Anisotropic local laws for random matrices

Introduction

for large $M$ and with high probability. Here $m(z)$ is the Stieltjes transform of the asymptotic eigenvalue density, which we denote by $\varrho$. We call an estimate of the form (1.1) an averaged law.

As may be easily seen by taking the imaginary part of (1.1), control of the convergence of $M^{-1}\operatorname{Tr}R(z)$ yields control of an order $\eta M$ eigenvalues around the point $E$. A local law is an estimate of the form (1.1) for all $\eta\gg M^{-1}$. Note that the approximation (1.1) cannot be correct at or below the scale $\eta\asymp M^{-1}$, at which the behaviour of the left-hand side of (1.1) is governed by the fluctuations of individual eigenvalues. Such local laws have become a cornerstone of random matrix theory, starting from the work ESY2 where a local law was first established for Wigner matrices. Well known corollaries of a local law include bounds on the eigenvalue counting function as well as eigenvalue rigidity. Moreover, local laws constitute the main tool needed to analyse (a) the distribution of eigenvalues (including the universality of the local spectral statistics), (b) eigenvector delocalization, (c) the distribution of eigenvectors, and (d) finite-rank deformations of $Q$.

In fact, for all of the applications (a)–(d), the averaged local law from (1.1) is not sufficient. One has to control not only the normalized trace of $R(z)$ but the matrix $R(z)$ itself, by showing that $R(z)$ is close to some deterministic matrix depending on $z$, provided that $\eta\gg M^{-1}$. Such control was first obtained for Wigner matrices in EYY1 , where the closeness was established in the sense of individual matrix entries: $R_{ij}(z)\approx m(z)\delta_{ij}$. We call such an estimate an entrywise local law. More generally, in KY2 ; BEKYY this closeness was established in the sense of generalized matrix entries:

Analogous results for uncorrelated sample covariance matrices were obtained in PY ; BEKYY . The estimate (1.2) states that for large $M$ the resolvent $R(z)$ is approximately isotropic (i.e. proportional to the identity matrix), and we accordingly call an estimate of the form (1.2) an isotropic local law. We remark that the basis-independent control in (1.2) is crucial for many applications, including the distribution of eigenvectors and the study of finite-rank deformations of $Q$.

Unlike in the case of Wigner matrices and uncorrelated sample covariance matrices mentioned above, the resolvent $R(z)$ is in general not close to a multiple of the identity matrix, but rather to some general deterministic matrix $P(z)$. In that case (1.2) is to be replaced with

We call an estimate of the form (1.3) an anisotropic local law. The goal of this paper is to develop a method yielding (anisotropic) local laws for many matrix models built from sums and products of deterministic or independent random matrices. Applications include all the four (a)–(d) listed above, some of which we illustrate in this paper.

2. Sample covariance matrices

3. Outline of results

For simplicity, we focus on the matrix $Q$, bearing in mind that similar results also apply for $\dot{Q}$ (see Section 11.2). We assume that the three matrix dimensions $M,\widehat{M},N$ are comparable, and that the entries of $\sqrt{N}X$ possess a sufficient number of bounded moments. Moreover, we assume that $\lVert\Sigma\rVert$ is bounded, and that the spectrum of $\Sigma$ satisfies certain regularity conditions, given Definition 2.7 below, which essentially state that the connected components of the support of $\varrho$ are separated by some positive constant, and that the density of $\varrho$ has square root decay near its edges in $(0,\infty)$. Note that we do not assume that $T$ is square, and in particular $T$ may have many vanishing singular values; this allows us to cover e.g. the general linear model of multivariate statistics.

Our main result is the anisotropic local law for $Q$. Roughly, it states that (1.3) holds with

where $m(z)$ is the Stieltjes transform of the asymptotic density $\varrho$. In fact, we prove a more general anisotropic local law that is more useful in applications. Its formulation is most transparent under the additional assumption that $T=T^{*}=\Sigma^{1/2}$, although this assumption is a mere convenience and may be easily removed (see Section 11.1). We prove an anisotropic local law of the form

A simple application of Schur’s complement formula to (1.5) yields the anisotropic local law for the resolvent of $Q=TXX^{*}T^{*}$ and a similar result for the resolvent of the companion matrix $X^{*}T^{*}TX$. The estimate (1.5) holds with high probability, and we give an explicit and optimal error bound. We remark that the anisotropic local law holds under very general assumptions on the distribution of $X$, the dimensions of $X$ and $T$, and the spectrum of $TT^{*}$. In particular, we make no assumptions on the singular vectors of $T$. We remark that, previously, an anisotropic global law, valid for $\eta\asymp 1$, was derived in HLNP for a different matrix model.

As an application of the anisotropic local law, we prove the edge universality of the eigenvalues near the soft spectral edges, whereby the joint distribution of the eigenvalues is asymptotically governed by the Tracy-Widom-Airy statistics of random matrix theory. More precisely, we prove that the asymptotic distribution of the eigenvalues near the soft edges depends only on the nonzero spectrum of $TT^{*}$. This may be regarded as a universality result in both the distribution of the entries of $X$ and the (left and right) singular vectors of $T$, including their dimensions. We then conclude that the Tracy-Widom-Airy statistics hold near the soft edges by noting that they have been previously established HHN ; ElK2 ; Ona ; SL for Gaussian $X$ and diagonal $T$.

We also prove the rigidity of eigenvalues, as well as the complete delocalization of the eigenvectors with respect to an arbitrary deterministic basis. Further applications of the anisotropic local law, such as the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations, will appear elsewhere.

Finally, we also apply our method to deformed Wigner matrices of the form $W+A$, where $W$ is a Wigner matrix and $A$ a bounded Hermitian matrix. This model describes Wigner matrices whose entries may have arbitrary expectations. As for $Q$, we establish the Tracy-Widom-Airy statistics near the spectral edges of $W+A$. More precisely, we prove that the asymptotic distribution of the eigenvalues near the edges depends only on the asymptotic spectrum of $A$, which may be regarded as a universality result in the distribution of $W$ and the eigenvectors of $A$. We then conclude that the Tracy-Widom-Airy statistics hold near the edges by noting that they have been previously established LSY for diagonal $A$.

4. Overview of the proof

We conclude this section by outlining some ideas of the proof of the anisotropic local law. Roughly, the proof proceeds in three steps: (A) the entrywise local law for Gaussian $X$ and diagonal $\Sigma$, (B) the anisotropic local law for Gaussian $X$ and general $\Sigma$, and (C) the anisotropic local law for general $X$ and general $\Sigma$. Steps (A) and (B) may be performed by adapting the methods of BEKYY , and we do not comment on them any further.

The main argument, and the bulk of the proof, is Step (C). Its core is a self-consistent comparison method, which yields the anisotropic local law for general $X$ assuming it has been proved for Gaussian $X$. Up to now, all interpolation or Lindeberg replacement arguments in random matrix theory have crucially relied on a local law as input. Since the local law is exactly what we are trying to prove, a different approach is clearly needed – one that does not need a local law to work. Our method yields a new way of deriving local laws for general random matrices expressed as polynomials of deterministic and random matrices whose entries are independent. It relies on two key novel ideas: (a) continous Green function comparison argument where the errors are controlled self-consistently, and (b) a bootstrapping of the Green function comparison on the spectral scale $\eta$.

In the remainder of this subsection we give a few details of our method, and in particular explain the ideas (a) and (b) in more detail. We construct a continuous family $(X^{\theta})_{\theta\in}$ of matrices, whereby $X^{0}$ is a Gaussian ensemble and $X^{1}$ is the ensemble in which we are interested. Then we operate in the $(\theta,\eta)$-plane and perform simultaneously a continuous interpolation in $\theta$ and a discrete bootstrapping in $\eta$. (See Figure 1.1 below.)

