We consider two classical ensembles of the random matrix theory: the Wigner matrices and sample covariance matrices, and prove Central Limit Theorem for linear eigenvalue statistics under rather weak (comparing with results known before) conditions on the number of derivatives of the test functions and also on the number of the entries moments. Moreover, we develop a universal method which allows one to obtain automatically the bounds for the variance of differentiable test functions, if there is a bound for the variance of the trace of the resolvent of random matrix. The method is applicable not only to the Wigner and sample covariance matrices, but to any ensemble of random matrices.
Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, and let $μ_1(M_n), μ_2(M_n), ..., μ_n(M_n)$ denote the unordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics $$ \sum_{i=1}^{n-k} f(μ_i(M_n)) $$ as $n \rightarrow \infty$ for sufficiently nice test functions $f$. We consider both the case when $k$ is fixed and when $\min{k,n-k}$ tends to infinity with $n$.
It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions a…
We prove that the linear statistics of the eigenvalues of a Wigner matrix converge to a universal Gaussian process on all mesoscopic spectral scales, i.e. scales larger than the typical eigenvalue spacing and smaller than the global extent of the spectrum.
We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v, (X^* X - z)^{-1} w \rangle - \langle v,w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w \in \mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be…
We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics.
This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erd{\H o}s-Schlein-Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle. This observable also provides a simple and unified proof of universality in the bulk and at the edge, which is quantitative. To illustrate this, we give the first explicit rate …
These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled …
We consider the single eigenvalue fluctuations of random matrices of general Wigner-type, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue around its classical location are Gaussian with a universal variance. Our method is based on a dynamical approach to mesoscopic linear spectral statistics which reduces their behavior on short scales to that on larger scales. We prove a central limit theorem for linear spectral statistics on larger scales via resolvent techniques and show that for certain class…
In this paper we consider Wigner random matrices -- symmetric n by n random matrices whose entries are independent identically distributed real random variables. We prove that the probability distribution of one or several eigenvalues close to the center of the spectrum does not depend on the probability distribution of the entries of the matrix and is the same as for the Gaussian Orthogonal Ensemble. We make only mild smoothness assumptions on the probability distribution of the entries and assume that the probability distribution of the entries decays polynomially with sufficiently large pow…
We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly $O(N^{-1/3})$, as the matrix dimension $N$ tends to infinity. We allow the variances of the entries of $H$ to have distinct values but of comparable sizes such that $\sum_{i} \mathbb{E}|h_{ij}|^2=1$. Our result improves the previous rate $O(N^{-2/9})$ by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.
We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let $x_k$ denote eigenvalue number $k$. Under the condition that both $k$ and $n-k$ tend to infinity with $n$, we show that $x_k$ is normally distributed in the limit. We also consider the joint limit distribution of $m$ eigenvalues from the GOE or GSE with similar conditions on the indices. The result is an $m$-dimensional normal dist…
We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size $N$ converge to the Tracy--Widom laws at a rate $O(N^{-1/3+ω})$, as $N$ tends to infinity. For Wigner matrices this improves the previous rate $O(N^{-2/9+ω})$ obtained by Bourgade [5] for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős, Yau and Yin [19] to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow indu…
In this paper, we study the complex Wigner matrices $M_n=\frac{1}{\sqrt{n}}W_n$ whose eigenvalues are typically in the interval $[-2,2]$. Let $λ_1\leq λ_2...\leqλ_n$ be the ordered eigenvalues of $M_n$. Under the assumption of four matching moments with the Gaussian Unitary Ensemble(GUE), for test function $f$ 4-times continuously differentiable on an open interval including $[-2,2]$, we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold $u$ in the bulk of the Wigner semicircle law as $\mathcal{A}_n[f; u]=\…
We consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one. We investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large. Under some conditions on moments of the sample distribution, we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex Gaussian sample with the same true covariance. The real setting is also considered.
By the Moutard transformation method we construct two-dimensional Schrodinger operators with real smooth potential decaying at infinity and with a multiple positive eigenvalue. These potentials are rational functions of spatial variables and their sines and cosines.
We show that the linear symplectic and anti-symplectic transformations form the maximal covariance group for both the Wigner transform and Weyl operators. The proof is based on a new result from symplectic geometry which characterizes symplectic and anti-symplectic matrices, and which allows us, in addition, to refine a classical result on the preservation of symplectic capacities of ellipsoids.
