Extremal singular values of random matrix products and Brownian motion on GL(N,C)

TL;DR

This paper proves universality for the largest singular values of large products of random matrices with invariant distributions, linking their behavior to Dyson Brownian motion and Brownian motion on GL(N,C) through advanced mathematical tools.

Contribution

It introduces a novel approach using multivariate Bessel generating functions to analyze the extremal singular values of matrix products in the large size limit.

Findings

Largest singular values follow Dyson Brownian motion with specific drift.

Universality holds for products of random matrices with invariant distributions.

Behavior matches that of Brownian motion on GL(N,C) in the large N limit.

Abstract

We establish universality for the largest singular values of products of random matrices with right unitarily invariant distributions, in a regime where the number of matrix factors and size of the matrices tend to infinity simultaneously. The behavior of the largest log singular values coincides with the large N limit of Dyson Brownian motion with a characteristic drift vector consisting of equally spaced coordinates, which matches the large N limit of the largest log singular values of Brownian motion on GL(N, C). Our method utilizes the formalism of multivariate Bessel generating functions, also known as spherical transforms, to obtain and analyze combinatorial expressions for observables of these processes.