Interpolation methods have been extensively used in random matrix theory, in the form of both discrete Lindeberg-type replacement schemes Chat ; TV1 ; EYY1 ; EYY3 and continuous interpolations SL ; LSY ; LP . Moreover, Dyson Brownian motion, as used e.g. in SL ; LSY ; ESY4 ; ESYY , may be regarded as a form of continuous interpolation for the case that $X^{0}$ is Gaussian. The usual choice of interpolation, as used in SL ; LP and also resulting from Dyson Brownian motion, is $X^{\theta}=\sqrt{1-\theta}\,X^{0}+\sqrt{\theta}\,X^{1}$. In this paper, we instead interpolate using i.i.d. Bernoulli random variables:

where $K_{n,\alpha}^{\theta}$ is defined through the formal power series

Next, we explain the basic strategy behind the self-consistent comparison method mentioned in (a) above. We choose a function

The next section is devoted to the definition of the model and the statement of results. We give an outline of the structure of the paper in Section 3.5 below.

Conventions

The fundamental large parameter is $N$. All quantities that are not explicitly constant may depend on $N$; we almost always omit the argument $N$ from our notation.

We use $\tau>0$ in various assumptions to denote a positive constant that may be chosen arbitrarily small. A smaller value of $\tau$ corresponds to a weaker assumption. All constants may $C,c$ depend on $\tau$, and we neither indicate nor track this dependence.

Model

In this section we define our model, list our key assumptions, and explain the basic structure of the asymptotic eigenvalue density $\varrho$.

We consider the $M\times M$ matrix $Q\mathrel{\vbox{\hbox{.}\hbox{.}}}=TXX^{*}T^{*}$, where $T$ is a deterministic $M\times\widehat{M}$ matrix and $X$ a random $\widehat{M}\times N$ matrix. We regard $N$ as the fundamental parameter and $M\equiv M_{N}$ and $\widehat{M}\equiv\widehat{M}_{N}$ as depending on $N$. Here, and throughout the following, we omit the index $N$ from our notation, bearing in mind that all quantities that are not explicitly constant (such as the constant $\tau$) may depend on $N$. For simplicity, we always make the assumption that

(This assumption may relaxed to $\log M\asymp\log\widehat{M}\asymp\log N$ with some extra work; see KYY . We do not pursue this direction here.) We introduce the dimensional ratio

provided $\tau>0$ is chosen small enough.

We assume that the entries $X_{i\mu}$ of the $\widehat{M}\times N$ matrix $X$ are independent (but not necessarily identically distributed) real-valued random variables satisfying

The population covariance matrix is defined as

denote the empirical spectral density of $\Sigma$. We suppose that

This latter assumption means that the spectrum of $\Sigma$ cannot be concentrated at zero.

Sometimes it will be convenient to make the following stronger assumption on $T$:

The assumption (2.9) will frequently simplify the presentation and the proofs. Thanks to our general assumptions (2.7) and (2.8), it will always be relatively easy to remove (2.9). In particular, we emphasize that the assumption $\Sigma>0$ is purely qualitative in nature, and is made in order to simplify expressions involving the inverse of $\Sigma$. The case of $\Sigma\geqslant 0$ may always be easily obtained by considering $\Sigma+\varepsilon I_{M}$ and then taking $\varepsilon\downarrow 0$ at fixed $N$. We refer to Section 11.1 below for the details on how to remove the assumption (2.9).

To avoid repetition, we summarize our basic assumptions for future reference.

We suppose that (2.1), (2.4), (2.5), (2.7), and (2.8) hold.

2. Asymptotic eigenvalue density

Moreover, $m(z)$ is the Stieltjes transform of a probability measure $\varrho$ with bounded support in $[0,\infty)$.

The rest of this subsection is devoted to a discussion of the basic properties of the asymptotic eigenvalue density $\varrho$. Much this discussion is well known; see e.g. BS2 ; SC ; HHN . The reader interested only in the principal components of $Q$, i.e. its top eigenvalues, can skip this subsection and proceed directly to the results in Theorem 3.14 (with $k=1$) and Corollary 3.19 (i).

Let $n\mathrel{\vbox{\hbox{.}\hbox{.}}}=\lvert\operatorname{supp}\pi\setminus\{0\}\rvert$ be the number of distinct nonzero eigenvalues of $\Sigma$, and write

In Figures 2.1 and 2.2, we illustrate the graph of $f$ for the cases $\phi<1$ and $\phi>1$ respectively. In Figure 2.3 we plot the density of $\varrho$ for the examples from Figures 2.1 and 2.2. The behaviour of $\varrho$ may be entirely understood by an elementary analysis of $f$.

We have $\lvert\mathcal{C}\cap I_{0}\rvert=\lvert\mathcal{C}\cap I_{1}\rvert=1$ and $\lvert\mathcal{C}\cap I_{i}\rvert\in\{0,2\}$ for $i=2,\dots,n$.

We deduce from Lemma 2.4 that $\lvert\mathcal{C}\rvert=2p$ is even. We denote by $x_{1}\geqslant x_{2}\geqslant\cdots\geqslant x_{2p-1}$ the $2p-1$ critical points in $I_{1}\cup\dots\cup I_{n}$, and by $x_{2p}$ the unique critical point in $I_{0}$. For $k=1,\dots,2p$ we define the critical values $a_{k}\mathrel{\vbox{\hbox{.}\hbox{.}}}=f(x_{k})$.

The following result gives the basic structure of $\varrho$.

Our assumptions on $\pi$ (i.e. on the spectrum of $\Sigma$) take the form of the following regularity conditions.

We say that the edge $k=1,\dots,2p$ is regular if

The edge regularity condition from Definition 2.7 (i) has previously appeared (in a slightly different form) in several works on sample covariance matrices. For the rightmost edge $k=1$, it was introduced in ElK2 and was subsequently used in the works SL ; Ona ; BaoPanZhou on the distribution of eigenvalues near the top edge $a_{1}$. For general $k$, it was introduced in HHN . The second condition of (2.14) states that the gap in the spectrum of $\varrho$ adjacent to the edge $a_{k}$ does not close for large $N$; the third condition of (2.14) ensures a regular square-root behaviour of the spectral density near $a_{k}$, and in particular rules out outliers.

We conclude this subsection with a couple of examples verifying the regularity conditions of Definition 2.7.

We suppose that $n$ is fixed, and that $s_{1},\dots,s_{n}$ and $\phi\pi(\{s_{1}\}),\dots,\phi\pi(\{s_{n}\})$ all converge in $(0,\infty)$ as $N\to\infty$. We suppose that all critical points of $\lim_{N}f$ are nondegenerate, and that $\lim_{N}a_{i}>\lim_{N}a_{i+1}$ for $i=1,\dots,2p$. Then it is easy to check that, for small enough $\tau$, all edges $k=1,\dots,2p$ and bulk components $k=1,\dots,p$ are regular in the sense of Definition 2.7.

Results

In this section we state our main results for the sample covariance matrix $Q$ defined in Section 2. To ease readability, we state the anisotropic local law under three different sets of increasingly weak assumptions. In Section 3.2, we assume that all edges and bulk components are regular in the sense of Definition 2.7. In Section 3.3, we assume only regularity of a single edge or bulk component, and state the anisotropic local law in the vicinity of the corresponding edge or bulk component. As an application, we prove the edge universality near a regular edge. Finally, in Section 3.4 we give a general anisotropic law, where the concrete regularity assumptions from Definition 2.7 are replaced with a more general but abstract stability condition. The reader interested only in the principal components of $Q$ can proceed directly to the results in Theorem 3.14 (with $k=1$) and Corollary 3.19 (i).

In this preliminary subsection we introduce some basic notations and definitions. Our main results take the form of local laws, which relate the resolvents

to the Stieltjes transform $m$ of the asymptotic density $\varrho$. These local laws may be formulated in a simple, unified fashion under the assumption (2.9) using an $(M+N)\times(M+N)$ block matrix, which is a linear function of $X$.

We consistently use the letters $i,j\in\mathcal{I}_{M}$, $\mu,\nu\in\mathcal{I}_{N}$, and $s,t\in\mathcal{I}$. We label the indices of the matrices according to

The motivation behind this definition is that, assuming (2.9), a control of $G$ immediately yields control of the resolvents $R_{N}$ and $R_{M}$ via the identities

for $\mu,\nu\in\mathcal{I}_{N}$. Both of these identities may be easily checked using Schur’s complement formula.

Next, we introduce a deterministic matrix $\Pi$, which we shall prove is close to $G$ with high probability (in the sense of Definition 3.4 (iii) below).

