The local relaxation flow approach to universality of the local statistics for random matrices

Introduction

A central question concerning random matrices is the universality conjecture which states that local statistics of eigenvalues of large $N\times N$ square matrices $H$ are determined by the symmetry type of the ensembles but are otherwise independent of the details of the distributions. In particular they coincide with that of the corresponding Gaussian ensemble. The most commonly studied ensembles are

(i) hermitian, symmetric and quaternion self-dual matrices with identically distributed and centered entries that are independent (subject to the natural restriction of the symmetry);

(ii) sample covariance matrices of the form $H=A^{*}A$, where $A$ is an $M\times N$ matrix with centered real or complex i.i.d. entries.

There are two types of universalities: the edge universality and the bulk universality concerning energy levels near the spectral edges and in the interior of the spectrum, respectively. Since the works of Sinai and Soshnikov , the edge universality is commonly approached via the fairly robust moment method ; very recently an alternative approach was given in .

The bulk universality is a subtler problem. In the simplest case of the hermitian Wigner ensemble, it states that, independent of the distribution of the entries, the local $k$-point correlation functions of the eigenvalues (see (2.3) for the precise definition later), after appropriate rescaling and in the $N\to\infty$ limit, are given by the determinant of the sine kernel

Similar statement is expected to hold for all other ensembles mentioned above but the explicit formulas are somewhat more complicated. Detailed formulas for the different Wigner ensembles can be found e.g., in . The various sample covariance ensembles have the same local statistics for their singular values as the local eigenvalue statistics of the corresponding Wigner ensembles.

For ensembles of hermitian, symmetric or quaternion self-dual matrices that remain invariant under the transformations $H\to U^{*}HU$ for any unitary, orthogonal or symplectic matrix $U$, respectively, the joint probability density function of all the $N$ eigenvalues can be explicitly computed. These ensembles are typically given by the probability density

where $V$ is a real function with sufficient growth at infinity and ${\rm d}H$ is the flat Lebesgue measure on the corresponding symmetry class of matrices. The eigenvalues are strongly correlated and they are distributed according to a Gibbs measure with a long range logarithmic interaction potential. The joint probability density of the eigenvalues of $H$ with distribution (1.2) can be computed explicitly:

where $\beta=1,2,4$ for hermitian, symmetric and symplectic ensembles, respectively, and const. is a normalization factor. The formula (1.3) defines a joint probability density of $N$ real random variables for any $\beta\geq 1$ even when there is no underlying matrix ensemble. This ensemble is called the invariant $\beta$-ensemble. Quadratic $V$ corresponds to the Gaussian ensembles; we note that these are the only ensembles that are simultaneously invariant and have i.i.d. matrix entries. These are called the Gaussian Orthogonal, Unitary and Symplectic Ensembles (GOE, GUE, GSE for short) in case of $\beta=1,2,4$, respectively. Somewhat different choices of $V$ lead to two other classical ensembles, the Laguerre and the Jacobi ensembles, that also have matrix interpretation for $\beta=1,2,4$ (e.g., the Laguerre ensemble corresponds to the Gaussian sample covariance matrices which are also called Wishart matrices), see for more details. The local statistics can be obtained via a detailed analysis of orthogonal polynomials on the real line with respect to the weight function $\exp(-V(x))$. This approach was originally applied to classical ensembles by Dyson , Mehta and Gaudin and Mehta that lead to classical orthogonal polynomials. Later general methods using orthogonal polynomials were developed to tackle a very general class of invariant ensembles by Deift et.al., see and references therein, and also by Bleher and Its and Pastur and Schcherbina .

Many natural matrix ensembles are typically not unitarily invariant; the most prominent examples are the Wigner matrices or the sample covariance matrices mentioned in (i) and (ii). For these ensembles, apart from the identically distributed Gaussian case, no explicit formula is available for the joint eigenvalue distribution. Thus the basic algebraic connection between eigenvalue ensembles and orthogonal polynomials is missing and completely new methods needed to be developed.

The bulk universality for hermitian Wigner ensembles has been established recently in , by Tao and Vu in and in . These works rely on the Wigner matrices with Gaussian divisible distribution, i.e., ensembles of the form

where $\widehat{H}$ is a Wigner matrix, $V$ is an independent standard GUE matrix and $s$ is a positive constant. Johansson (see also Ben Arous and Péché and the recent paper ) proved the bulk universality for the eigenvalues of such matrices by an asymptotic analysis on an explicit formula for the correlation functions adapted from Brézin-Hikami . Unfortunately, the similar formula for symmetric or quaternion self-dual Wigner matrices, as well as for real sample covariance matrices, is not very explicit and the technique of cannot be extended to prove universality. Complex sample covariance matrices can however be handled with an analogous formula and universality without any Gaussian component is a work in progress .

A key observation of Dyson is that if the matrix $\widehat{H}+\sqrt{s}V$ is embedded into a stochastic matrix flow, i.e. one considers $\widehat{H}+V(s)$ where the matrix elements of $V(s)$ are independent standard Brownian motions with variance $s/N$, then the evolution of the eigenvalues is given by a system of coupled stochastic differential equations (SDE), commonly called the Dyson Brownian motion (DBM) . If we replace the Brownian motions by the Ornstein-Uhlenbeck processes to keep the variance constant, then the resulting dynamics on the eigenvalues, which we still call DBM, has the GUE eigenvalue distribution as the invariant measure. Similar stochastic processes can be constructed for symmetric, quaternion self-dual and sample covariance type matrices, and, in fact, on the level of eigenvalue SDE they can be extended to other values of $\beta$ (see (5.5) and (5.8) for the precise formulas).

The result of can be interpreted as stating that the local statistics of GUE is reached via DBM for time of order one. In fact, by analyzing the dynamics of DBM with ideas from the hydrodynamical limit, we have extended Johansson’s result to $s\gg N^{-3/4}$ . The key observation of is that the local statistics of eigenvalues depend exclusively on the approach to local equilibrium which in general is faster than reaching the global equilibrium. Unfortunately, the identification of local equilibria in still uses explicit representations of correlation functions by orthogonal polynomials (following e.g. ), and the extension to other ensembles is not a simple task.

In we introduced an approach based on a new stochastic flow, the local relaxation flow, which locally behaves like DBM, but has a faster decay to equilibrium. This method completely circumvented explicit formulas and it resulted in proving universality for symmetric Wigner matrices (the method applies to hermitian and quaternion self-dual Wigner matrices as well). As an input of this method, we needed a fairly detailed control on the local density of eigenvalues that could be obtained from our previous works on Wigner matrices .

In this paper we will prove a general theorem which states that as long as the eigenvalues are at most $N^{-1/2-\varepsilon}$ distance near their classical location on average, the local statistics is universal and in particular it coincides with the Gaussian case for which explicit formulas have been computed. To introduce this flow, denote by $\gamma_{j}$ the location of the $j$-th eigenvalue that will be defined in (2.12). We first define the pseudo equilibrium measure by

where $\mu_{N}$ is the probability measure for the eigenvalue distribution of the corresponding Gaussian ensemble. In case of Wigner matrices, $\mu_{N}$ is the measure for the general $\beta$ ensemble ($\beta\geq 1$ and $\beta=2$ for GUE):

In this setting, it is natural to view eigenvalues as random points and their equilibrium measure as Gibbs measure with a Hamiltonian ${\mathcal{H}}$. We will freely use the terminology of statistical mechanics. Note that the additional term $W_{j}$ in ${\omega}_{N}$ confines the $j$-th point $x_{j}$ near its classical location, but the probability w.r.t. the equilibrium measure $\mu_{N}$ of the event that $x_{j}$ near its classical location will be shown to be very close to $1$. Furthermore, we will prove that the local statistics of the measures ${\omega}_{N}$ and $\mu_{N}$ are identical in the limit $N\to\infty$ and this justifies the term pseudo equilibrium measure.