After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy--Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Peche.
We extend the results about the fluctuations of the matrix entries of regular functions of Wigner matrices to the case of sample covariance random matrices.
The existence of potentials for relativistic Schrodinger operators allowing eigenvalues embedded in the essential spectrum is a long-standing open problem. We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which an embedded eigenvalue exists. We show that in the non-relativistic limit these potentials converge to the classical Neumann-Wigner and Moses-Tuan potentials, respectively. For the massless operator in one dimension we construct two families of potentials, different by the parities of the (generalized) eigenfun…
We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix $\widetilde{W}$ and its minor $W$. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of $\widetilde{W}$ and $W$. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices the fluctuation may en…
For the eigenvalues of principal submatrices of stochastically evolving Wigner matrices, we construct and study the edge scaling limit: a random decreasing sequence of continuous functions of two variables, which at every point has the distribution of the Airy point process. The analysis is based on the methods developed by Soshnikov to study the extreme eigenvalues of a single Wigner matrix.
We investigate to what extent the microscopic Wigner-Gaudin-Mehta-Dyson (WGMD) (or sine kernel) statistics of random matrix theory remain valid on mesoscopic scales. To that end, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. In the mesoscopic regime, density correlations are much weaker than in the microscopic regime. Our result is an explicit formula for the two-point function. This formula implies that the WGMD statistics are valid to leading order on all mesoscopic scales, that in the real symmetric case there are…
We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $ν(x)= e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $ν(x) \le C e^{- C |x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.
In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\in C^5(\mathbb R)$, we show that the convergence rate is either $N^{-1/2+\varepsilon}$ or $N^{-1+\varepsilon}$, depending on the first Chebyshev coefficient of $f$ and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, …
We consider Hermitian random matrices of the form $H = W + λV$, where $W$ is a Wigner matrix and $V$ a diagonal random matrix independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $λ\sim 1$ so that the eigenvalues of $W$ and $λV$ are of the same order in the bulk of the spectrum. In this paper, we prove for a large class of diagonal matrices $V$ that the local deformed semicircle law holds for $H$, which is an analogous result to the local semicircle law for Wigner matrices. We also prove complete delocalization of eigenvectors and other results about t…
We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from [22,10], by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate $(\log N)/N$ and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bou…
We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp constants. Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components. We then extend this control to all generalized Wigner matrices by comparison arguments based on a framework of regularized eigenvectors, level repulsion, and …
We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $ν_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $ν_{ij}$ c…
In this paper, we extend results of Eigenvector Thermalization to the case of generalized Wigner matrices. Analytically, the central quantity of interest here are multiresolvent traces, such as $Λ_A:= \frac{1}{N} \text{Tr }{ GAGA}$. In the case of Wigner matrices, as in \cite{cipolloni-erdos-schroder-2021}, one can form a self-consistent equation for a single $Λ_A$. There are multiple difficulties extending this logic to the case of general covariances. The correlation structure prevents us from deriving a self-consistent equation for a single matrix $A$; this is due to the introduction of new…
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales $η\gg N^{-1}$. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result \cite{E…
The Wigner-von Neumann method, which was previously used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary $T$-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, $λ$, into the operator's absolutely continuous spectrum. Introducing a new rational function, $C(λ;T)$, related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum …
We analyse the spectrum of additive finite-rank deformations of $N \times N$ Wigner matrices $H$. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue $d_i$ of the deformation crosses a critical value $\pm 1$. This transition happens on the scale $|d_i| - 1 \sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N \to \infty$, we …
Consider $N\times N$ hermitian or symmetric random matrices $H$ with independent entries, where the distribution of the $(i,j)$ matrix element is given by the probability measure $ν_{ij}$ with zero expectation and with variance $σ_{ij}^2$. We assume that the variances satisfy the normalization condition $\sum_{i} σ^2_{ij} = 1$ for all $j$ and that there is a positive constant $c$ such that $c\le N σ_{ij}^2 \le c^{-1}$. We further assume that the probability distributions $ν_{ij}$ have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution…
In the first part of these notes, we review some of the recent developments in the study of the spectral properties of Wigner matrices. In the second part, we present a new proof of a Wegner estimate for the eigenvalues of a large class of Wigner matrices. The Wegner estimate gives an upper bound for the probability to find an eigenvalue in an interval $I$, proportional to the size $|I|$ of the interval.
