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

TL;DR

This paper proves that the spectral measures of certain Hermitian random matrices with dependent entries concentrate around their mean, extending previous results to larger block sizes and applying to various structured matrices.

Contribution

It extends concentration results for spectral measures of Hermitian matrices with dependent entries to larger block sizes and broadens applications to structured and heavy-tailed matrices.

Findings

Spectral measures concentrate around their mean for block sizes up to o(n^2).

Weak convergence to the semicircle law is established for matrices with dependent entries.

Applications include Toeplitz, Hankel, circulant matrices, and heavy-tailed distributions.

Abstract

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 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 a recent result by Kemp and Zimmerman, where the size of 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.