The local relaxation flow is defined to be the reversible flow (or the gradient flow) generated by the pseudo-equilibrium measure. The main advantage of the local relaxation flow is that it has a faster decay to global equilibrium (Theorem 4.2) compared with the DBM. The idea behind this construction can be related to the treatment of metastability in statistical physics. Imagine that we have a double well potential and we wish to treat the dynamics of a particle in one of the two wells. Up to a certain time, say $t_{0}$, the particle will be confined in the well where the particle initially located. However, the potential of this particle, given by the double well, is not convex. A naive idea is to regain the convexity before the time $t_{0}$ is to modify the potential to be a single well! Now as long as we can prove that the particle was confined in the initial well up to $t_{0}$, there is no difference between these two dynamics. But the modified dynamics, being w.r.t. a convex potential, can be estimated much more precisely and this estimate can be carried over to the original dynamics up to the time $t_{0}$.

In our case, the convexity of the equilibrium measure $\mu_{N}$ is rather weak and in fact, it comes from the quadratic confining potential $\beta x_{i}^{2}/4$ of (1.6). So the potential is convex, just not “convex enough”. There is no sharp transition like jumping from one metastable state to another as in the double well case. Instead, there are two time scales: in short time the local equilibrium is formed, on longer time, it approaches the global equilibrium. The approach to the local equilibrium is governed by a strong intrinsic convexity in certain directions due to the interactions (see (2.10) later for a precise formula). To reveal this additional convexity, in our previous paper we introduced a pseudo equilibrium measure where we replaced the long range part of the interaction by a mean-field potential term using the classical locations of far away particles. This potential term inherited the intrinsic convexity of the interaction and it could be directly used to enhance the decay to the local statistics. One technical difficulty with this approach was that we needed to handle the singular behavior of the logarithmic interaction potential. In this paper we show that the pseudo equilibrium measure can be defined by adding a Gaussian term. This simple modification turns out to be sufficient and is also model-independent. Since the Gaussian modification is regular, we no longer need to deal with singularities. The price to pay is that we need a slightly stronger local semicircle law which will be treated in Section 8.

The method of local relaxation flow itself proves universality for Wigner matrices with a small Gaussian component $\sqrt{s}V$ (typically of variance $s\geq N^{-\gamma}$ with some $0<\gamma<1$). In other words, we can prove universality for a Wigner ensemble whose single entry distribution (the distribution of its matrix elements) is given by $e^{tB}u_{0}$, where $B$ is the generator of the Ornstein-Uhlenbeck process and $u_{0}$ is any initial distribution (We remark that in our approach of decay to equilibrium, the Brownian motion in the construction of DBM is always replaced by the Ornstein-Uhlenbeck process). To obtain universality for Wigner matrices without any Gaussian component, it remains to prove that for a given Wigner matrix ensemble with a single entry distribution $\nu$ we can find $u_{0}$ and $t$ such that the eigenvalue distributions of the ensembles given by $\nu$ and $e^{tB}u_{0}$ are very close to each other. By the method of reverse heat flow introduced in , we choose $u_{0}$ to be an approximation of $e^{-tB}\nu$. Although the Ornstein-Uhlenbeck evolution cannot be reversed, we can approximately reverse it provided that $\nu$ is sufficient smooth and the time is short. This enables us to compare local statistics of Wigner ensembles with and without small Gaussian components assuming that the single entry distribution is sufficiently smooth (see Section 6).

As an application, we will use this method to prove the bulk universality of sample covariance ensembles. The necessary apriori control on the location of eigenvalues will be obtained by a local semicircle law. In addition to sample covariance ensembles, we will outline the modifications needed for proving the bulk universality of symplectic ensembles.

Universality for the local relaxation flow

We consider time dependent permutational symmetric probability measures with density $f_{t}({\bf{x}})$, $t\geq 0$, with respect to the measure $\mu({\rm d}{\bf{x}})=\mu({\bf{x}}){\rm d}{\bf{x}}$. The dynamics is characterized by the forward equation

with a given permutation symmetric initial data $f_{0}$. Here the generator $L$ is defined via the Dirichlet form as

Formally, we have $L=\frac{1}{2N}\Delta-\frac{1}{2}(\nabla{\mathcal{H}})\nabla$. In Appendix A we will show that under general conditions on ${\mathcal{H}}$ the generator can be defined as a self-adjoint operator on an appropriate domain and the dynamics is well defined for any $f_{0}\in L^{1}({\rm d}\mu)$ initial data. Strictly speaking, we will consider a sequence of Hamiltonians ${\mathcal{H}}_{N}$ and corresponding dynamics $L_{N}$ and $f_{t,N}$ parametrized by $N$, but the $N$-dependence will be omitted. All results will concern the $N\to\infty$ limit.

With a slight abuse of notations, we will sometimes also use $\mu$ to denote the density of the measure $\mu$ with respect to the Lebesgue measure. The correlation functions of the equilibrium measure are denoted by

We now list our main assumptions on the initial distribution $f_{0}$ and on its evolution $f_{t}$. We first define the subdomain

of ordered sets of points ${\bf{x}}$. In the application to the sample covariance matrices, we will use the subdomain

Assumption I. The Hamiltonian ${\mathcal{H}}$ of the equilibrium measure has the form

The condition $U^{\prime\prime}\geq 0$ can be relaxed to $\inf U^{\prime\prime}>-\infty$, see remark after (4.11).

Alternatively, in order to discuss the case of the sample covariance matrices, we will also consider the following modification of Assumption I.

Assumption I’. The Hamiltonian ${\mathcal{H}}$ of the equilibrium measure has the form

where $\beta\geq 1$ and $c_{N}\geq 1$. The function $U$ satisfies the same conditions as in Assumption I.

It is easy to check that the condition (2.7) guarantees that the following bound holds for the normalization constant

with some exponent $m$ depending on $\delta$.

It follows from Assumption I (or I’) that the Hessian matrix of ${\mathcal{H}}$ satisfies the following bound:

This convexity bound is the key assumption; our method works for a broad class of general Hamiltonians as long as (2.10) holds. In particular, an arbitrary many-body potential function $V({\bf{x}})$ can be added to the Hamiltonians (2.6), (2.8), as long as $V$ is convex on the open sets $\Sigma_{N}$ and $\Sigma_{N}^{+}$, respectively. The argument in the proof of the main Theorem 2.1 remains unchanged, but the technical details of the regularization of the singular dynamics (Appendix B) becomes more involved. We do not pursue this direction here since we do not need it for the application for Wigner and sample covariance matrices.

Let $\gamma_{j}=\gamma_{j,N}$ denote the location of the $j$-th point under the limiting density, i.e., $\gamma_{j}$ is defined by

We will call $\gamma_{j}$ the classical location of the $j$-th point. Note that $\gamma_{j}$ may not be uniquely defined if the support of $\varrho$ is not connected but in this case the next Assumption III will not be satisfied anyway.