We also extend $\Sigma$ to an $\mathcal{I}\times\mathcal{I}$ matrix

The following notion of a high-probability bound was introduced in EKY2 , and has been subsequently used in a number of works on random matrix theory. It provides a simple way of systematizing and making precise statements of the form “$\xi$ is bounded with high probability by $\zeta$ up to small powers of $N$”.

be two families of nonnegative random variables, where $U^{(N)}$ is a possibly $N$-dependent parameter set.

We say that $\xi$ is stochastically dominated by $\zeta$, uniformly in $u$, if for all (small) $\varepsilon>0$ and (large) $D>0$ we have

If $\xi$ is stochastically dominated by $\zeta$, uniformly in $u$, we use the notation $\xi\prec\zeta$. Moreover, if for some complex family $\xi$ we have $\lvert\xi\rvert\prec\zeta$ we also write $\xi=O_{\prec}(\zeta)$.

Because of (2.1), all (or some) factors of $N$ in Definition 3.4 could be replaced with $M$ or $\widehat{M}$ without changing the definition of stochastic domination.

2. The full spectrum

As explained at the beginning of this section, we first state the anisotropic local law for all arguments $z$ under the assumption that all edges and bulk components are regular in the sense of Definition 2.7. In the next subsection, we relax these assumptions by restricting the domain of $z$ to the vicinity of an edge or bulk component, and only requiring the regularity of the corresponding edge or bulk component.

Our main result is the following anisotropic local law. We introduce the fundamental control parameter

and the Stieltjes transform of the empirical eigenvalue density of $X^{*}T^{*}TX$,

Fix $\tau>0$. Suppose that (2.9) and Assumption 2.1 hold. Suppose moreover that every edge $k=1,\dots,2p$ satisfying $a_{k}\geqslant\tau$ and every bulk component $k=1,\dots,p$ is regular in the sense of Definition 2.7. Then

(Note that the presence of the factors $\underline{\Sigma}\!\,$ in (3.10) strengthens the result; by (2.7), they can be trivially dropped to obtain a weaker estimate. They ensure that the control of the error is stronger in directions where the covariance $\Sigma$ is small.)

Outside the support of the asymptotic spectrum, one has stronger control all the way down to the real axis.

Fix $\tau>0$. Suppose that (2.9) and Assumption 2.1 hold. Suppose moreover that every edge $k=1,\dots,2p$ satisfying $a_{k}\geqslant\tau$ is regular in the sense of Definition 2.7 (i). Then

uniformly in $z\in[\tau,\tau^{-1}]\times(0,\tau^{-1}]$ satisfying $\operatorname{dist}(E,\operatorname{supp}\varrho)\geqslant N^{-2/3+\tau}$.

An explicit expression for the error term in (3.12) may be obtained from (A.7) below.

Theorem 3.7 can be used to obtain a complete picture of the outlier eigenvalues of $Q$ in the case where a bounded number of eigenvalues of $\Sigma$ are changed to some arbitrary values (in particular possibly violating the regularity assumption from Definition 2.7 (i)). We note that the outliers may also lie between bulk components of $\varrho$. The analysis is similar to the one performed in KYY for the case $\Sigma=I_{M}$; we omit the details.

In the remainder of this subsection, we state several corollaries of Theorem 3.6 where the assumption (2.9) is removed. From Theorem 3.6 it is not hard to deduce the following result on the resolvents $R_{N}$ and $R_{M}$, defined in (3.1).

Fix $\tau>0$. Suppose that Assumption 2.1 holds. Suppose moreover that every edge $k=1,\dots,2p$ satisfying $a_{k}\geqslant\tau$ and every bulk component $k=1,\dots,p$ is regular in the sense of Definition 2.7. Then

Theorem 3.7 has an analogous corollary for $R_{N}$ and $R_{M}$ for $z$ satisfying $\operatorname{dist}(E,\operatorname{supp}\varrho)\geqslant N^{-2/3+\tau}$, whereby the right-hand sides of (3.13) and (3.14) are replaced with the right-hand side of (3.12); we omit the precise statement.

The following consistency check may be applied to the deterministic matrices on the left-hand sides of (3.13) and (3.14). If, in the identity $\frac{1}{M}\operatorname{Tr}R_{M}=\frac{1}{M}\operatorname{Tr}R_{N}+\frac{\phi-1}{\phi}\frac{1}{z}$, we replace $\operatorname{Tr}R_{M}$ and $\operatorname{Tr}R_{N}$ with the corresponding deterministic matrices from the left-hand sides of (3.13) and (3.14), we recover (2.11).

We conclude this subsection with another consequence of Theorem 3.6 – eigenvalue rigidity. We denote by

the nontrivial eigenvalues of $Q$, and by $\gamma_{1}\geqslant\gamma_{2}\geqslant\cdots\geqslant\gamma_{M\wedge N}$ the classical eigenvalue locations defined through

If $p\geqslant 1$, it is convenient to relabel $\lambda_{i}$ and $\gamma_{i}$ separately for each bulk component $k=1,\dots,p$. To that end, for $k=1,\dots,p$ we define the classical number of eigenvalues in the $k$-th bulk component through

Fix $\tau>0$. Suppose that Assumption 2.1 holds. Suppose moreover that every edge $k=1,\dots,2p$ satisfying $a_{k}\geqslant\tau$ and every bulk component $k=1,\dots,p$ is regular in the sense of Definition 2.7. Then we have for all $k=1,\dots,p$ and $i=1,\dots,N_{k}$ satisfying $\gamma_{k,i}\geqslant\tau$ that

As in (BEKYY, , Theorem 2.8), Corollary 3.9 and Theorem 3.12 imply the complete delocalization, with respect to an arbitrary deterministic basis, of the eigenvectors of $TXX^{*}T^{*}$ and $X^{*}T^{*}TX$ associated with eigenvalues $\lambda_{i}$ satisfying $\gamma_{i}\geqslant\tau$.

Note that Theorem 3.12 in particular implies an exact separation of the eigenvalues into connected components, whereby the number of eigenvalues in the $k$-th connected component is with high probability equal to the deterministic number $N_{i}$. This phenomenon of exact separation was first established in BS2 ; BS3 .

3. Individual spectral regions and edge universality

In this subsection we elaborate on the results of Subsection 3.2 by only requiring the regularity of the edge or bulk component near which the spectral parameter $z$ lies. To that end, for fixed $\tau,\tau^{\prime}>0$ we define the subdomains

Fix $\tau>0$. Suppose that Assumption 2.1 holds. Suppose that the edge $k=1,\dots,2p$ is regular in the sense of Definition 2.7 (i). Then there exists a constant $\tau^{\prime}>0$, depending only on $\tau$, such that the following holds.

Let $\hat{k}\mathrel{\vbox{\hbox{.}\hbox{.}}}=\lfloor{(k+1)/2}\rfloor$ be the bulk component to which the edge $k$ belongs. Then for all $i=1,\dots,N_{\hat{k}}$ satisfying $\gamma_{\hat{k},i}\in[a_{k}-\tau^{\prime},a_{k}+\tau^{\prime}]$ we have

Fix $\tau,\tau^{\prime}>0$. Suppose that Assumption 2.1 holds. Suppose that the bulk component $k=1,\dots,2p$ is regular in the sense of Definition 2.7 (ii).

Suppose that at least one of the two edges $2k$ and $2k-1$ is regular in the sense of Definition 2.7 (i). Then for all $i=1,\dots,N_{k}$ satisfying $\gamma_{k,i}\in[a_{2k}+\tau^{\prime},a_{2k-1}-\tau^{\prime}]$ we have $\lvert\lambda_{k,i}-\gamma_{k,i}\rvert\prec N^{-1}$.

Fix $\tau,\tau^{\prime}>0$. Suppose that Assumption 2.1 holds.

As in Section 3.2, if $\operatorname{dist}(E,\operatorname{supp}\varrho)\geqslant N^{-2/3+\tau}$ then the error parameters $\Psi(z)$ on the right-hand sides of (3.10), (3.13), and (3.14) in (i) and (ii) of Theorems 3.14 and 3.16 can be replaced with the smaller quantity $\sqrt{\operatorname{Im}m(z)/(N\eta)}$ and the lower bound $\eta\geqslant N^{-1+\tau}$ relaxed to $\eta>0$. (See Theorem 3.7.) We omit the detailed statement.

