Convergence of the empirical spectral measure of unitary Brownian motion

TL;DR

This paper provides explicit bounds on how quickly the empirical spectral measure of a unitary Brownian motion converges to its expected and limiting spectral measures, with implications for understanding spectral measure dynamics over time.

Contribution

It introduces explicit bounds on Wasserstein distances for spectral measures of unitary Brownian motion, advancing convergence rate analysis in random matrix theory.

Findings

Explicit bounds on Wasserstein distance to limiting measures

Controlled convergence rates over compact time intervals

Application of advanced tools for spectral measure analysis

Abstract

Let $\{U^N_t\}_{t\ge 0}$ be a standard Brownian motion on $\mathbb{U}(N)$. For fixed $N\in\mathbb{N}$ and $t>0$, we give explicit bounds on the $L_1$-Wasserstein distance of the empirical spectral measure of $U^N_t$ to both the ensemble-averaged spectral measure and to the large-$N$ limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.