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

Under Assumption II, the typical spacing between neighboring points is of order $1/N$ away from the spectral edges, i.e., in the vicinity of any energy $E$ with $\varrho(E)>0$. Assumption III guarantees that typically the random points $x_{j}$ remain in the $N^{-1/2-\varepsilon}$ vicinity of their classical location.

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

The main general theorem is the following:

for $\tau=N^{-2\varepsilon+\delta}$ where $\varepsilon>0$ is the exponent from Assumption III.

Suppose in addition to the Assumption I-IV, that there exists an $A>0$ such that, for any $c\,^{\prime}>0$

for some constants $c$ and $C$ only depending on $c\,^{\prime}$. Then for $\tau=N^{-2\varepsilon+\delta}$ we have

This theorem shows that the local statistics of the points $x_{j}$ in the bulk with respect to the time evolved distribution $f_{t}$ coincides with the local statistics with respect to the equilibrium distribution $\mu$ as long as $t\gg N^{-2\varepsilon}$. In many applications, the local equilibrium statistics can be explicitly computed and in the $b\to 0$ limit it becomes independent of $E$, in particular this is the case for the classical matrix ensembles (see next section). The restriction on the time $t\gg N^{-2\varepsilon}$ will be removed by the reverse heat flow argument (see Section 6) for matrix ensembles.

Since the eigenvalues fluctuate at least on a scale $1/N$, the best possible exponent in Assumption III is $2\varepsilon\sim 1$, but we will only be able to prove it for some $\varepsilon>0$ for the ensembles considered in this paper. Similarly, the optimal exponent in (2.16) is $A\sim 0$. If we use these optimal estimates, $2\varepsilon\sim 1$, $A\sim 0$, and we choose $\delta=2\varepsilon\sim 1$, thus $\tau\sim 1$, then we can choose $b\sim N^{-1}$, i.e., we obtain the universality with essentially no averaging in $E$. On the other hand, the error estimate is the strongest, of order $\sim N^{-1/2}$, for an averaging on an energy window of size $b\sim 1$. These errors become weaker if time $\tau$ is reduced. These considerations are not important in this paper, but will be useful when good estimates on $\varepsilon$ and $A$ can be obtained.

Convention: Throughout the paper the letters $C,c$ denote positive constants whose values may change from line to line and they are independent of the relevant parameters. Since we will always take the $N\to\infty$ limit at the end, all estimates are understood for sufficiently large $N$.

Universality for Matrix Ensembles

Now we specialize Theorem 2.1 to Wigner and sample covariance matrices with i.i.d. entries. In the next sections we give the precise definitions of these ensembles; formulas for the equilibrium measure and the dynamics will be deferred until Section 5.

In order to apply Theorem 2.1 to Wigner and sample covariance matrices, we need to check that Assumptions I-IV are satisfied for these ensembles. Assumptions I or I’ are satisfied by the definition of the Hamiltonian, the precise formulas are given in Section 5. Assumption II is satisfied since the density of eigenvalues is given by the Wigner semicircle law (3.7) for Wigner matrices . In case of the sample covariance matrices, the singular values of $A$ will play the role of $x_{j}$’s and their density is given by the Marchenko-Pastur law (3.14) after an obvious transformation (3.15) . In fact, in Section 8 we prove a local version of the Marchenko-Pastur law in analogy with our previous work on the local semicircle law for Wigner matrices . In Section 9 (Theorem 9.1) we will show that Assumption III is satisfied for these ensembles (more precisely, we will prove that Assumption III is satisfied for sample covariance matrices; the proof for Wigner matrices is analogous, and will not be given in details). Assumption IV will be proved in Lemma 8.1 for the sample covariance matrices, for Wigner matrices the proof was given, e.g., in Theorem 4.6 of . We remark that the assumption that the matrix entries are identically distributed, will only be used in checking Assumptions III and IV. Assumption II holds under much more general conditions on the matrix entries. Finally, the apriori estimate on the entropy $S_{\mu}(f_{t_{0}})$ follows from the smoothing property of the OU-flow (see Section 5).

To fix the notation, we assume that in the case of real symmetric matrices, the matrix elements of $H$ are given by

Finally, for the quaternion self-dual case we assume that $H$ is a $2N$ by $2N$ complex matrix that can be viewed as an $N\times N$ matrix with elements consisting of $2\times 2$ blocks of the form

The dual of the quaternion $q$ is defined to be $q^{+}:=a-b{\bf i}-c{\bf j}-d{\bf k}$ which corresponds to the hermitian conjugate of the matrix (3.3).

where $x_{kk}$ has a law $\widetilde{\nu}$ with zero expectation and variance $\frac{1}{2}$. The spectrum of $H$ is doubly degenerate and we will neglect this degeneracy, i.e., we consider only $N$ real (typically distinct) eigenvalues, $x_{1}<x_{2}<\ldots<x_{N}$.

In particular, the typical spacing between neighboring eigenvalues is of order $1/N$ in the bulk of the spectrum.

We will often need to assume that the distributions $\nu$ and $\widetilde{\nu}$ have Gaussian decay, i.e., there exists $\delta_{0}>0$ such that

In several statements we can relax this condition to assuming only subexponential decay, i.e., that there exists $\delta_{0}>0$ and $\gamma>0$ such that

For some statements we will need to assume that the measures $\nu$, $\widetilde{\nu}$ satisfy the logarithmic Sobolev inequality, i.e., for any density $h\geq 0$ with $\int h{\rm d}\nu=1$ it holds that

and a similar bound holds for $\widetilde{\nu}$. We remark that (3.10) implies (3.8), see, e.g. .

2 Sample Covariance Matrix

The real sample covariance matrix ensemble consists of symmetric $N\times N$ matrices of the form $H=A^{*}A$. Here $A$ is an $M\times N$ real matrix with $d=N/M$ fixed and we assume that $0<d<1$. The elements of $A$ are given by

Moreover, analogously to (3.6), the empirical density of eigenvalues converges weakly in probability to the Marchenko-Pastur law

Most of the analysis will be done for the singular values of $A$ that are denoted by ${\bf{x}}=(x_{1},\ldots,x_{N})$. They are supported asymptotically in $[\sqrt{\lambda_{-}},\sqrt{\lambda_{+}}]$ and therefore the typical spacing between neighboring singular values is of order $1/N$. Their empirical density converges to

We remark that the assumption that $\nu$ is symmetric is used only at one technical step, namely when we refer to the large deviation result for the extreme eigenvalues of the sample covariance matrices in (see Lemma 9.2 below). The similar result for Wigner matrices has been proven without the symmetry condition, see Theorem 1.4 in .

3 Main Theorems

With the remarks at the beginning of Section 3, Theorem 2.1 applies directly to prove universality for Wigner and sample covariance ensembles with a small Gaussian component; we will not state these theorems separately. To remove the small time restriction from Theorem 2.1, we will apply the reverse heat flow argument. This will give our main result:

Here $\mu$ denotes the probability measure of the eigenvalues of the appropriate Gaussian ensemble, i.e. GUE, GOE, GSE for the case of hermitian, symmetric, and, respectively, quaternion self-dual Wigner matrices; and the ensembles of real or complex sample covariance matrices with Gaussian entries (Wishart ensemble) in case of the covariance matrices $A^{*}A$. These measures are given in (1.6), with $\beta=1,2,4$, for Wigner matrices, and, expressed in terms of singular values, in (5.6), with $\beta=1,2$, for sample covariance matrices.