Like Theorem 3.18, Corollary 3.19 (ii) also holds for the joint distribution at several regular edges. In particular, for the case $\beta=2$ and under the assumption that the top and bottom edges of $\varrho$ are regular, we obtain the universality of the condition number of $Q$.

Corollary 3.19 (i) for $\beta=1,2$ was previously established in SL under the assumption that $\Sigma$ is diagonal, corresponding to uncorrelated population entries. Before that, Corollary 3.19 (i) for $\beta=2$ and diagonal $\Sigma$ was established in BaoPanZhou , following the results of ElK2 ; Ona in the complex Gaussian case. Corollary 3.19 (ii) was recently established for Gaussian $X$ in HHN .

4. The general anisotropic local law

In this subsection we conclude the statement of our results with a general anisotropic local law, which takes the form of a black box yielding the anisotropic local law for $Q$ assuming it has been established for a much simpler matrix. This latter result may be proved independently. In particular, this black box formulation may be used to establish the anisotropic local law in cases where the regularity assumptions from Definition 2.7 fail; we do not pursue such generalizations here. Aside from its great generality, this black box formulation also makes precise the three Steps (A) – (C) mentioned in the introduction, which constitute the basic strategy of our proof.

We begin by introducing some basic terminology.

The main conclusion of this paper is that the anisotropic local law holds for general $X$ and $T$ provided that the entrywise local law holds for Gaussian $X$ and diagonal $T$. This latter case may be established independently, as we illustrate in Section 5 and Appendix A.

Aside from Assumption 2.1, the only assumption that we shall need is

This assumption holds for instance under the regularity assumptions of Definition 2.7 (see Lemmas A.4, A.6, and A.8 below). Clearly, we always have $\lvert 1+m(z)\sigma_{i}\rvert>0$ (see (2.11)), and (3.20) is a uniform version of this bound. Generally, the assumption (3.20) is necessary to guarantee that the generalized matrix entries of $(Q-z)^{-1}$ (or, alternatively, of $G(z)$) remain bounded. Indeed, in Corollary 3.9 we saw that the generalized entries of $(Q-z)^{-1}$ are close to those of $-z^{-1}(1+m(z)\Sigma)^{-1}$.

The next theorem shows that the hypotheses in (i) and (ii) of Theorem 3.21 may be verified under a stability condition on the spectrum of $\Sigma$, made precise in Definition 5.4 below.

5. Outline of the paper

Having completed the proof of the anisotropic local law, we prove the averaged local law (Theorem 3.21 (ii)) in Section 9. This will conclude the proof of Theorems 3.21 and 3.22. At the end of Section 9, we explain how to deduce Theorem 3.6, Corollary 3.9, and Theorems 3.14 (i)–(ii), 3.15 (i)–(ii), and 3.16.

In Section 10 we prove the rigidity of the eigenvalues (Theorems 3.12, 3.14 (iii), and 3.15 (iii)) and the universality of their joint distribution near the edges (Theorem 3.18). Next, in Section 11 we explain how to remove the assumption (2.9) and how to extend all of our results from the matrix $Q$ to the matrix $\dot{Q}$.

In Section 12, as a further illustration of the self-consistent comparison method, we present and prove analogous results for deformed Wigner matrices.

Basic tools

The rest of this paper is devoted to the proofs. In this preliminary section we collect various identities from linear algebra and simple estimates that we shall use throughout the paper.

We always use the following convention for matrix multiplication.

Suppose (2.9). Define the $\mathcal{I}\times\mathcal{I}$ matrices

as well as the $\mathcal{I}_{M}\times\mathcal{I}_{M}$ matrix

and the $\mathcal{I}_{N}\times\mathcal{I}_{N}$ matrix

Throughout the following we frequently omit the argument $z$ from our notation.

Since $H(z)$ and $G(z)$ are only defined under the assumption (2.9), we shall always tacitly assume (2.9) whenever we use them. Note that under the assumption (2.9) we have $R_{N}=G_{N}$.

For $S\subset\mathcal{I}$ we define the minor $H^{(S)}\mathrel{\vbox{\hbox{.}\hbox{.}}}=(H_{st}\mathrel{\vbox{\hbox{.}\hbox{.}}}s,t\in\mathcal{I}\setminus S)$. We also write $G^{(S)}\mathrel{\vbox{\hbox{.}\hbox{.}}}=(H^{(S)})^{-1}$. The matrices $G_{N}^{(S)}$ and $G_{M}^{(S)}$ are defined similarly. We abbreviate $(\{s\})\equiv(s)$ and $(\{s,t\})\equiv(st)$.

Suppose that $\Sigma$ is diagonal. Then for $i\in\mathcal{I}_{M}$ we have

For $i\in\mathcal{I}_{M}$ and $\mu\in\mathcal{I}_{N}$ we have

In addition, if $\Sigma$ is diagonal, we have

For $r\in\mathcal{I}$ and $s,t\in\mathcal{I}\setminus\{k\}$ we have

All of the identities from (i)–(v) hold for $G^{(S)}$ instead of $G$ if $S\subset\mathcal{I}_{N}$ or $S\subset\mathcal{I}$ and $\Sigma$ is diagonal.

The identities (4.3), (4.4), and (4.6) follow from Schur’s complement formula. The remaining identities follow easily from resolvent identities that have been previously derived in EYY1 ; EKYY2 ; they are summarized e.g. in (EKYY4, , Lemma 4.5). ∎

Next, we introduce the spectral decomposition of $G$. We use the notation

for the singular value decomposition of $\Sigma^{1/2}X$, where

Moreover, for $i\in\mathcal{I}_{M}$ and $\mu\in\mathcal{I}_{N}$ we have

Finally, the estimates (4.16)–(4.20) remain true for $G^{(S)}$ instead of $G$ if $S\subset\mathcal{I}_{N}$ or $S\subset\mathcal{I}$ and $\Sigma$ is diagonal.

In order to prove (4.18), we use (4.3) to write

In order to prove (4.19), we use (4.3) and (4.12) to get

Finally, the same estimates for $G^{(S)}$ instead of $G$ follow using a trivial modification of the above argument. ∎

The following result may be used to estimate the factors $\lVert X^{*}X\rVert$ in Lemma 4.6 with high probability. It follows from (BEKYY, , Theorem 2.10).

Under the assumptions (2.1), (2.4), and (2.5), there exists a constant $C>0$ such that $\lVert X^{*}X\rVert\leqslant C$ with high probability.

Using Lemma 4.8, we observe that we may improve (4.16) provided we settle for a high-probability instead of a deterministic statement.

The claim is an easy consequence of the first identity of (4.3) combined with (4.2). ∎

We conclude this section with the following basic properties of $m$, which can be proved as in BaoPanZhou and the references therein.

Fix $\tau>0$ and suppose that (2.3), (2.7), and (2.8) hold. Then there exists a constant $C>0$ such that

In particular, from the upper bound in (4.21) we deduce that $\varrho$ has a bounded density on $[\tau,\infty)$.

Entrywise local law for diagonal ΣΣ\Sigma

In this section we prove Theorem 3.22, hence performing Step (A) of the proof mentioned in the introduction. The proof of Theorem 3.22 is similar to previous proofs of local entrywise laws, such as PY ; BEKYY . We follow the basic approach of (BEKYY, , Section 4), and only give the details where the argument departs significantly from that of BEKYY .

The main novel observation of this section is that the equation (2.11) arises very easily from the random matrix model by a double application of Schur’s complement formula. Heuristically, this may be seen using the identities (4.4) and (4.6). Indeed, suppose that $G_{\mu\mu}\approx m$ for $\mu\in\mathcal{I}_{N}$. We ignore the random fluctuations in (4.4) to get

Similarly, ignoring the random fluctuations in (4.6), we get

Plugging (5.2) into (5.1) yields (2.11). In this section we give a rigorous justification of these approximations.

In this subsection we establish the following weaker version of Theorem 3.22. It is analogous to (BEKYY, , Proposition 4.2).

