Random Matrices: Theory and Applications

We study the empirical spectral distribution (ESD) for complex $n\times n$ matrix polynomials of degree k under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) $n\to \infty$ with k constant and (2) $k\to \infty$ with n bounded by $O(k^P)$ for some $P>0$; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on k (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

We consider linear spectral statistics built from the block-normalized correlation matrix of a set of M mutually independent scalar time series. This matrix is composed of $M^2$ blocks. Each block has size $L\times L$ and contains the sample cross-correlation measured at L consecutive time lags between each pair of time series. Let N denote the total number of consecutively observed windows that are used to estimate these correlation matrices. We analyze the asymptotic regime where $M,L,N\to +\infty$ while $ML/N\to c_{\star }$, $0<c_{\star }<\infty$. We study the behavior of linear statistics of the eigenvalues of this block correlation matrix under these asymptotic conditions and show that the empirical eigenvalue distribution converges to a Marcenko–Pastur distribution. Our results are potentially useful in order to address the problem of testing whether a large number of time series are uncorrelated or not.

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures $\mathrm{\Xi }$ on a space S with measure $\lambda$, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, $H_{\ell },\ell =1,2$, which are assumed to be realized as $L^2$-spaces, $L^2(S_{\ell },{\lambda }_{\ell })$, $\ell =1,2$, and introduce a bounded linear operator ${𝒲}:H_1\to H_2$ and its adjoint ${{𝒲}}^{\ast }:H_2\to H_1$. We show that if ${𝒲}$ is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP $({\mathrm{\Xi }}_1,K_1,{\lambda }_1)$ associated with ${{𝒲}}^{\ast }{𝒲}$. In addition, if ${{𝒲}}^{\ast }$ is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, $({\mathrm{\Xi }}_{\ell },K_{\ell },{\lambda }_{\ell })$, $\ell =1,2$. We also give a practical framework which makes ${𝒲}$ and ${{𝒲}}^{\ast }$ satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ($d\in \mathbb{N}$) series of infinite DPPs on $S={ℝ}^d$ and ${ℂ}^d$ are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

In this paper, we consider the spectrum of a Laplacian matrix, also known as Markov matrices where the entries of the matrix are independent but have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that the variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution converges. We give an expression for the moments of the limiting measure in terms of graph homomorphisms. In some special cases, we identify the limit explicitly. We also study the spectral norm and derive the order of the maximum eigenvalue. We show that our results cover Laplacians of various random graphs including inhomogeneous Erdős–Rényi random graphs, sparse W-random graphs, stochastic block matrices and constrained random graphs.

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix ${{𝒞}}_n$ of dimension $n$ under the existence of the moment generating function of the random entries is $\kappa ({{𝒞}}_n)=\text{O}(\frac{1}{𝜀}{n}^{\rho +1/2}{(logn)}^{1/2})$ with probability $1-\text{O}(({𝜀}^2+𝜀)n^{-2\rho }+n^{-1/2+\text{o}(1)})$ for any $𝜀>0$, $\rho \in (0,1/4)$. Moreover, if the random entries only have the second moment, the condition number satisfies $\kappa ({{𝒞}}_n)=\text{O}(\frac{1}{𝜀}{n}^{\rho +1/2}logn)$ with probability $1-\text{O}(({𝜀}^2+𝜀)n^{-2\rho }+{(logn)}^{-1/2})$. Also, we analyze the condition number of a random symmetric circulant matrix ${{𝒞}}_n^{\mathrm{\text{sym}}}$. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix ${{𝒯}}_n$ we establish $\kappa ({{𝒯}}_n)\le \kappa ({{𝒞}}_{2n}){({\sigma }_{min}(C_{2n}){\sigma }_{min}({{𝒮}}_n))}^{-1}$, where ${\sigma }_{min}(A)$ is the minimum singular value of the matrix $A$. The matrix $C_{2n}$ is a random circulant matrix and ${{𝒮}}_n:=F_{2,n}^{\ast }{D}_{1,n}^{-1}{F}_{2,n}+F_{4,n}^{\ast }{D}_2^{-1}{F}_{4,n}$, where $F_{2,n},F_{4,n}$ are deterministic matrices, $F^{\ast }$ indicates the conjugate transpose of $F$ and $D_{1,n},D_{2,n}$ are random diagonal matrices. From random experiments, we conjecture that ${{𝒮}}_n$ is well-conditioned if the moment generating function of the random entries of ${{𝒞}}_{2n}$ exists.

In this paper, a new concept for a $3\times 3$ block relation matrix is studied in a Banach space. It is shown that, under certain condition, we can investigate the Frobenius–Schur decomposition of relation matrices. Furthermore, we present some conditions which should allow the multivalued $3\times 3$ matrices linear operator to be closable.

The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form $S_n=\frac{1}{n}{B}_n{X}_n{X}_n^{\ast }{B}_n^{\ast }$, where $B_n$ is a $p\times m$ nonrandom matrix and $X_n$ is an $m\times n$ matrix consisting of i.i.d standard complex entries. $p/n\to c\in (0,\infty )$ as $n\to \infty$ while m can be arbitrary but no smaller than p. We first prove that under some mild assumptions, with probability 1, for all large n, there will be no eigenvalues in any closed interval contained in an open interval which is outside the supports of the limiting distributions for all sufficiently large n. Then we get the strong convergence result for the extreme eigenvalues as an extension of Bai-Yin law.

Given $n,m\in \mathbb{N}$, we study two classes of large random matrices of the form

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our results for the sparse SYK random matrices resembles the formulas for higher order freeness for ordinary GUE random matrices.

For $N,n\in \mathbb{N}$, consider the sample covariance matrix $S_N(T)=\frac{1}{N}X{X}^{\ast }$ from a data set $X=C_N^{1/2}Z{T}_n^{1/2}$, where $Z=(Z_{i,j})$ is a $N\times n$ matrix having i.i.d. entries with mean zero and variance one, and $C_N,T_n$ are deterministic positive semi-definite Hermitian matrices of dimension $N$ and $n$, respectively. We assume that ${(C_N)}_N$ is bounded in spectral norm, and $T_n$ is a Toeplitz matrix with its largest eigenvalues diverging to infinity. The matrix $X$ can be viewed as a data set of an $N$-dimensional long memory stationary process having separable dependence structure. As $N,n\to \infty$ and $N{n}^{-1}\to r\in (0,\infty )$, we establish the asymptotics and the joint CLT for $({\lambda }_1(S_N(T)),\dots ,{\lambda }_m{(S_N(T))}^{\top }$, where ${\lambda }_j(S_N(T))$ denotes the $j$th largest eigenvalue of $S_N(T)$ and $m$ is a fixed integer. For the CLT, we first study the case where the entries of $Z$ are Gaussian, and then we generalize the result to some more generic cases. This result substantially extends our previous result in Merlevède et al. 2019, where we studied ${\lambda }_1(S_N(T))$ in the case where $m=1$ and $X=Z{T}_n^{1/2}$ with $Z$ having Gaussian entries. In order to establish this CLT, we are led to study the first order asymptotics of the largest eigenvalues and the associated eigenvectors of some deterministic Toeplitz matrices related to long memory stationary processes. We prove multiple spectral gap properties for the largest eigenvalues and a delocalization property for their associated eigenvectors.