Remark 1.: In the case of symmetric and hermitian Wigner matrices, the condition (3.16) can be removed by applying the Four-moment theorem of Tao and Vu (Theorem 15 of ) as in the proof of Corollary 2.4 of . Similar remark applies to the sample covariance ensembles and to the quaternion self-dual Wigner ensemble provided the corresponding Four-moment theorem is established.

We also remark that a manuscript by Ben-Arous and Péché with a similar statement is in preparation for complex sample covariance matrices that holds for a fixed $E^{\prime}$, i.e., without averaging over the energy parameter in (3.17).

Remark 2.: After the first version of this manuscript was posted on the arxiv, the question that whether the four moment theorem for sample covariance matrices holds was settled in . In particular, gives an alternative proof of the universality of local statistics for the complex sample covariance ensemble when combined with the result of . For the real sample covariance ensemble the universality was established for distributions whose first four moments match the standard Gaussian variable. An important common ingredient to both our approach and that of is the local Marchenko-Pastur law, established in Proposition 8.1; a slightly different version suitable for the application to prove the four moment theorem is proved in .

The four moment theorem in compares the distributions of individual eigenvalues for two different ensembles. For our application to the correlation functions and gap distributions, an alternative approach is to use the recent Green function comparison theorem . This will also remove the smoothness and logarithmic Sobolev inequality restrictions in Theorem 3.1.

We now state our result concerning the eigenvalue gap distribution both for Wigner and sample covariance ensembles. For any $s>0$ and $E$ with $\rho(E)>0$ we define the density of eigenvalue pairs with distance less than $s/N\varrho(E)$ in the vicinity of $E$ by

Consider an $N\times N$ Wigner or sample covariance matrix as in Theorem 3.1 such that the probability measure ${\rm d}\nu=u_{0}{\rm d}x$ of the matrix elements satisfies the logarithmic Sobolev inequality (3.10) and, additionally, $\nu$ is symmetric in the sample covariance matrix case. Suppose that the initial density $u_{0}$ satisfies

where $\mu$ is the probability measure of the eigenvalues of the appropriate Gaussian ensemble, as in Theorem 3.1.

Theorem 3.2 shows that, in particular, the probability to find no eigenvalue in the interval $[E,E+\alpha/(\varrho(E)N)]$ is asymptotically the same as in the corresponding classical Gaussian ensemble. Theorems 3.1 and 3.2 will follow from Theorem 2.1 and the reverse heat flow argument that we present in Section 6. We remark that the additional condition on the symmetry of $\nu$ in the case of sample covariance matrices stems from using a result from on the lowest eigenvalue of these matrices, see Lemma 9.2.

Theorem 3.2 can be proven directly from Theorem 4.1 since the test functions of the form

determine the distribution of the random variable $\Lambda(E;s)$ uniquely. Here we take $J$ to be the set

Local Relaxation Flow

Suppose that the Hamiltonian ${\mathcal{H}}$ given in (2.6) satisfies the convexity bound (2.10) with $\beta\geq 1$. Let $f_{t}$ be the solution of the forward equation (2.1) with an initial density $f_{0}$. Fix a positive $\varepsilon>0$, set $t_{0}=N^{-2\varepsilon}$ and define

Then for any sufficiently small $\varepsilon^{\prime}>0$, there exist constants $C,c$, depending only on $\varepsilon^{\prime}$ and $G$ such that for any $J\subset\{1,2,\ldots,N-m_{n}\}$ and for any $\tau\geq 3t_{0}=3N^{-2\varepsilon}$, we have

where $|J|$ is the number of the elements in $J$.

The proof of this theorem is similar but much simpler than that of Theorem 2.1 of . The estimate (4.3) improves slightly over the similar estimate in by a factor $|J|/N$ due to the improvement in (4). Theorem 2.1 will follow from the fact that in case $\tau\geq N^{-2\varepsilon+\delta}$, the assumption (2.13) guarantees that

with the choice $\varepsilon^{\prime}=\delta/3$ and using $|J|\leq N$. More precise error bound will be obtained by relating $b$ to $|J|$. Therefore the local statistics of observables involving eigenvalue differences coincide in the $N\to\infty$ limit. To complete the proof of Theorem 2.1, we will have to show that the convergence of the observables ${\mathcal{G}}_{i,{\bf{m}}}$ is sufficient to identify the correlation functions of the $x_{i}$’s in the sense prescribed in Theorem 2.1. The details will be given in Section 7.

Proof of Theorem 4.1. Without loss of generality we can assume in the sequel that $f_{0}\in L^{\infty}({\rm d}\mu)$. To see this, note that any $f_{0}\in L^{1}({\rm d}\mu)$ can be approximated by a sequence of bounded functions $f_{0}^{(k)}$ in $L^{1}$-norm with arbitrary precision and the dynamics is a contraction in $L^{1}$ (see Appendix A), thus $f_{\tau}$ and $f_{\tau}^{(k)}$ are arbitrarily close in $L^{1}$. Since $G$ is bounded on the left hand side of (4.3), this is sufficient to pass to the limit $k\to\infty$.

Every constant in this proof depends on $\varepsilon^{\prime}$ and $G$, and we will not follow the precise dependence. We can assume that $\varepsilon^{\prime}<\varepsilon$. Given $\tau>0$, we define

Notice that the choice of $R$ depending on $\tau$ which is the main reason that $\tau$ appears in the denominator on the right hand side of (4.3). We now introduce the pseudo equilibrium measure, ${\omega}_{N}={\omega}=\psi\mu_{N}$, defined by

where $\widetilde{Z}$ is chosen such that ${\omega}$ is a probability measure, in particular $\omega=e^{-N\widetilde{\mathcal{H}}}/\widetilde{Z}$ with

Note that the additional term $W_{j}$ confines the $j$-th point $x_{j}$ near its classical location. We will prove that the probability w.r.t. the equilibrium measure $\mu_{N}$ of the event that $x_{j}$ near its classical location is very close to $1$. Thus there is little difference between the two measures ${\omega}_{N}$ and $\mu_{N}$ and in fact, we will prove that their local statistics are identical in the limit $N\to\infty$. The main advantage of the pseudo equilibrium measure comes from the fact that it has a faster decay to global equilibrium as shown in Theorem 4.2.

Since the additional potential $W_{j}$ is uniformly convex with

Here we have used $U^{\prime\prime}\geq 0$ in the last estimate. If this assumption is replaced by

for some constant $M$ independent of $N$, then there will be an extra term $-M\|{\bf{v}}\|^{2}$ in (4.10). Assuming $\tau\leq N^{\varepsilon^{\prime}}$, we have $R\leq N^{-\varepsilon^{\prime}/2}$, then this extra term can be controlled by the $R^{-2}$ term and the same proof will go through. Since for the applications in this paper, the condition $U^{\prime\prime}\geq 0$ is satisfied, we will not use this remark here.

The $R^{-2}$ in the first term comes from the additional convexity of the local interaction and it enhances the “local Dirichlet form dissipation”. In particular we have the uniform lower bound

This guarantees that the relaxation time to equilibrium ${\omega}$ for the $\widetilde{L}$ dynamics is bounded above by $CR^{2}$. We recall the definition of the relative entropy of $f$ with respect to any probability measure ${\rm d}\lambda$

The first ingredient to prove Theorem 4.1 is the analysis of the local relaxation flow which satisfies the logarithmic Sobolev inequality and the following dissipation estimate. Its proof follows the standard argument in (used in this context in Section 5.1 of ). In Appendix B we will explain how to extend this argument onto the subdomain $\Sigma_{N}$. Here we only remark that the key inputs are the convexity bounds (4.10, 4.12) on the Hessian of $\widetilde{\mathcal{H}}$ (4.10).