The rest of this subsection is devoted to the proof of Proposition 5.1. For each $i\in\mathcal{I}_{M}$ we define

Recalling (2.11), we find that the functions $m$ and $m_{i}$ satisfy

Next, we define the random control parameters

We extend the definitions of $\sigma_{i}$ and $m_{i}$ for $i\in\mathcal{I}_{M}$ by setting $\sigma_{\mu}\mathrel{\vbox{\hbox{.}\hbox{.}}}=1$ and $m_{\mu}\mathrel{\vbox{\hbox{.}\hbox{.}}}=m$ for $\mu\in\mathcal{I}_{N}$. We may therefore write

Moreover, we define the averaged control parameters

For $s\in\mathcal{I}$ we introduce the conditional expectation

Using (4.6) we get for $i\in\mathcal{I}_{M}$

and using (4.4) we get for $\mu\in\mathcal{I}_{N}$

In analogy to (BEKYY, , Section 4), we define the $z$-dependent event \Xi\mathrel{\vbox{\hbox{.}\hbox{.}}}=\bigl{\{}{\Lambda\leqslant(\log N)^{-1}}\bigr{\}} and the control parameter

The following estimate is analogous to (BEKYY, , Lemma 4.4).

The proof relies on the identities from Lemma 4.4 and large deviation estimates, like that of (BEKYY, , Lemma 4.4) and (PY, , Theorems 6.8 and 6.9). Note first that (5.4), combined with (4.21) and (5.3), yields

Using (4.11) and a simple induction argument, it is not hard to conclude that

for any $S\subset\mathcal{I}$ and $t\in\mathcal{I}\setminus S$ satisfying $\lvert S\rvert\leqslant C$.

Let us first estimate $\Lambda_{o}$ in (5.9). We shall in fact prove that

Let us start with $G_{ij}$ for $i\neq j\in\mathcal{I}_{M}$. Using (4.7), (5.12), and a large deviation estimate (see (BEKYY, , Lemma 3.1)), we find

where in the first step we used (4.11) and (5.12). This yields (5.13) for $s,t\in\mathcal{I}_{M}$.

An analogous argument for $Z_{s}$ (see e.g. (EKYY4, , Lemma 5.2)) completes the proof of (5.9).

In order to prove (5.10), we proceed similarly. For $\eta\geqslant 1$, we proceed as above to get $\lvert Z_{s}\rvert\prec N^{-1/2}$, where we used that $\lVert G^{(s)}\rVert\leqslant C$ by (4.16). Similarly, as in (5.14) we get

where in the last step we used that $\operatorname{Im}G_{\mu\mu}^{(ij)}\leqslant C$, by (4.16). Moreover, from (4.1) and (4.3) we get immediately that $\lvert G_{ii}\rvert\leqslant C\sigma_{i}$, and a similar argument for $G^{(i)}$ implies that $\lvert G_{jj}^{(i)}\rvert\leqslant C\sigma_{j}$. This concludes the proof. ∎

As in (BEKYY, , Lemma 4.7), it is easy to derive from (5.16) and Lemma 5.2 that

Plugging this and (5.15) into (5.16) yields

Suppose that the assumptions of Theorem 3.22 hold. Define

If $\operatorname{Im}z<1$ suppose also that

2. Fluctuation averaging and proof of Theorem 3.22

The estimate (5.21) is a trivial extension of (BEKYY, , Lemma 4.9) and (EKYY4, , Theorem 4.7). The estimate (5.22) may be proved using exactly the same method, explained in (EKYY4, , Appendix B). The only complication in the proof is that the coefficients $\frac{\sigma_{i}^{2}}{(1+m_{N}\sigma_{i})^{2}}$ are random and depend on $i$. Using (4.11), this is dealt with in the proof of (EKYY4, , Appendix B) by writing, for any $j\in\mathcal{I}_{N}$,

Anisotropic local law for Gaussian X𝑋X

We now begin the proof of Theorem 3.21, which consists of Sections 6–9. In this section we perform the first step of the proof, by establishing Theorem 3.21 for the special case that $X=X^{\text{\rm{Gauss}}}$ is Gaussian. This corresponds to Step (B) of the proof mentioned in the introduction.

Theorem 3.21 holds if $X=X^{\text{\rm{Gauss}}}$ is Gaussian.

The rest of this section is devoted to the proof of Proposition 6.1. We shall in fact prove the following result.

In this proof we abbreviate $G^{D}\equiv G$. Using (6.1) with $O_{M}=U^{*}$ and $O_{N}=1$, we find that it suffices to prove

for all $s,t\in\mathcal{I}$. In components, this reads

for $i,j\in\mathcal{I}_{M}$ and $\mu,\nu\in\mathcal{I}_{N}$.

The estimate (6.4) is trivial by assumption. What remains is the proof of (6.2) and (6.3). It is based on the polynomialization method developed in (BEKYY, , Section 5). The argument is very similar to that of BEKYY , and we only outline the differences.

Let us begin with (6.2). By the assumption $\lvert G_{kk}-m_{k}\rvert\prec\Psi\sigma_{k}^{2}$ and orthogonality of $U$, we have

for $T\subset\mathcal{I}_{M}$ (which follows from (4.7)), (4.6), and

as follows from (4.6), (5.4), and $(XG^{(iT)}X^{*})_{ii}-m=O_{\prec}(\sigma_{i}^{2}\Psi)$ (which may itself be deduced from (4.6)). We omit further details.

The rest of the argument is the same as before. ∎

Self-consistent comparison I: the main argument

In this section we establish Theorem 3.21 (i) under the additional assumption that the third moment of all entries of $X$ is zero.

Suppose that the assumptions of Theorem 3.21 hold. Suppose moreover that $X$ satisfies the additional condition

The rest of this section is devoted to the proof of Proposition 7.1. This is the heart of the proof of the anisotropic local law.

Computing the derivatives on the left-hand side of (7.3) leads to a product of terms of the form

and their complex conjugates (here we omit the argument $X^{\theta}$). Terms of type (i) are simply kept as they are; they will be put into the second term on the right-hand side of (7.3). Terms of type (ii) are the key to the gain that allows us to compensate losses from several other terms, such as the terms of type (iii) (see blow). The gain is obtained in combination with the summation over $i$ and $\mu$ on the left-hand side of (7.3), according to estimates of the form

which allows us to estimate the right-hand side of (7.5) by $N^{C\delta}\Psi^{2}$. The a priori bound (7.6) will be obtained from a bootstrapping, as explained below.

The preceding discussion was rather cavalier with the various constants $C$ in the factors $N^{C\delta}$. In fact, some care has to be taken to ensure that the factors of $N^{C\delta}$ arising from the use of the a priori bounds (7.6) and (7.7) are compensated by a sufficiently large number of factors of the form (7.5). Moreover, in order to be able to iterate this argument from large to small scales, as explained in the next paragraph, we need an explicit bound of the constant $C$ in (7.3) in terms of the constant $C$ in (7.6).

Since $\delta>0$ is fixed, the bootstrapping consists of $O(\delta^{-1})$ steps. The resulting estimates contain an extra factor $N^{C\delta}$. Since $\delta>0$ can be made arbitrarily small, the claim will follow on all scales.

2. Bootstrapping on the spectral scale

The proof consists of a bootstrap argument from larger scales to smaller scales in multiplicative increments of $N^{-\delta}$. Here

is fixed, where $C_{0}>0$ is a universal constant that will be chosen large enough in the proof ($C_{0}=25$ will work). For any $\eta\geqslant N^{-1}$ we define $\eta_{0}\leqslant\eta_{1}\leqslant\dots\leqslant\eta_{L}$, where

The bootstrapping is started by the following result.

What remains is the proof of Lemma 7.4. We shall estimate the random variable

3. Interpolation

We use the interpolation outlined in Section 7.1.

Introduce the notations $X^{0}\mathrel{\vbox{\hbox{.}\hbox{.}}}=G^{\text{\rm{Gauss}}}$ and $X^{1}\mathrel{\vbox{\hbox{.}\hbox{.}}}=X$. For $u\in\{0,1\}$, $i\in\mathcal{I}_{M}$, and $\mu\in\mathcal{I}_{N}$, denote by $\rho^{u}_{i\mu}$ the law of $X^{u}_{i\mu}$. For $\theta\in$ we define the law

Let $(X^{0},X^{\theta},X^{1})$ be a triple of independent $\mathcal{I}_{M}\times\mathcal{I}_{N}$ random matrices, where for $u\in\{0,\theta,1\}$ the matrix $X^{u}=(X^{u}_{i\mu})$ has law

We shall prove Lemma 7.5 by interpolation between the ensembles $X^{0}$ and $X^{1}$. The bound for the Gaussian case $X^{0}$ is given by the following result.

