Empirical Spectral Distribution Research Articles

Local Semicircle Law Under Fourth Moment Condition

We consider a random symmetric matrix $$\mathbf{X}= [X_{jk}]_{j,k=1}^n$$ with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that $$ \max _{jk} {{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4+\delta } 0$$ , it was proved in Gotze et al. (Bernoulli 24(3):2358–2400, 2018) that with high probability the typical distance between the Stieltjes transforms $$m_n(z)$$ , $$z = u + i v$$ , of the empirical spectral distribution (ESD) and the Stieltjes transforms $$m_{\text {sc}}(z)$$ of the semicircle law is of order $$(nv)^{-1} \log n$$ . The aim of this paper is to remove $$\delta >0$$ and show that this result still holds if we assume that $$ \max _{jk} {{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4} < \infty $$ . We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.

On optimal bounds in the local semicircle law under four moment condition

We consider symmetric random matrices with independent mean zero and unit variance entries in the upper triangular part. Assuming that the distributions of matrix entries have finite moment of order four, we prove optimal bounds for the distance between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law. Application concerning the convergence rate in probability of the empirical spectral distribution to the semicircle law is discussed as well.

Local Law for Eigenvalues of Random Regular Bipartite Graphs

In this paper, we study the local law for eigenvalues of large random regular bipartite graphs with degree growing moderately fast. We prove that the empirical spectral distribution of the adjacency matrix converges in probability to a scaled down copy of the Marchenko–Pastur distribution on intervals of short length.

Sum rules and large deviations for spectral matrix measures

In the paradigm of random matrices, one of the most classical object under study is the empirical spectral distribution. This random measure is the uniform distribution supported by the eigenvalues of the random matrix. In this paper, we give large deviation theorems for another popular object built on Hermitian random matrices: the spectral measure. This last probability measure is a random weighted version of the empirical spectral distribution. The weights involve the eigenvectors of the random matrix. We have previously studied the large deviations of the spectral measure in the case of scalar weights. Here, we will focus on matrix valued weights. Our probabilistic results lead to deterministic ones called “sum rules” in spectral theory. A sum rule relative to a reference measure on $\mathbb{R}$ is a relationship between the reversed Kullback–Leibler divergence of a positive measure on $\mathbb{R}$ and some non-linear functional built on spectral elements related to this measure. By using only probabilistic tools of large deviations, we extend the sum rules to the case of Hermitian matrix-valued measures.

Structure of eigenvectors of random regular digraphs

Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and sloping with many levels. As a corollary of this theorem, we show that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: all subsets of coordinates with the same value have cardinality less than $Cn\ln^2 d/\ln n$. For a large constant $d$ this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with $n$, and -- combined with additional arguments -- that the rank of the adjacency matrix $M$ is at least $n-1$, with probability going to 1 as $n\to\infty$. Moreover, our main result for $d$ slowly growing to infinity with $n$ contributes a crucial step towards establishing the circular law for the empirical spectral distribution of sparse random $d$-regular directed graphs. As a key technical device we introduce a decomposition of ${\mathcal C}^n$ into vectors of different degree of structuredness, which is an alternative to the decomposition based on the least common denominator in the regime when the underlying random matrix is very sparse.

Analysis on the Empirical Spectral Distribution of Large Sample Covariance Matrix and Applications for Large Antenna Array Processing

This paper addresses the asymptotic behavior of a particular type of information-plus-noise-type matrices, where the column and row number of the matrices are large and of the same order, while signals are diverged and time delays of the channel are fixed. We prove that the empirical spectral distribution (ESD) of the large dimension sample covariance matrix and a well-studied spiked central Wishart matrix converge to the same distribution. As an application, an asymptotic power function is presented for the general likelihood ratio statistics for testing the presence of signal in large array signal processing.

Convergence of the empirical spectral distribution of Gaussian matrix-valued processes

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(\lambda _{1}^{(n)}(t),\dots , \lambda _{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $\mu ^{(n)}=(\mu _{t}^{(n)}; t\geq 0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $\mu ^{(n)}$ converges in probability to a deterministic limit $\mu $, in the topology of uniform convergence over compact sets. We show that the process $\mu $ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers’ equation. Our results extend those of Rogers and Shi [14] for the free Brownian motion and Pardo et al. [12] for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.

Concentration of the empirical spectral distribution of random matrices with dependent entries

We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_{n}$ be a Hermitian random matrix of the size $n\times n$ that can be split into independent blocks of the size at most $d_{n}=o(n^{2})$. We prove that under some mild conditions on the distribution of the entries of $X_{n}$, the empirical spectral measure of $X_{n}$ concentrates around its mean. The main theorem is a strengthening of the recent result by Kemp and Zimmerman, where the size of the blocks grows as $o(\log n)$. As an application, we are able to upgrade the results of Schenker and Schulz on the convergence in expectation to the semicircle law of a class of random matrices with dependent entries to weak convergence in probability. Other applications include patterned random matrices, e.g. matrices of Toeplitz, Hankel or circulant type and matrices with heavy tailed entries in the domain of attraction of the Gaussian distribution.

On the local semicircular law for Wigner ensembles

We consider a random symmetric matrix $\mathbf{X}=[X_{jk}]_{j,k=1}^{n}$ with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that $\mathbb{E}|X_{11}|^{4+\delta}=:\mu_{4+\delta} 0$. The aim of this paper is to significantly extend a recent result of the authors Gotze, Naumov and Tikhomirov (2015) and show that with high probability the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}}\mathbf{X}$ and Wigner’s semicircle law is of order $(nv)^{-1}\log n$, where $v$ denotes the distance to the real line in the complex plane. We apply this result to the rate of convergence of the ESD to the distribution function of the semicircle law as well as to rigidity of eigenvalues and eigenvector delocalization significantly extending a recent result by Gotze, Naumov and Tikhomirov (2015). The result on delocalization is optimal by comparison with GOE ensembles. Furthermore the techniques of this paper provide a new shorter proof for the optimal $O(n^{-1})$ rate of convergence of the expected ESD to the semicircle law.