Suppose (4.10) holds. Consider the forward equation

with an initial condition $q_{0}$ and with the reversible measure $\omega$. Assume that $q_{0}\in L^{\infty}({\rm d}{\omega})$. Then we have the following estimates

with a universal constant $C$. Thus the time to equilibrium is of order $R^{2}$:

The estimate (4.15) on the second term in (4.10) plays a key role in the next theorem.

Proof. For simplicity, we will consider the case when ${\bf{m}}=(1,2,\ldots n)$, the general case easily follows by appropriately redefining the function $G$. Let $q_{t}$ satisfy

with an initial condition $q$. Thanks to the exponential decay of the entropy on time scale $\tau\gg R^{2}$, see (4.17), and the entropy bound on the initial state $q$, the difference between the local statistics w.r.t. $q_{\tau}{\omega}$ and $q_{\infty}{\omega}={\omega}$ is subexponentially small in $N$,

giving the second term on the r.h.s. of (4.18). To compare $q$ with $q_{\tau}$, by differentiation, we have

Here we used the definition of $\widetilde{L}$ from (4.7) and note that the $1/N$ factor present in (4.7) cancels the factor $N$ from the argument of $G$ (4.2). From the Schwarz inequality and $\partial q=2\sqrt{q}\partial\sqrt{q}$, the last term is bounded by

since $G$ is smooth and compactly supported. This proves Theorem 4.3.

As a comparison to Theorem 4.3, we state the following result which can be proved in a similar way.

Notice that by exploiting the local Dirichlet form dissipation coming from the second term on the r.h.s. of (4.14), we have gained the crucial factor $N^{-1/2}$ in the estimate (4.18) compared with (4.20).

The final ingredient to prove Theorem 4.1 is the following entropy and Dirichlet form estimates.

Suppose that (2.10) holds and recall $\tau=R^{2}N^{\varepsilon^{\prime}}\geq 3t_{0}$ with $t_{0}=N^{-2\varepsilon}$. Let $g_{t}=f_{t}/\psi$ so that $S_{\mu}(f_{t}|\psi)=S_{\omega}(g_{t})$. Assume that $S_{\mu}(f_{t_{0}})\leq CN^{m}$ with some fixed $m$. Then the entropy and the Dirichlet form satisfy the estimates:

Proof. Recall that $\partial_{t}f_{t}=Lf_{t}$. The standard estimate on the entropy of $f_{t}$ with respect to the invariant measure is obtained by differentiating the entropy twice and using the logarithmic Sobolev inequality. The entropy and the Dirichlet form in (4.21) are, however, computed with respect to the measure ${\omega}$. This yields the additional second term in the following identity that holds for any probability density $\psi_{t}$:

where $g_{t}=f_{t}/\psi_{t}$. In our application we set $\psi_{t}$ to be time independent, $\psi_{t}=\psi={\omega}/\mu$, hence we have

Since ${\omega}$ is invariant, the middle term on the right hand side vanishes, and from the Schwarz inequality

To obtain the first inequality in (4.21), we integrate (4.24) from $t_{0}=N^{-2\varepsilon}$ to $\tau/2$, using that $\tau=R^{2}N^{\varepsilon^{\prime}}$ and $S_{\omega}(g_{t_{0}})\leq CN^{m}+N^{2}Q$ with some finite $m$, depending on $\varepsilon$. This apriori bound follows from

where we used (2.9) and (4.6). The second inequality in (4.21) can be obtained from the first one by integrating (4.22) from $t=\tau/2$ to $t=\tau$ and using the monotonicity of the Dirichlet form in time.

Finally, we complete the proof of Theorem 4.1. Recall that $\tau=R^{2}N^{\varepsilon^{\prime}}$ and $t_{0}=N^{-2\varepsilon}$. Choose $q_{\tau}:=g_{\tau}=f_{\tau}/\psi$ as density $q$ in Theorem 4.3. The condition $q_{\tau}\in L^{\infty}$ can be guaranteed by the approximation argument from the beginning of the proof of Theorem 4.1. Then Theorem 4.5, Theorem 4.3 together with (4.25) and the fact that $\Lambda\tau=Q\tau^{-1}N^{2\varepsilon^{\prime}}$ directly imply that

i.e., the local statistics of $f_{\tau}\mu$ and ${\omega}$ can be compared. Clearly, equation (4.26) also holds for the special choice $f_{0}=1$ (for which $f_{\tau}=1$), i.e., local statistics of $\mu$ and ${\omega}$ can also be compared. This completes the proof of Theorem 4.1.

Equilibrium measure and Dyson Brownian motion

where $\beta\geq 1$ is an arbitrary parameter, i.e., this corresponds to choosing $U(x)=x^{2}/4$ in (2.6). With a slight abuse of notations we will use $\mu$ for both the measure ${\rm d}\mu$ and its density $e^{-N{\mathcal{H}}_{\beta}}/Z_{\beta}$ with respect to the Lebesgue measure. The specific value $\beta=1,2,4$ correspond to the GUE, GOE and GSE ensembles, respectively.

acting on $L^{2}(\mu)$. The measure $\mu$ is invariant and reversible with respect to the dynamics generated by $L$. Define the Dirichlet form and entropy by

Let $f_{t}{\rm d}\mu$ denote the probability measure on the set $\Sigma_{N}$ at the time $t$ with the given generator $L$. Then $f_{t}$ satisfies the forward equation

with initial condition $f_{0}$. This dynamics is the Dyson Brownian motion.

The Dyson Brownian motion is the corresponding system of stochastic differential equations for the vector ${\bf x}(t)$ that is given by

The treatment of the sample covariance ensembles is fully analogous, but the formulas change slightly. We use the convention in the sample covariance case that $x_{i}$ denotes the singular values of $A$ and $\lambda_{i}=x_{i}^{2}$ are the eigenvalues of $A^{*}A$. Most of the formulas will be in terms of $x_{i}$’s; in particular we consider the joint distribution function $f_{0}({\bf{x}})$ of the singular values. The invariant measure for the singular values is given by (c.f. (5.1)):

where $d=M/N$ and $\beta=1$ when $X$ is a real matrix, $\beta=2$ when $X$ is a complex matrix. This formula can be obtained by direct calculation (see also Proposition 2.16 of or Fig. 1 of after appropriate rescaling). Define the generator (c.f. (5.2))

Finally, the stochastic differential equation is given by (c.f. (5.5))

In applications to Wigner matrices ($\beta=1,2,4$), $f_{0}{\rm d}\mu$ will be the joint probability density of the eigenvalues of the initial hermitian, symmetric or quaternion self-dual Wigner matrix $\widehat{H}$. The limiting density is the Wigner semicircle law given in (3.7). The Dyson Brownian motion describes the eigenvalues of the matrix valued process

with $H_{0}=\widehat{H}$. Here ${\bf B}_{t}$ is a symmetric, hermitian or quaternion self-dual matrix-valued process whose offdiagonal elements are standard real, complex or quaternion Brownian motions with variance one and the diagonal elements of are real Brownian motions with variance $2,1$ and $\frac{1}{2}$, in case $\beta=1,2,4$, respectively. More precisely, let $u_{t}$ denote the density function of the distribution of one real component of the $(ij)$-th entry of $H_{t}$, $i<j$ (there are two real components for the hermitian matrices and four for the quaternion matrices), then