Lemma 7.5 holds if $X$ is replaced with $X^{0}$.

The basic interpolation formula is given by the following lemma, which follows from the fundamental theorem of calculus.

In order to prove Lemma 7.10, we compare the ensembles $X^{\theta,X^{0}_{i\mu}}_{(i\mu)}$ and $X^{\theta,X^{1}_{i\mu}}_{(i\mu)}$ via $X^{\theta,0}_{(i\mu)}$. Clearly, it suffices to prove the following result.

where we recall the definitions of $L(\eta)$ and $\eta_{l}$ from (7.10) and (7.11).

It suffices to estimate the first term. Setting $\eta_{-1}\mathrel{\vbox{\hbox{.}\hbox{.}}}=0$ and $\eta_{L+1}\mathrel{\vbox{\hbox{.}\hbox{.}}}=\infty$, we define the subsets of indices

and treat each $l$ separately. For $l=1,\dots,L$ we find

where in the second step we used that $\lambda_{k}\prec 1$, as follows from (2.7) and Lemma 4.8. This concludes the proof. ∎

where in the second step we used (3.20) and the definition of $\Pi$ from (3.5). The estimate (7.19) now follows easily using Lemma 7.14.

4. Expansion

Let $i\in\mathcal{I}_{M}$ and $\mu\in\mathcal{I}_{N}$. Define the $\mathcal{I}\times\mathcal{I}$ matrix $\Delta^{\lambda}_{(i\mu)}$ through

The following result provides a priori bounds for the entries of $G^{\theta,\lambda}_{(i\mu)}$.

Suppose that $y$ is a random variable satisfying $\lvert y\rvert\prec N^{-1/2}$. Then

for all $i\in\mathcal{I}_{M}$ and $\mu\in\mathcal{I}_{N}$.

We use (7.21) with $K\mathrel{\vbox{\hbox{.}\hbox{.}}}=10$, $\lambda^{\prime}\mathrel{\vbox{\hbox{.}\hbox{.}}}=y$, and $\lambda\mathrel{\vbox{\hbox{.}\hbox{.}}}=X_{i\mu}^{\theta}$, so that $G_{(i\mu)}^{\theta,\lambda}=G^{\theta}$. By (3.20), we have

To simplify notation, we introduce the function

The following result is easy to deduce from (7.21) and Lemma 7.14.

where we used that $X_{i\mu}^{1}$ has vanishing first and third moments, by (7.1), and its variance is equal to $1/N$. Recalling our goal (7.18), we therefore find that we only have to prove

for $n=4,\dots,4p$. Here we used the bounds (2.5).

holds for $n=4,\dots,4p$. Then (7.27) holds for $n=4,\dots,4p$.

To simplify notation, we abbreviate $f_{(i\mu)}\equiv f$ and $X^{\theta}_{i\mu}\equiv\xi$. The proof consists of a repeated application of the identity

for $l\leqslant 4p$, which follows from Lemmas 7.7 and 7.15. Fix $n=4,\dots,4p$. Using (7.29) we get

The claim now follows easily using (2.5). ∎

What therefore remains is to prove (7.28). Since it only involves the matrix ensemble $X^{\theta}$, for the remainder of the proof we abbreviate $X^{\theta}\equiv X$. Recalling the notation (7.24), we find from Lemma 7.16 that it suffices to prove the following result.

We conclude this subsection with a general digression on the algebra of the Bernoulli interpolation method underlying the proof of Proposition 7.1, and in particular establish the formula (1.8) from the introduction. Let $(X_{\alpha}^{0})_{\alpha}$ and $(X_{\alpha}^{1})_{\alpha}$ be arbitrary independent finite families of independent random variables. Define $X^{\theta}=(X^{\theta}_{\alpha})_{\alpha}$ as in (1.6). Then we get

where $X^{\theta,x}_{(\alpha)}$ denotes the family obtained from $X^{\theta}$ by replacing $X_{\alpha}^{\theta}$ with $x$. Fix $\alpha$ and abbreviate f(x)\mathrel{\vbox{\hbox{.}\hbox{.}}}=F\bigl{(}{X^{\theta,x}_{(\alpha)}}\bigr{)}, $\zeta\mathrel{\vbox{\hbox{.}\hbox{.}}}=X_{\alpha}^{1}$, $\zeta^{\prime}\mathrel{\vbox{\hbox{.}\hbox{.}}}=X_{\alpha}^{0}$, and $\xi\mathrel{\vbox{\hbox{.}\hbox{.}}}=X_{\alpha}^{\theta}$. We may assume that $\zeta$, $\zeta^{\prime}$, and $\xi$ are independent. We want to compute the difference

Since we are only interested in the algebra of the interpolation, we assume for simplicity that all random variables have finite exponential moments and that $f$ is analytic. (Otherwise, as in the computations above, the expansions have to be truncated.) Repeating the steps of the proof of Lemma 7.16, we find

5. Introduction of words and conclusion of the proof

In order to prove (7.30), we shall have to exploit the detailed structure of the derivatives in on the left-hand side of (7.30). The following definition introduces the basic algebraic objects that we shall use.

Next, we assign to each letter $*$ its value $[*]\equiv[*]_{i,\mu}\in\mathcal{I}$ through

(Our choice of the names of the two letters is suggestive of their value. Note, however, that it is important to distinguish the abstract letter from its value, which is an index in $\mathcal{I}$ and may be used as a summation index.)

If $n(w)\geqslant 1$ with $w$ as in (7.31) we define

for any $n=0,1,2,\dots$. This may be easily deduced from (7.21). Using Leibnitz’s rule we conclude that

To prove (7.30), it therefore suffices to prove that

for $4\leqslant n\leqslant 4p$ and words $w_{r}\in\mathcal{W}$ satisfying $\sum_{r}n(w_{r})=n$. (The proof of (7.33) is the same with slightly heaver notation.) Treating words $w_{r}$ with $n(w_{r})=0$ separately, we find that it suffices to prove

for $4\leqslant n\leqslant 4p$, $1\leqslant q\leqslant p$, and words $w_{r}\in\mathcal{W}$ satisfying $\sum_{r}n(w_{r})=n$, $n(w_{0})=0$, and $n(w_{r})\geqslant 1$ for $r\geqslant 1$. Note that we also have the bound $q\leqslant n$.

In order to estimate (7.35), we introduce the quantity

For $w\in\mathcal{W}$ we have the rough bound

Finally, for $n(w)=1$ we have the sharper bound

In addition to the high-probability bounds from Lemma 7.20, we have the rough estimate

for some $C>0$ and all $w\in\mathcal{W}$ satisfying $n(w)\leqslant 4p$; this follows easily from Definition 7.19 and Lemma 4.9.

By pigeonholing on the words appearing on the left-hand side of (7.35), if $n\leqslant 2q-2$ then there exist at least two words $w_{r}$ satisfying $n(w_{r})=1$. Using Lemma 7.20 we therefore get

From Lemma 4.6 combined with Lemma 4.8 we get

where in the second step we used (7.20), and in the last step the definition of $\Psi$. Inserting (7.41) into (7.40), we get using Lemma 7.7 and (7.39) that the left-hand side of (7.35) is bounded by

Using that $\Psi\geqslant cN^{-1/2}$, we find that the left-hand side of (7.35) is bounded by

where we used that $q\leqslant n$ and $n\geqslant 4$. Choose $C_{0}\geqslant 25$. Then, by assumption (7.9) on $\delta$ we have $N^{(C_{0}/2+12)\delta}\Psi\leqslant 1$. Moreover, if $n\geqslant 4$ and $n\geqslant 2q-1$ then $n\geqslant q+2$. We conclude that the left-hand side of (7.35) is bounded by

Now (7.35) follows using Young’s inequality. This concludes the proof of (7.30), and hence of Lemma 7.11. The proof of Proposition 7.1 is therefore complete.

