Concentration and convergence rates for spectral measures of random matrices

TL;DR

This paper investigates the typical behavior and convergence rates of spectral measures of large random matrices from various ensembles, providing estimates and concentration results for the Wasserstein distance to deterministic measures.

Contribution

It introduces new estimates and concentration inequalities for spectral measures of diverse random matrix ensembles, establishing almost sure convergence results.

Findings

Expected Wasserstein distance bounds for spectral measures

Concentration inequalities for spectral measure distances

Almost sure convergence of empirical spectral measures

Abstract

The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.