Let $\gamma({\rm d}x)=\gamma(x){\rm d}x:=(\beta/2\pi)^{1/2}e^{-\beta x^{2}/2}{\rm d}x$ denote the reversible measure for this process. The diagonal elements evolve according to an OU process with twice variance. For any $t\geq 0$, the solution to (5.9), $H_{t}$, has the same distribution as

The generator of the induced stochastic process on the eigenvalues is given by (5.2). The equilibrium measure $\mu$ is the GUE, GOE or GSE eigenvalue distribution. Theorem 2.1 thus says in this case that the local eigenvalue statistics of a Wigner random matrix with a small Gaussian component coincides with the local statistics of the corresponding Gaussian ensemble. The entropy condition on $S_{\mu}(f_{t_{0}})$ in Theorem 2.1 can be easily obtained by

In the real or complex sample covariance case ($\beta=1,2$), the matrix elements of $A$ evolve according to the OU process (5.10), i.e. $A_{t}$ has the same distribution as

where $W$ is an $M\times N$ matrix whose elements are i.i.d real or complex Gaussian variables with mean 0 and variance $1/\beta$.

Reverse heat flow

To remove the short time restriction from Theorem 2.1 in case of Wigner and sample covariance ensembles and to prove Theorems 3.1 and 3.2, we apply the reverse heat flow argument, presented first in and used also in Corollary 2.4 of .

For fixed $\beta=1,2$ or 4, recall the Ornstein-Uhlenbeck process from (5.10) with the reversible Gaussian measure $\gamma({\rm d}x)$. Let $u$ be a positive density with respect to $\gamma$, i.e., $\int u{\rm d}\gamma=1$ and we write $u(x)=\exp(-V(x))$. Suppose that for any $K$ fixed there are constants $C_{1},C_{2}$ depending on $K$ such that

and the measure ${\rm d}\nu=u{\rm d}\gamma$ satisfies the subexponential decay condition. We will apply this for the initial distribution ${\rm d}\nu=u_{0}(x){\rm d}x$, so $u$ and $u_{0}$ differ by a Gaussian factor.

Suppose that $\nu=u\gamma$ satisfies the subexponential decay condition and (6.1) for some $K$. Then there is a small constant $\alpha_{K}$ depending on $K$ such that for $t\leq\alpha_{K}$ there exists a probability density $g_{t}$ with mean zero and variance $\frac{1}{2}$ such that

for some $C>0$ depending on $K$. Furthermore, $g_{t}$ can be chosen such that if the logarithmic Sobolev inequality (3.10) holds for the measure $\nu=u\gamma$, then it holds for $g_{t}\gamma$ as well, with the logarithmic Sobolev constant changing by a factor of at most $2$.

Furthermore, let ${\mathcal{B}}=B^{\otimes n}$, $F=u^{\otimes n}$ with some $n\leq CN^{2}$. Denote by $G_{t}=g_{t}^{\otimes n}$. Then we also have

In order to prove Theorems 3.1 and 3.2, appropriate observables $J$ need to be chosen that depend on the matrix elements via the eigenvalues to express the quantities in (3.17) and (3.20). It is easy to see that $\|J\|_{\infty}$ may grow at most polynomially in $N$. But we can always choose $K$ large enough to compensate for it with the choice $t=N^{-2\varepsilon+\delta}$ allowed in Theorem 2.1. Here the verifications of the Assumptions I-IV of Theorem 2.1 were explained at the beginning of Section 3. This completes the proof of our main theorems.

Proof of Proposition 6.1. Define $\theta(x)=\theta_{0}(t^{\alpha}x)$ with some small positive $\alpha>0$ depending on $K$, where $\theta_{0}$ is a smooth cutoff function satisfying $\theta_{0}(x)=1$ for $|x|\leq 1$ and $\theta_{0}(x)=0$ for $|x|\geq 2$. Set

By assumption (6.1), $h_{s}$ is positive and

for any $s\leq t$ if $t$ is small enough. To see this, take, e.g., $K=2$ and we have

where we have used $\alpha\ll 1$, $s\leq t$ and the assumption (6.1).

Define $v_{s}=e^{sB}h_{s}$ and by definition, $v_{0}=u$. Then

Since the Ornstein-Uhlenbeck is a contraction in $L^{1}({\rm d}\gamma)$, together with (6.1), we have

for sufficiently small $t$. To estimate the last two terms, we also used that on the support of $\theta-1$ the measure ${\rm d}\gamma$ decays subexponentially in $t$.

Notice that $h_{t}$ may not be normalized as a probability density w.r.t. $\gamma$ but this can be easily adjusted. To compute this normalization, take for example, $K=1$ and we have, by using $s\leq t^{\alpha}$,

The last term is bounded by $O(t^{M})$ for any $M>0$ due to that $u(x)\gamma$ has a subexponential decay and using the assumption (6.1) on $V$.

We have proved that there is a constant $c_{t}=1+O(t^{M})$, for any $M>0$ positive, such that $c_{t}h_{t}$ is a probability density. Clearly,

and the same formulas hold if $h_{t}$ is replaced by $v_{t}$ since the OU flow preserves expectation and variance. Let $g_{t}$ be defined by

Then $g_{t}$ is a probability density w.r.t. $\gamma$ with zero mean and variance $\beta^{-1}$. It is easy to check that the total variation norm of $h_{t}-g_{t}$ is smaller than any power of $t$. Using again the contraction property of $e^{tB}$ and (6.5), we get

Now we check the LSI constant for $g_{t}$. Recall that $g_{t}$ was obtained from $h_{t}$ by translation and dilation. By definition of the LSI constant, the translation does not change it. The dilation changes the constant, but since our dilation constant is nearly one, the change of LSI constant is also nearly one. So we only have to compare the LSI constants between ${\rm d}\nu=u{\rm d}\gamma$ and $c_{t}h_{t}{\rm d}\gamma$. From (6.4) and that $c_{t}$ is nearly one, the LSI constant changes by a factor less than $2$. This proves the claim on the LSI constant.

and this completes the proof of Proposition 6.1.

Proof of Theorem 2.1

We will set $Y_{i,{\bf{m}}}=0$ if $i+m_{n}>N$. Our goal is to estimate the difference

Let $M$ be an $N$-dependent parameter chosen at the end of the proof, in fact, $M$ will be chosen as $M=N^{c}$ with some small positive exponent $c>0$, depending on $n$. Let

and note that $|S_{n}(M)|\leq M^{n-1}$. We have the simple bound $\Theta\leq\Theta_{M}^{(1)}(\tau)+\Theta_{M}^{(2)}(\tau)+\Theta_{M}^{(2)}(\infty)$ where

Note that $\Theta_{M}^{(2)}(\infty)$ is the same as $\Theta_{M}^{(2)}(\tau)$ but with $f_{\tau}$ replaced by the constant $1$, i.e., $f_{\infty}{\rm d}\mu$ is the equilibrium.

