Spectral measures of powers of random matrices

TL;DR

This paper studies the spectral distribution of powers of random matrices from classical groups, providing bounds on Wasserstein distances, convergence rates, and establishing a sharp logarithmic Sobolev inequality on the unitary group.

Contribution

It offers new sharp bounds on spectral measures of matrix powers and proves a key inequality on the unitary group, advancing understanding of spectral convergence.

Findings

Sharp bounds on Wasserstein distances between spectral measures and uniform measure

Identification of a smooth transition in spectral behavior with increasing power

Proof of the sharp logarithmic Sobolev inequality on the unitary group

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.