The recent paper "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" by G.S. Dhesi and M. Ausloos [Phys. Rev. E 93 (2016), 062115] uses the replica method to compute the $1/N$ correction to the Wigner semi-circle law for the ensemble of real symmetric random matrices with $0$'s down the diagonal, and upper triangular entries independently chosen from the values $\pm w$ with equal probability. We point out that the results obtained are inconsistent with known results in the literature, as well as with known large $N$ series expansions for the t…
The distributions of the smallest and largest eigenvalues for the matrix product $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$ determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently for the same distributions.
We consider $N\times N$ Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales $η\gg N^{-1} (\log N)^8$. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the sp…
We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the eigenvalue densities for these ensembles. In all cases the joint eigenvalue density exhibits a biorthogonal structure. A determinantal representation, based on a generalization of Andréief's integration formula, is used to compactly express the $r$-point correlation function of eigenvalues. This representation circumvents the complications encountered in the u…
Nowadays, strict finite size effects must be taken into account in condensed matter problems when treated through models based on lattices or graphs. On the other hand, the cases of directed bonds or links are known as highly relevant, in topics ranging from ferroelectrics to quotation networks. Combining these two points leads to examine finite size random matrices. To obtain basic materials properties, the Green function associated to the matrix has to be calculated. In order to obtain the first finite size correction a perturbative scheme is hereby developed within the framework of the repl…
We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related, and in fact in competition, with the possibility of damping the interferences which arise due to the quadratic nature of the distributions. We also show that the well known fully covariance property of the Wigner distribution in fact characterizes it (up to a constant factor) among the quadratic distributions $L^{2}(\mathbb{R}^{n})\rightarrow C_{0}({ \mathbb{R}^{2n}})$. A similar characterization for the closely related Weyl …
We consider an arbitrary square integrable function $F$ on the phase space and look for the Wigner function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate t…
We consider $N\times N$ symmetric or hermitian random matrices with independent, identically distributed entries where the probability distribution for each matrix element is given by a measure $ν$ with a subexponential decay. We prove that the local eigenvalue statistics in the bulk of the spectrum for these matrices coincide with those of the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE), respectively, in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. We also show tha…
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance $N^{-3/4+β}$ for some positive $β>0$. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.
We consider $n^2\times n^2$ real symmetric and hermitian matrices $M_n$, which are equal to sum of $m_n$ tensor products of vectors $X^μ=B(Y^μ\otimes Y^μ)$, $μ=1,\dots,m_n$, where $Y^μ$ are i.i.d. random vectors from $\mathbb R^n (\mathbb C^n)$ with zero mean and unit variance of components, and $B$ is an $n^2\times n^2$ positive definite non-random matrix. We prove that if $m_n/n^2\to c\in [0,+\infty)$ and the Normalized Counting Measure of eigenvalues of $BJB$, where $J$ is defined below, converges weakly, then the Normalized Counting Measure of eigenvalues of $M_n$ converges weakly in proba…
We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup $U_q[gl(m|n)]$. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients.
We revisit the classic Wigner semi-circle from two different angles. One consists in studying the Stieltjes transform directly on the real axis, which does not converge to a fixed value but follows a Cauchy distribution that depends on the local eigenvalue density. This result was recently proven by Aizenman \& Warzel for a wide class of eigenvalue distributions. We shed new light onto their result using a Coulomb gas method. The second angle is to derive a Langevin equation for the full (matrix) resolvent, extending Dyson's Brownian motion framework. The full matrix structure of this equation…
Real non-symmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudo-symmetric as $ηM η^{-1} = M^t$, where the metric $η$ could be secular (a constant matrix) or depending upon the matrix elements of $M$. Here, we construct ensembles of a large number $N$ of pseudo-symmetric $n \times n$ ($n$ large) matrices using ${\cal N}$ $(n(n+1)/2 \le {\cal N} \le n^2)$ independent and identically distributed (iid) random numbers as their elements. Based on our numerical calculations, we conjecture that for these ensembles the Nearest Level Spacing Dis…
We derive the joint distribution of the moments $\mathrm{Tr}\, Q^κ$ ($κ\geq0$) of the Wigner-Smith matrix for a chaotic cavity supporting a large number of scattering channels $n$. This distribution turns out to be asymptotically Gaussian, and we compute explicitly averages and covariances. The results are in a compact form and have been verified numerically. The general methodology of proof and computations has a wide range of applications.