After performing the ${\rm d}E^{\prime}$ integration, we will eventually apply Theorem 4.1 to the function

For any $E$ and $0<\xi<b$ define sets of integers $J=J_{E,b,\xi}$ and $J^{\pm}=J^{\pm}_{E,b,\xi}$ by

where $\gamma_{i}$ was defined in (2.12). Clearly $J^{-}\subset J\subset J^{+}$. With these notations, we have

The error term $\Omega^{+}_{J,{\bf{m}}}$, defined by (7.6) indirectly, comes from those $i\not\in J^{+}$ indices, for which $x_{i}\in[E-b+O(N^{-1}),E+b+O(N^{-1})]$ since $Y_{i,{\bf{m}}}(E^{\prime},{\bf{x}})=0$ unless $|x_{i}-E^{\prime}|\leq C/N$, the constant depending on the support of $O$. Thus

for any sufficiently large $N$, assuming $\xi\gg 1/N$ and using that $O$ is a bounded function. The additional $N^{-1}$ factor comes from the ${\rm d}E^{\prime}$ integration. Taking the expectation with respect to the measure $f_{\tau}{\rm d}\mu$, we get

using Assumption III (2.13). We can also estimate

where the error term $\Xi^{+}_{J,{\bf{m}}}$, defined by (7.9), comes from indices $i\in J^{-}$ such that $x_{i}\not\in[E-b,E+b]+O(1/N)$. It satisfies the same bound (7.8) as $\Omega^{+}_{J,{\bf{m}}}$.

By the continuity of $\varrho$, the density of $\gamma_{i}$’s is bounded by $CN$, thus $|J^{+}\setminus J^{-}|\leq CN\xi$ and $|J\setminus J^{-}|\leq CN\xi$. Therefore, summing up the formula (7.5) for $i\in J$, we obtain from (7.6) and (7.9)

for each ${\bf{m}}\in S_{n}$. A similar lower bound can be proved analogously and we obtain

Adding up (7.12) for all ${\bf{m}}\in S_{n}(M)$, we get

and the same estimate holds for the equilibrium, i.e., if we set $\tau=\infty$ in (7.13). We now subtract the these two formulas and apply (4.3) from Theorem 4.1 to each summand on the second term in (7.13). Choosing $\xi=N^{-(1+2\varepsilon)/3}$ to minimize the two error terms involving $\xi$, we conclude that

where we have used $\tau=N^{-2\varepsilon+\delta}$ and that $|J|\leq CNb$.

Together with (7.14), we have thus proved that

Choosing $M$ such that $M^{n}=N^{\varepsilon^{\prime}}$ and then choose $k$ large enough so that the last term $\frac{C_{a}}{M^{k-1}}$ is smaller than, say, $N^{-2}$. We have thus proved that

for $\tau=N^{-2\varepsilon+\delta}$ and this concludes (2.15).

For the proof of (2.17), we choose $\xi\geq 2N^{-1+A}$, and then by using (2.16) we can estimate $\Omega^{+}_{J,{\bf{m}}}$ directly as

for any $K>0$, instead of (7.8). Therefore, the estimate on the right hand side of (7.12) and the subsequent estimates can be replaced by

provided $\xi\geq 2N^{-1+A}$. Choosing $\xi=2N^{-1+A}$ and following the same proof, we can improve the estimate (7.18) to

for $\tau=N^{-2\varepsilon+\delta}$. This proves (2.17) and we have completed the proof of Theorem 2.1.

Local Marchenko-Pastur law

In this section we establish that the empirical density of eigenvalues for sample covariance matrices is close to the Marchenko-Pastur law even on short scale. We do this by controlling the difference of the Stieltjes transform, establishing results analogous to Theorem 4.1. and Proposition 4.2 of . In this section, we focus on $0<d=N/M<1$, in particular the lower spectral edge $\lambda_{-}>0$. The constants appearing in this subsection may depend on $d$.

Before the detailed proof, we explain the main steps of the argument which is similar to the method we have successively developed in . The proof given here is somewhat complicated by fact that the matrix elements themselves are not independent but are generated as a quadratic expression of independent random variables. The first step, Lemma 8.1, is an apriori bound on the local density on short scales, $\eta\gg 1/N$, using resolvent expansion and a large deviation principle for quadratic forms. Expressing the resolvent of $H$ in terms the resolvents of its minors, we obtain a self-consistent equation (8.20) for the Stieltjes transform $m_{N}$ of the eigenvalues. This equation is very close to the defining quadratic equation of the Stieltjes transform $m_{W}$ of the Marchenko-Pastur law, see (8.7), with a perturbation term $Y(z)$. This term can be estimated by large deviation arguments and using the a-priori bound on the local density. Then in Lemma 8.3 we investigate the stability of the self-consistent equation for $m_{W}$. Although the perturbed equation has two solutions, only one of them can be close to $m_{N}$. To select the correct solution, we use a continuity argument in the spectral parameter $z$. For $z=z_{0}$ with a large imaginary part, say $z_{0}=10+5i$, the explicit formula (8.27) for the solution can be directly analyzed. For $z$ approaching to the real axis, we prove that the two unperturbed solutions remain far away from each other (8.35). Since the perturbed solutions are also continuous in the spectral parameter, for a sufficiently small perturbation they must remain in the vicinity of the correct solution of the unperturbed equation.

Let $0<E<10$ and $0<d<1$. Consider the interval $I_{\eta}=[E-\eta,E+\eta]$. Let ${\mathcal{N}}_{I}$ denote the number of eigenvalues of $H=A^{*}A$ in the interval $I_{\eta}$. Suppose that $N^{-1+\varepsilon}\leq\eta\leq E/2$, for some $\varepsilon>0$. Then there exist constants $C,c>0$ such that

for all $N,K$ large enough (independent of $E$).

We remark that the assumption on $\eta$ can be relaxed to $CN^{-1}\leq\eta\leq E/2$. But we do not need this result here. For details, one can refer to Theorem 5.1 of .

Proof of Lemma 8.1. We observe, first of all, that

where we defined $z=E+i\eta$. It follows that

Denoting by $a_{1}$ the first column of $A$ and by $B$ the $M\times(N-1)$ matrix consisting of the last $N-1$ column of $A$, we have

Denote $\mu_{\alpha}$’s $(\alpha=1,\ldots,N-1)$ the eigenvalues of the $(N-1)\times(N-1)$ matrix $B^{*}B$. The $\mu_{\alpha}$’s are also the eigenvalues of $M\times M$ matrix $BB^{*}$ and the other eigenvalues of $BB^{*}$ are zeros. Then define $v_{\alpha}$ $(\alpha=1,\ldots,N-1)$ as the normalized eigenvectors of $BB^{*}$ associated with non-zero eigenvalues $\mu_{\alpha}$, i.e., the matrix elements of $BB^{*}$ are given by

where we used the assumption that $\eta<E/2$. Because the eigenvalues $\mu_{\alpha}$’s are the eigenvalues of the $(N-1)\times(N-1)$ matrix $B^{*}B$, they are interlaced with the eigenvalues of $H$ and $|\{\alpha:|\mu_{\alpha}-E|\leq\eta/2\}|\geq{\mathcal{N}}_{I}-1$. It follows from (8.2) that

where in the last step we used Lemma 4.7 from . The claim follows by the assumption that $N\eta\geq N^{\varepsilon}$ and that $N$ and $K$ are large enough.