Self-consistent comparison II: general X𝑋X

In this section and the next we prove the following result, which is Proposition 7.1 without the condition (7.1).

The proof of Proposition 8.1 builds on that of Proposition 7.1. Throughout this section, we take over the notations of Section 7 without further comment. In particular, as in (7.9), $C_{0}$ is some large enough constant (which may depend only on $\tau$) and $\delta>0$ is fixed and small, satisfying (7.9).

The assumption (7.1) was used in the proof of Proposition 7.1 only in (7.26), where it ensured that the summation over $n$ starts from $4$ instead of $3$. Without the assumption (7.1), we in addition have to estimate the term $n=3$ in (7.26). It therefore suffices to prove the following result.

As in Lemma 7.16, one may easily replace the matrix $X^{\theta,0}_{i\mu}$ in the definition of $f^{(3)}_{i\mu}(0)$ with $X^{\theta}$. As in Lemma 7.17, from now on we abbreviate $X^{\theta}\equiv X$. Thus, we find that in order to prove Lemma 8.2 it suffices to prove the following result, which complements Lemma 7.17.

Next, recall the definition of words from Definition 7.19. As in (7.33)–(7.35), Lemma 8.3 is proved provided we can show the following result, which is analogous to (7.35).

Beyond the factor of $N^{-1/2}$, the need to obtain the bound from (7.34) that is strong enough to close the self-consistent estimate presents significant difficulties, which we outline briefly. These difficulties are roughly of two types. (a) The off-diagonal entries of $G^{(\mu)}$ are in general not small; only entries of $G^{(\mu)}-\Pi$ are small. (b) A priori, using the estimate (7.19), the entries of $G^{(\mu)}$ are not bounded by $O_{\prec}(1)$ but by $O_{\prec}(N^{2\delta})$. This would yield a bound on the coefficients of the polynomial in $X_{\mu}$ that is a high power of $N^{2\delta}$, which may become too large to conclude the proof.

2. Tagged words

Lemma 8.4 holds provided it holds with $A$ replaced by $\widehat{A}$ in (7.34).

Next, from Definition 7.19 we conclude, analogously to the proof of Lemma 7.20, that if $n(w_{r})\geqslant 1$ for $1\leqslant r\leqslant q$ and $\sum_{r=1}^{q}n(w_{r})=3$ then

where in the last step we used (7.41). We therefore conclude using Lemma 7.7 that

If $n=0$ (i.e. $w$ is the empty word) we define

where we introduced the matrix \Pi^{(\mu)}\mathrel{\vbox{\hbox{.}\hbox{.}}}=\bigl{(}{\Pi_{st}\mathrel{\vbox{\hbox{.}\hbox{.}}}s,t\in\mathcal{I}\setminus\{\mu\}}\bigr{)}.

From (8.4), (8.3), and Definition 8.7 we find that for $w\in\mathcal{W}_{n}$ we have

Recalling Lemma 8.5, we conclude that, in order to prove Lemma 8.4, it suffices to prove the following result.

We shall require the following rough bounds, which refine those of Lemma 7.20.

Suppose that (8.1) holds. Let $(w,\sigma)$ be a tagged word. Then

Using a large deviation estimate (see (BEKYY, , Lemma 3.1)) combined with Lemmas 4.6 and 4.8 we get

where in the second step we used (8.1). Therefore,

for all $\sigma=0,1,2$. This concludes the proof. ∎

Using the large deviation estimates from (BEKYY, , Lemma 3.1), we find $\lvert Z_{\mu}\rvert\prec N^{-\tau/2+2\delta}$. Since $\lvert G_{\mu\mu}\rvert\prec N^{2\delta}$ by (8.1), we therefore deduce using $-\tau/2+2\delta<-2\delta$ that

Hence there exists a constant $K\equiv K(\tau)$ such that

Plugging in the definition of $Z_{\mu}$, we find

Now we replace the factors $G_{\mu\mu}$ on the left-hand side of (8.7) with the leading term in (8.15). From (8.15), (7.19), (8.16), and Lemma 7.7 we get

where the coefficients $\mathcal{Z}_{\mu,k}$ are $X^{(\mu)}$-measurable and satisfy the bound

Moreover, for any tagged word $(w,\sigma)$ we split

4. Degree counting

For the following we choose $q$, $w_{1},\dots,w_{p}$, and $\sigma_{1},\dots,\sigma_{r}$ as in Lemma 8.8. We introduce the abbreviation

Here the constant $C_{p}$ accounts for the immaterial constants depending on $p$ arising from the combinatorics of all partitions. For $L$, $\{d_{l}\}$, and $\{k_{l}\}$ as above, we may estimate

where we used (2.5). Now we use the estimate

which follows from (8.11), (8.12), and (8.1). Hence we get

where the maxima are subject to the same conditions as above.

Next, from $\sum_{l}(d_{l}+k_{l})=2k+d$ and $d_{l}+k_{l}\geqslant 2$ we deduce that

Together with $\sum_{l}[d_{l}-2]_{+}\leqslant\sum_{l}d_{l}=d$, this gives

where in the last step we used that $[d_{l}+k_{l}-2]_{+}\geqslant[d_{l}-2]_{+}$, and $\sum_{l}(2\wedge d_{l})+\sum_{l}[d_{l}-2]_{+}=d$. For the case $d=1$, we also used that $\sum_{l}(2\wedge d_{l})+\sum_{l}[d_{l}+k_{l}-2]_{+}-1\geqslant 1$.

We conclude that the left-hand side of (8.21) is bounded by

Plugging (8.33) and (8.34) into (8.32), and recalling Lemma 7.7, yields

5. Power counting: proof of (8.36)

as may be easily checked from Definition 8.7. (Recall that $n(w_{r})=0$ for $r\geqslant q+1$.) We conclude that (8.36) holds provided we can show that

The proof of (8.21), and hence of Lemma 8.8, is therefore complete. This concludes the proof of Lemma 8.4.

We have hence proved Proposition 8.1. Recalling Proposition 6.1, we conclude the proof of Theorem 3.21 (i).

Averaged local law

for some constant $c>0$. In particular, we recover Definition 3.20 (iii) by setting $\Phi=(N\eta)^{-1}$. In this section we prove the following result, which also completes the proof of Theorem 3.21.

The proof is similar to that of Propositions 7.1 and 8.1, and we only explain the differences. Note that now there is no bootstrapping, since the necessary a priori bounds are obtained from the anisotropic local law. In analogy to (7.15), we define

Following the argument leading up to (7.30) and Lemma 8.3 to the letter, we find that it suffices to prove that

For $n\geqslant 4$, the claim (9.2) easily follows from (9.3) and an application of Young’s inequality to the factors $r$ satisfying $n(w_{r})=0$ and $n(w_{r})\geqslant 1$; we omit the details. What remains, therefore, is to verify (9.2) for $n=3$.

which may be regarded as an improvement of Lemma 8.11. The proof of (9.5) is similar to that of Lemma 8.11, and we merely give a sketch. We have to estimate an expression of the form

where $s_{l}\in\{i,\nu_{1},\dots,\nu_{p}\}$ and $b_{i}\in\{0,1\}$. (This is simply the general form of a polynomial in $X$ of the correct degree.) The proof of (9.5) relies on the crucial observations that, by Theorem 3.21 (i),

Next, let $r\geqslant q+1$ satisfy $\sigma_{r}=0$, so that the left-hand side of (9.5) does not depend on $\nu_{r}$. There are $p-q-\sum_{s=q+1}^{p}\lvert\sigma_{s}\rvert$ such indices $r$, and for each such $r$ we find

As in the argument following Lemma 8.11, we conclude that the claim holds provided that

This concludes the proof of Theorem 3.21. We conclude this section by drawing consequences from Theorems 3.21 and 3.22.

Similarly, Theorems 3.15 (i) and 3.16 (i) follow from Lemmas A.6 and A.8 respectively, combined with Theorems 3.21 and 3.22 ∎

These follow immediately from Theorem 3.6 and Theorems 3.14 (i), 3.15 (i), and 3.16 (i) respectively, combined with (3.3), (3.4), and (3.5). ∎

How to remove the assumption (2.9) in the above proof is explained in Section 11.1.

