Strong Convergence of Multiplicative Brownian Motions on the General Linear Group

TL;DR

This paper proves that multiplicative Brownian motions on the general linear group strongly converge to free multiplicative Brownian motion as dimension increases, extending previous results and including deterministic matrices.

Contribution

It establishes the almost sure strong convergence of a broad family of multiplicative Brownian motions on GL(n) to their free counterparts, generalizing prior special cases.

Findings

Almost sure strong convergence of finite-dimensional marginals

Convergence of noncommutative distribution and operator norm

Extension to deterministic matrices alongside Brownian motions

Abstract

We consider the family of multiplicative Brownian motions $G_{\lambda,\tau}$ on the general linear group introduced by Driver-Hall-Kemp. They are parametrized by the real variance $\lambda\in \mathbb{R}$ and the complex covariance $\tau \in \mathbb{C}$ of the underlying elliptic Brownian motion. We show the almost sure strong convergence of the finite-dimensional marginals of $G_{\lambda,\tau}$ to the corresponding free multiplicative Brownian motion introduced by Hall-Ho: as the dimension tends to infinity, not only does the noncommutative distribution converge almost surely, but the operator norm does as well. This result generalizes the work of Collins-Dahlqvist-Kemp for the special case $(\lambda,\tau)=(1,0)$ which corresponds to the Brownian motion on the unitary group. Actually, this strong convergence remains valid when the family of multiplicative Brownian motions $G_{\lambda,\tau}$ is considered alongside a family of strongly converging deterministic matrices.