with the properties that

for $n=(\log N)(\log\log N)$. Under the condition (2.6) the same truncation can be performed, but the estimates in (7.20) will be somewhat weaker; instead of the exponent $n=(\log N)(\log\log N)$ we get $n=D\log N$ for any fixed $D>0$. The conclusion of the same proof is that, assuming only (2.6), we have

for any positive number $D$ and for any $N\geqslant N_{0}(D)$. This guarantees that $\phi_{2}\lVert H\rVert\prec 0$. Together with the estimate $\phi_{1}\lVert H\rVert\leqslant 3\phi_{1}\prec 0$ established above, this completes the proof of Theorem 7.3. ∎

The estimate of $\lVert H\rVert$ with $X=M^{-1/6}$ follows from the proof of part (2) of Lemma 7.2 in EYY , by choosing $k=M^{-1/6-\varepsilon}$ with any small $\varepsilon>0$ in (7.32) of EYY . This argument can be improved to $X=M^{-1/4}$ by the remark after (7.18) in EYY . Finally, the bound with $X=M^{-2/3}$ under the symmetry condition on the entries of $H$ is proved in Theorem 3.4 of EK2 . ∎

Next, we establish an estimate on the normalized counting function ${\mathfrak{n}}_{N}$ defined in (7.1). As above, the exponents are not expected to be optimal, but the estimate is in general sharp if $\delta_{+}\geqslant c$.

Suppose that $\delta_{-}\geqslant c$ (so that (7.5) holds). Then

where we introduced the control parameter

First we prove the bound (7.22) for any fixed $E\in$. Define the dyadic energies $E_{k}\mathrel{\vbox{\hbox{.}\hbox{.}}}=-2-2^{k}(\delta_{+}+M^{-1/5})$. By (7.6) we have for all $k\geqslant 0$

A similar bound holds for $E_{k}^{\prime}\mathrel{\vbox{\hbox{.}\hbox{.}}}=-2+2^{k}(\delta_{+}+M^{-1/5})$. For any $E\in$, we express ${\mathfrak{n}}_{N}(E)-n(E)$ as a telescopic sum and use (7.4) to get

Here we used that $n(-10)=0$ and ${\mathfrak{n}}_{N}(-10)\leqslant{\mathfrak{n}}_{N}(-3)\prec 0$ by (7.21). In fact, (7) easily extends to any $E<-10$. By an analogous dyadic analysis near the upper spectral edge, we also get (7.21) for any $E\geqslant 0$. Since this holds for any $\gamma>0$, we thus proved

To prove the statement uniformly in $E$, we define the classical location of the $\alpha$-th eigenvalue $\gamma_{\alpha}$ through

Applying (7.25) for the $N$ energies $E=\gamma_{1},\dots,\gamma_{N}$, we get

uniformly in $\alpha=1,\dots,N$. Since ${\mathfrak{n}}_{N}(E)$ and $n(E)$ are nondecreasing and $Y\geqslant 1/N$, we find

uniformly in $\alpha=2,3,\dots$. Below $\gamma_{1}$ we use (7.27) to get

Finally, for any $E\geqslant\gamma_{N}$, we have ${\mathfrak{n}}_{N}(E)-n(E)={\mathfrak{n}}_{N}(E)-1\leqslant 0$ deterministically. Thus we have proved

Next, we derive rigidity bounds on the locations of the eigenvalues. Recall the definition of $\gamma_{\alpha}$ from (7.26).

Suppose that $\delta_{-}\geqslant c$ (so that (7.5) holds) and that (7.11) and (7.22) hold with some positive control parameters $X,Y\leqslant C$. Define $\widehat{\alpha}\mathrel{\vbox{\hbox{.}\hbox{.}}}=\min\{{\alpha,N+1-\alpha}\}$ and let $\varepsilon>0$ be arbitrary. Then

To simplify notation, we assume that $\alpha\leqslant N/2$ so that $\widehat{\alpha}=\alpha$; the other eigenvalues are handled analogously. Without loss of generality we assume that $\lambda_{N/2}\leqslant 1$. Indeed, the condition $\lambda_{N/2}\leqslant 1$ is equivalent to ${\mathfrak{n}}(1)\geqslant 1/2$, which holds with very high probability by Theorem 7.5 and the fact that $n_{sc}(1)>1/2$.

where in the last step we used Theorem 7.5. By definition of $n(x)$ we have for $-2\leqslant x\leqslant 1$ that

Suppose first that $\alpha\geqslant\alpha_{0}\mathrel{\vbox{\hbox{.}\hbox{.}}}=M^{\varepsilon}NY$. Then $n(\gamma_{\alpha})\geqslant M^{\varepsilon}Y$, so that the relation (7.30) implies

which yields $n(\gamma_{\alpha})\asymp n(\lambda_{\alpha})$. By (7.31), we we therefore get that $n^{\prime}(\gamma_{\alpha})\asymp n^{\prime}(\lambda_{\alpha})$ as well. Since $n^{\prime}$ is nondecreasing, we get $n^{\prime}(x)\asymp n^{\prime}(\gamma_{\alpha})\asymp n^{\prime}(\lambda_{\alpha})$ for any $x$ between $\gamma_{\alpha}$ and $\lambda_{\alpha}$. Therefore, by the mean value theorem, we have

where in the last step we used (7.30) and (7.32). This proves (7.28) for $\alpha\geqslant M^{\varepsilon}NY$.

For the remaining indices, $\alpha<\alpha_{0}$, we get from (7.30) the upper bound