Spectral measures of powers of random matrices

Elizabeth Meckes,Mark Meckes

doi:10.1214/ecp.v18-2551

Elizabeth Meckes, Mark Meckes

Open Access

https://doi.org/10.1214/ecp.v18-2551

Copy DOI

Abstract

This paper considers the empirical spectral measure of a power of a random matrix drawn uniformly from one of the compact classical matrix groups. We give sharp bounds on the $L_p$-Wasserstein distances between this empirical measure and the uniform measure on the circle, which show a smooth transition in behavior when the power increases and yield rates on almost sure convergence when the dimension grows. Along the way, we prove the sharp logarithmic Sobolev inequality on the unitary group.

Highlights

The eigenvalues of large random matrices drawn uniformly from the compact classical groups are of interest in a variety of fields, including statistics, number theory, and mathematical physics; see e.g. [7] for a survey
This paper considers the empirical spectral measure of a power of a random matrix drawn uniformly from one of the compact classical matrix groups
We give sharp bounds on the Lp-Wasserstein distances between this empirical measure and the uniform measure on the circle, which show a smooth transition in behavior when the power increases and yield rates on almost sure convergence when the dimension grows

Summary

Introduction

The eigenvalues of large random matrices drawn uniformly from the compact classical groups are of interest in a variety of fields, including statistics, number theory, and mathematical physics; see e.g. [7] for a survey. In this paper we quantify in a precise way the degree of uniformity of the eigenvalues of U m when U is drawn uniformly from any of the classical compact groups U (N ), SU (N ), O (N ), SO (N ), and Sp (2N ) We do this by bounding, for any p ≥ 1, the mean and tails of the Lp-Wasserstein distance Wp between the empirical spectral measure μN,m of U m and the uniform measure ν on S1 (see Section 4 for the definition of Wp). We prove the logarithmic Sobolev inequality on U (N ) with a constant of optimal order

A miraculous representation of the eigenvalue counting function

Means and variances

Wasserstein distances

Other groups