Concentration for Coulomb gases and Coulomb transport inequalities

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann–Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Maïda and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.

The semicircle law for matrices with ergodic entries

We study the empirical spectral distribution (ESD) of symmetric random matrices with ergodic entries on the diagonals. We observe that for entries with correlations that decay to 0, when the distance of the diagonal entries becomes large the limiting ESD is the well known semicircle law. If it does not decay to 0 (and have the same sign) the semicircle law cannot be the limit of the ESD. This is in good agreement with results on exchangeable processes analyzed in Friesen and Löwe (2013) and Hochstättler et al. (2016).

The Traffic Distribution of the Squared Unimodular Random Matrix and a Formula for the Moments of its ESD

The k-th moment of the mean empirical spectral distribution (ESD) of the squared unimodular random matrix of dimension N can be expressed in the form N−2k—1Qk(N), where Qk(x) is a polynomial of degree k + 1 with integer coefficients. We use tools from traffic-free probability to express the coefficients of this polynomial in terms of the number of quotients, with a certain property, of some colored directed graphs. The obtained result disproves the formula conjectured in A. Lakshminarayan, Z. Puchala, K. Życzkowski [3] .

Optimal Filter Design for Signal Processing on Random Graphs: Accelerated Consensus

In graph signal processing, filters arise from polynomials in shift matrices that respect the graph structure, such as the graph adjacency matrix or the graph Laplacian matrix. Hence, filter design for graph signal processing benefits from knowledge of the spectral decomposition of these matrices. Often, stochastic influences affect the network structure and, consequently, the shift matrix empirical spectral distribution. Although the joint distribution of the shift matrix eigenvalues is typically inaccessible, deterministic functions that asymptotically approximate the matrix empirical spectral distribution can be found for suitable random graph models using tools from random matrix theory. We employ this information regarding the density of eigenvalues to develop criteria for optimal graph filter design. In particular, we consider filter design for distributed average consensus and related problems, leading to improvements in short-term error minimization or in asymptotic convergence rate.

The spectral norm of random inner-product kernel matrices

We study an “inner-product kernel” random matrix model, whose empirical spectral distribution was shown by Xiuyuan Cheng and Amit Singer to converge to a deterministic measure in the large n and p limit. We provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a Marchenko–Pastur law. By comparing the tracial moments of this matrix to an additive deformation of a Wigner matrix, we establish that for odd kernel functions, the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum. Our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components, and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting.

Circular law for the sum of random permutation matrices

Let $P_n^1,\dots , P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum _{\ell =1}^d P_n^\ell $. We show that if $\log ^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d} $ converges weakly to the circular law in probability as $n \to \infty $.

Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs

For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution $\mu_n^Y$ of the rescaled entry-wise product \[ Y_n = \left(\frac1{\sqrt{n}} \sigma_{ij}X_{ij}\right). \] For our main result we provide a deterministic sequence of probability measures $\mu_n$, each described by a family of Master Equations, such that the difference $\mu^Y_n - \mu_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger--Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.

Large Deviation Theorem for Zeros of Polynomials and Hermitian Random Matrices

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled.

Empirical Spectral Distribution of a Matrix Under Perturbation

We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first matrix are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are related either to the one-dimensional Gaussian free field or to free probability theory.

Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from n independent observations of a p-dimensional time series with iid components converge almost surely to (1+γ)2 and (1−γ)2, respectively, as n→∞, if p∕n→γ∈(0,1] and the truncated variance of the entry distribution is “almost slowly varying”, a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marčenko–Pastur law, which extends a result in Bai and Zhou (2008). We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study.In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.

Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees

We analyze the eigenvalues of a random graph ensemble, proposed by Chung and Lu, in which a given sequence of expected degrees, denoted by $\overline{w}_n = (w^{(n)}_1,\ldots, w^{(n)}_n)$ , is prescribed on the $n$ nodes of a random graph. We focus on the eigenvalues of the normalized (random) adjacency matrix of the graph ensemble, defined as $\mathbf {A}_n$ $=$ $\sqrt{n\rho _n}[\mathbf {a}^{(n)}_{i,j}]_{i,j=1}^{n}$ , where $\rho _n = 1/\sum _{i=1}^{n} w^{(n)}_i$ and $\mathbf {a}^{(n)}_{i,j} =1$ if there is an edge between the nodes $\lbrace i,j\rbrace$ , 0 otherwise. The empirical spectral distribution of $\mathbf {A}_n$ , denoted by $\mathbf {F}_n(\mathord {\cdot})$ , is the empirical measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the symmetric matrix $\mathbf {A}_n$ . Under some technical conditions on the expected degree sequence, we show that with probability one $\mathbf {F}_n(\mathord {\cdot})$ converges weakly to a deterministic distribution $F(\mathord {\cdot})$ as $n\rightarrow \infty$ . Furthermore, we fully characterize this deterministic distribution by providing explicit closed-form expressions for the moments of $F(\mathord {\cdot})$ . We illustrate our results with two well-known degree distributions, namely, the power-law and the exponential degree distributions. Based on our results, we provide significant insights about the bulk behavior of the eigenvalue spectrum; in particular, we analyze the quasi-triangular spectral distribution of power-law networks.