Random Matrices: Theory and Applications

Consider the random matrix $W_n=B_n+n^{-1}{X}_n^{\ast }{A}_n{X}_n$, where $A_n$ and $B_n$ are Hermitian matrices of dimensions $p\times p$ and $n\times n$, respectively, and $X_n$ is a $p\times n$ random matrix with independent and identically distributed entries of mean 0 and variance 1. Assume that $p$ and $n$ grow to infinity proportionally, and that the spectral measures of $A_n$ and $B_n$ converge as $p,n\to \infty$ towards two probability measures $\mathcal{𝒜}$ and $\mathcal{ℬ}$. Building on the groundbreaking work of [V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sb. 114(4) (1967) 507–536], which demonstrated that the empirical spectral distribution of $W_n$ converges towards a probability measure $F$ characterized by its Stieltjes transform, this paper investigates the properties of $F$ when $\mathcal{ℬ}$ is a general measure. We show that $F$ has an analytic density at the region near where the Stieltjes transform of $\mathcal{ℬ}$ is bounded. The density closely resembles $C\sqrt{|x-x_0|}$ near certain edge points $x_0$ of its support for a wide class of $\mathcal{𝒜}$ and $\mathcal{ℬ}$. We provide a complete characterization of the support of $F$. Moreover, we show that $F$ can exhibit discontinuities at points where $\mathcal{ℬ}$ is discontinuous.

We examine the spectral properties of balanced stochastic block models where the average degree grows slower than the number of nodes (sparse regime) or proportionally to it (dense regime). For both regimes, we demonstrate a phase transition in the extreme eigenvalues of the stochastic block model at a specific threshold, which aligns with the Kesten–Stigum threshold in the sparse regime. Additionally, we establish the central limit theorem for the linear spectral statistics in both regimes. Based on the central limit theorem for the linear spectral statistics, we propose a hypothesis test to determine the presence of communities in the graph.

This paper discusses the multivariate Behrens–Fisher problem, which tests the equality of two population mean vectors under heteroscedasticity. The classical Wald-type test is not applicable in high dimensions owing to the singularity of sample covariance matrices. A ridgelized Wald-type (RIWT) test is then proposed. Using the exact four-moment theorem, its asymptotic null distribution is derived under moment conditions. Thus, our test can accommodate non-Gaussian distributions with general covariance matrices. Simulation results demonstrate the good performance of the proposed test.

Let ${\mathrm{\Lambda }}_X(s)=det(I-s{X}^{†})$ be the characteristic polynomial of a Haar distributed unitary matrix $X$. It is believed that the distribution of values of ${\mathrm{\Lambda }}_X(s)$ model the distribution of values of the Riemann zeta-function $\zeta (s)$. This principle motivates many avenues of study. Of particular interest is the behavior of ${\mathrm{\Lambda }}_X^{\prime }(s)$ and the distribution of its zeros (all of which lie inside or on the unit circle). In this paper, we present several identities for the moments of ${\mathrm{\Lambda }}_X^{\prime }(s)$ averaged over $U(N)$, for $s\in ℂ$ as well as specialized to $\left|s\right|=1$. Additionally, we prove, for positive integer $k$, that the polynomial ${\int }_{U(N)}|{\mathrm{\Lambda }}_X(1){|}^{2k}dX$ of degree $k^2$ in $N$ divides the polynomial ${\int }_{U(N)}|{\mathrm{\Lambda }}_X^{\prime }(1){|}^{2k}dX$ which is of degree $k^2+2k$ in $N$ and that the ratio, $f(N,k)$, of these moments factors into linear factors modulo $4k-1$ if $4k-1$ is prime. We also discuss the relationship of these moments to a solution of a second-order nonlinear Painléve differential equation. Finally we give some formulas in terms of the ${}_3{F}_2$ hypergeometric series for the moments in the simplest case when $N=2$, and also study the radial distribution of the zeros of ${\mathrm{\Lambda }}_X^{\prime }(s)$ in that case.

The purpose of this note is to study the asymptotic volume of intersections of unit balls associated with two norms in ${ℝ}^n$ as their dimension $n$ tends to infinity. A general framework is provided and then specialized to the following cases. For classical ${\ell }_p^n$-balls the focus lies on the case $p=\infty$, which has previously not been studied in the literature. As far as Schatten $p$-balls are considered, we concentrate on the cases $p=2$ and $p=\infty$. In both situations we uncover an unconventional limiting behavior.

In this paper, we study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial leading coefficients of the monic OPs, the associated Hankel determinants and the squares of $L^2$-norm of the monic OPs. These results are derived from the combination of the ladder operator approach, Dyson’s Coulomb fluid approach and some recent results in the literature.