Strong Convergence of Unitary Brownian Motion

TL;DR

This paper proves that the unitary Brownian motion on matrices converges strongly to the free unitary Brownian motion, with implications for spectral measures and joint convergence with other ensembles.

Contribution

It establishes strong convergence of unitary Brownian motion to its free counterpart, including spectral edge behavior and joint convergence results.

Findings

Spectral measure has a hard edge with no outliers.

Strong convergence extends to joint ensembles independent of the Brownian motion.

Application to the Jacobi process demonstrates practical implications.

Abstract

The Brownian motion $(U^N_t)_{t\ge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{t\ge 0}$ as $N\to\infty$. In this paper, we prove that it converges strongly as a process: not only in distribution but also in operator norm. In particular, for a fixed time $t>0$, we prove that the spectral measure has a hard edge: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly jointly with the Brownian motion. We give an application of this strong convergence to the Jacobi process.