Eigenvalue rigidity and edge universality

In the first part of this section we establish eigenvalue rigidity (Theorems 3.12, 3.14 (iii), and 3.15 (iii)). As a consequence of Theorem 3.6 we prove Theorem 3.7. In the second part of this section we establish the edge universality from Theorem 3.18. We assume (2.9) throughout this section; how to remove it is explained in Section 11.1 below.

For simplicity, we first prove Theorem 3.12, i.e. we assume that all edges and bulk components are regular. The proof consists of three steps. First, we prove that with high probability there are no eigenvalues at a distance greater than $N^{-2/3+\varepsilon}$ from the support of $\varrho$. Second, we prove that a neighbourhood of the $k$-th bulk component of $\varrho$ contains with high probability exactly $N_{k}$ eigenvalues (recall the definition (3.16)). Third, we use the averaged local law from Theorem 3.6 together with the first two steps to complete the proof.

We begin with the first step. We define the distance to the spectral edge through

Fix $\varepsilon\in(0,2/3)$. Under (2.9) and the assumptions of Theorem 3.12 we have

with high probability (recall Definition 4.7).

The argument is similar to that of the previous works EYY3 ; PY , and we only explain how to adapt it. From (BEKYY, , Theorem 2.10), we find that $\lVert X^{*}X\rVert\leqslant C_{0}$ with high probability for some large enough constant $C_{0}>0$ depending on $\tau$. From (2.7) we therefore deduce that $\lambda_{1}\leqslant C$ with high probability for some constant $C>0$ depending on $\tau$. It therefore suffices to prove that, after a possible decrease of $\tau$,

In order to prove (10.2), it suffices to prove that $\operatorname{Im}m_{N}\prec N^{-\varepsilon/2}(N\eta)^{-1}$. By (A.7) and Lemmas A.4, A.6, and A.8, we have

It is easy to check that the control parameter on the right-hand side of (10.4) satisfies (9.1), so that by Proposition 9.1 and Theorem 3.6 it suffices to prove (10.4) for diagonal $\Sigma$. The proof is identical to that of Section 5, except that we use the strong stability of (2.11) from Definition A.2, which is established in Lemmas A.5, A.6, and A.8. This concludes the proof of (10.4), and hence of (10.2).

With applications to the deformed Wigner matrices in Section 12 in mind, we give another proof of (10.2), which is based on (BS2, , Section 6). Using a partial fraction decomposition, one easily finds that there exist universal constants $C_{n,k}$ such that

Similarly, we have (with the same $C_{n,k}$ and $z_{k}$)

for all $k=1,\dots,n$. Setting $n\mathrel{\vbox{\hbox{.}\hbox{.}}}=\lceil{2/\varepsilon}\rceil$, we get from (10.5) and (10.6) that

where in the last step we used that $\lvert x-E\rvert\geqslant\kappa$ for $x\in\operatorname{supp}\varrho$. This immediately implies that with high probability there is no eigenvalue in $[E-\eta,E+\eta]$. ∎

The second step represents most of the work. It is a counting argument, based on a continuous deformation of the matrix $Q$ to another matrix for which the claim is obvious. Since the eigenvalues depend continuously on the deformation parameter and each intermediate matrix satisfies a gap condition from Lemma 10.1, we shall be able to conclude that the number of eigenvalues in a neighbourhood of the $i$-th component does not change under the deformation. We shall in fact need two deformations: one which deforms the original matrix $Q$ to a Gaussian one, $Q^{\text{\rm{Gauss}}}$, with the same expectation $\Sigma$ as $Q$, and another which deforms the Gaussian matrix $Q^{\text{\rm{Gauss}}}$ to another Gaussian matrix where some eigenvalues of $\Sigma$ have been increased.

For $k=1,\dots,p-1$, we introduce the number of eigenvalues to the right of the $k$-th gap,

Under (2.9) and the assumptions of Theorem 3.12 we have $\Upsilon_{k}=\sum_{l\leqslant k}N_{l}$ with high probability.

As explained above, the first step in the proof of Proposition 10.2 is a deformation of the general matrix $X$ to a Gaussian one.

Let $X$ be general and $X^{\text{\rm{Gauss}}}$ a Gaussian matrix of the same dimensions. Suppose that (2.9) and the assumptions of Theorem 3.12 hold. If $\Upsilon_{k}=\sum_{l\leqslant k}N_{l}$ with high probability under the law of $X^{\text{\rm{Gauss}}}$, then $\Upsilon_{k}=\sum_{l\leqslant k}N_{l}$ with high probability under the law of $X$.

Let $X_{1}\mathrel{\vbox{\hbox{.}\hbox{.}}}=X$ and $X_{0}\mathrel{\vbox{\hbox{.}\hbox{.}}}=X^{\text{\rm{Gauss}}}$ be independent. For $t\in$ define $X(t)\mathrel{\vbox{\hbox{.}\hbox{.}}}=\sqrt{t}X_{1}+\sqrt{1-t}X_{0}$ and denote by $\lambda_{i}(t)$ the eigenvalues of $X(t)\Sigma X(t)^{*}$. We also write $\Upsilon_{k}(t)$ for $\Upsilon_{k}$ defined in terms of $\lambda_{i}(t)$. Note that Lemmas 4.8 and 10.1 hold for $X(t)$ uniformly in $t\in$. Recalling (2.7), we deduce that there exists a constant $C>0$ such that

with high probability, uniformly in $t,s\in$ and $i$. The claim now follows by considering $t$ in the lattice $0,1/K,2/K,\dots,1$, where $K$ is chosen large enough that $CK^{-1/2}\leqslant(a_{2k}-a_{2k+1})/10$ and $C$ is the constant from (10.8). Indeed, suppose $\Upsilon_{k}(t)=\sum_{l\leqslant k}N_{l}$ with high probability. Then we use (10.8) and the gap from Lemma 10.1 to deduce that $\Upsilon_{k}(t+K^{-1})=\sum_{l\leqslant k}N_{l}$ with high probability. The claim for $t=1=K/K$ therefore follows by induction. ∎

Using Lemma 10.3, in order to prove Proposition 10.2 it suffices to prove the following result.

Suppose that (2.9) and the assumptions of Theorem 3.12 hold, that $X$ is Gaussian, and that $\Sigma$ is diagonal. Then $\Upsilon_{k}=\sum_{l\leqslant k}N_{l}$ with high probability.

Abbreviate $d_{k}\mathrel{\vbox{\hbox{.}\hbox{.}}}=\sum_{l\leqslant k}N_{l}$. We write the diagonal matrix $\Sigma=\operatorname{diag}(\sigma_{1},\sigma_{2},\dots,\sigma_{M})$ in the block form $\Sigma=\operatorname{diag}(\Sigma_{1},\Sigma_{2})$, where $\Sigma_{1}$ contains the $d_{k}$ top eigenvalues of $\Sigma$. We introduce the deformed covariance matrix $\Sigma(t)\mathrel{\vbox{\hbox{.}\hbox{.}}}=\operatorname{diag}(t\Sigma_{1},\Sigma_{2})$. In particular, $\Sigma(1)=\Sigma$. The idea is to increase $t$ until the claim for $\Sigma$ replaced by $\Sigma(t)$ may be deduced from simple linear algebra. Then we use a continuity argument to compare $\Sigma(t)$ to $\Sigma$.

We add the argument $t$ to quantities to indicate that they are defined in terms of $\Sigma(t)$ instead of $\Sigma$. Using (A.5) (see also Figure 2.1), it is not hard to find that the gap $a_{2k}(t)-a_{2k+1}(t)$ is increasing in $t$, and that

for some constants $c,C>0$. Using that $\lVert\Sigma(t)-\Sigma(s)\rVert\leqslant C\lvert t-s\rvert$, we deduce from Lemma 4.8 that $\lVert Q(t)-Q(s)\rVert\leqslant C\lvert t-s\rvert$ with high probability. Now let $T$ be a fixed large time to be chosen later. By a continuity argument using Lemma 10.1 we therefore find that there exists a constant $K\equiv K(T)$ such that for any $t\in[1,T]$ we have