Consider sample covariance matrices $H=A^{*}A$ with $A$ an $M\times N$ matrix with independent and identically distributed complex entries. Let $0<d<1$. Recall $\lambda_{-}$ and $\lambda_{+}$ in (3.13) and define $\kappa$ as

We will often drop the argument $E$ from the notation of $\kappa$ for brevity. Then for any $E$, $\eta$ satisfying $N^{-1+\varepsilon}\leq\eta\leq\frac{1}{2}E$, $\frac{1}{2}\lambda_{-}\leq E\leq 10$, the Stieltjes transform,

of the empirical eigenvalue distribution of $H=A^{*}A$ satisfies

for any $\delta$ small enough (independent of $E$ and $\eta$) and $N\geq 2$. Here $m_{W}(z)$ is the unique solution of

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

Recall $\lambda_{\pm}=(1\pm d^{1/2})^{2}$ from (3.13). The function $m_{W}$ defined in (8.7) depends on $d$ and can be written as

where $\sqrt{\,\,}$ denotes the square root on complex plane whose branch cut is the negative real line. Explicit calculation shows $m_{W}(z)$ is the Stieltjes transform of the Marchenko-Pastur density given in (3.14).

Using (8.6) and (3.14), we have the local Marchenko-Pastur law for the number of eigenvalues in a small interval:

Consider an interval $I=[E-\eta,E+\eta]\subset[\lambda_{-},\lambda_{+}]$ within the bulk spectrum. Let $\delta$ be a suffiiciently small parameter. Suppose that $E$ and $\eta$ are chosen such that $\delta^{-2}N^{-1+\varepsilon}\leq\eta\leq C^{-1}\min\{\kappa,\delta^{1/2}\kappa^{3/4}\}$ with a large constant $C$ and with $\kappa=\kappa(E)$ given in (8.5). Then we have the convergence of the counting function, i.e.,

where $\mathcal{N}_{\eta}(E)=|\{\lambda_{\alpha}:|\lambda_{\alpha}-E|\leq\eta\}|$ denotes the number of eigenvalues of $H=A^{*}A$ in the interval $I=[E-\eta,E+\eta]$.

Proof of Corollary 8.2. The proof of (8.9) follows from the inequality (8.6) with a similar argument as the proof of Proposition 4.1 of .

We remark that, similarly to Theorem 3.1 in , the assumption on the lower bound $\eta\geq N^{-1+\varepsilon}$ can be relaxed to $\eta\geq KN^{-1}$ and obtain the local Marchenko-Pastur law on the shortest possible scale, at least away from the spectral edges.

Proof of Proposition 8.1. Let $a_{j}$ be the $j$-th column of $A$ and let $B^{(j)}$ be the remaining $M\times(N-1)$ matrix obtained from $A$ after removing the $j$-th column $a_{j}$. Let $\mu^{(j)}_{\alpha}$, $v^{(j)}_{\alpha}$ be the non-zero eigenvalues and the eigenvectors of the matrix $B^{(j)}[B^{(j)}]^{*}$ and we define $\xi_{\alpha}^{(j)}=M|a_{j}\cdot v^{(j)}_{\alpha}|^{2}$. Then we have the formula

Note that the vector $a_{j}$ is independent of $\mu^{(j)}_{\alpha}$ and $v^{(j)}_{\alpha}$. Therefore, we have

Define $m_{N-1}^{(j)}(z)\equiv\frac{1}{N-1}{\rm Tr}([B^{(j)}]^{*}B^{(j)}-z)^{-1}$, then

So with $\frac{1}{2}\lambda_{-}\leq E=Re(z)\leq 10$ and $N^{-1+\varepsilon}\leq\eta\leq E/2$, we have

with some fixed large $K$ (using dyadic decomposition and (8.1), similarly to the argument in Lemma 4.2 of ) apart from an event of probability $e^{-c\sqrt{N\eta}}$. Then with Proposition 4.5 of , we have

for sufficiently small $\delta>0$. Since the eigenvalues of $B^{*}B$ are interlaced with the eigenvalues of $H=A^{*}A$, we have

Combining (8.14), (8.15), and (8.16), we find that

for sufficiently small $\delta>0$. With the assumption $\eta\leq Re(z)/2=E/2$ and $N\eta\geq N^{\varepsilon}$, we obtain

On the other hand, with the definition of $Y^{(j)}$, for any $j$, $z$, $z^{\prime}$ such that, $|z|,|z^{\prime}|\leq 10$, $\mbox{Im}(z),\mbox{Im}(z^{\prime})\geq\eta$, we have

Together with (8.17), we obtain, for $N^{-1+\varepsilon}\leq\eta\leq\frac{1}{2}E$, $\frac{1}{2}\lambda_{-}\leq E\leq 10$ and sufficiently small $\delta$,

where $L(z,P_{10})$ is the line segment connecting points $z$ and $P_{10}$. Then the Proposition 8.1 follows from the next lemma.

Assume $H$ is a $N\times N$ positive semidefinite matrix with $\|H\|\leq 5$. For fixed $0<d<1$, we recall the notation $\lambda_{\pm}=(1\pm\sqrt{d})^{2}$. Let $z_{0}=E+i\eta$ and $N^{-1+\varepsilon}\leq\eta\leq\frac{1}{2}E$, $\frac{1}{2}\lambda_{-}\leq E\leq 10$. Denote $L(z_{0},P_{10})$ the line segment connecting $z_{0}$ and $P_{10}=10+5i$. Suppose that for any $z\in L(z_{0},P_{10})$, the Stieltjes transform $m_{N}(z)=\frac{1}{N}{\rm Tr}(H-z)^{-1}$ satisfies the following self-consistent relation:

for some $Y^{(j)}(z)$’s. Then there exists $\delta_{0}>0$ depending only on $d$, such that, whenever

with $\kappa=\kappa(E):=|(\lambda_{+}-E)(E-\lambda_{-})|$.

Proof of Lemma 8.3. We begin with a special case: $z_{0}=P_{10}$. In this case if $z\in L(z_{0},P_{10})$ then $z=z_{0}=P_{10}$. With the assumptions on $H$ and $0<d<1$, it is easy to see that:

Insert it into (8.20), we obtain when $z=P_{10}$,

by $S^{\Delta}_{\pm}(z)$. Explicit calculation shows

we note $m_{W}(z)=S_{+}(z)$ (see (8.7)). Then the following lemma implies, with ${\rm Im}(m_{N}(z))>0$, ${\rm Im}S_{+}(P_{10})>0$, ${\rm Im}S_{-}(P_{10})<0$, that (8.22) holds for $z_{0}=P_{10}$ if $\delta$ is small enough.

Let $S^{\Delta}_{\pm}(z)$ be the solutions of (8.26). Let $z=E+i\eta$ and $\frac{1}{2}\lambda_{-}\leq E\leq 10$. For sufficiently small $\Delta$, depending on $d$,

Proof of Lemma 8.4. First, when $\Delta$ is small enough, an easy calculation shows

Let $a=(z-\lambda_{-})(\lambda_{+}-z)$ and $b=(z-\lambda^{\Delta}_{-})(\lambda^{\Delta}_{+}-z)-(z-\lambda_{-})(\lambda_{+}-z)$. Note that $|b|\leq C\widetilde{\Delta}$ and therefore, by (8.31), $|b|\leq C\Delta$. Hence, (8.30) follows from $|(\lambda_{+}-z)(z-\lambda_{-})|\geq C\kappa(E)$ and from the inequality