Fluctuations of Brownian Motions on GL_N

TL;DR

This paper studies a family of invariant diffusion processes on GL_N, showing they exhibit Gaussian fluctuations in high dimensions with explicit covariance structure, extending previous results on unitary groups.

Contribution

It introduces a broad class of diffusion processes on GL_N, proves Gaussian fluctuation behavior with explicit covariance, and employs a geometric approach rather than combinatorial methods.

Findings

Gaussian fluctuations occur with error O(1/N)

Explicit covariance characterized via free multiplicative Brownian motions

Generalizes earlier results on unitary groups

Abstract

We consider a two parameter family of unitarily invariant diffusion processes on the general linear group $\mathbb{GL}_N$ of $N\times N$ invertible matrices, that includes the standard Brownian motion as well as the usual unitary Brownian motion as special cases. We prove that all such processes have Gaussian fluctuations in high dimension with error of order $O(1/N)$; this is in terms of the finite dimensional distributions of the process under a large class of test functions known as trace polynomials. We give an explicit characterization of the covariance of the Gaussian fluctuation field, which can be described in terms of a fixed functional of three freely independent free multiplicative Brownian motions. These results generalize earlier work of L\'evy and Ma\"ida, and Diaconis and Evans, on unitary groups. Our approach is geometric, rather than combinatorial.