Convergence of the empirical spectral distribution of Gaussian matrix-valued processes

TL;DR

This paper proves that the empirical spectral distribution of Gaussian matrix-valued processes converges to a deterministic limit characterized by a Burgers' equation, extending previous results to cases where eigenvalues may collide.

Contribution

It introduces a new convergence result for spectral measures of Gaussian matrix processes without requiring eigenvalue non-collision assumptions.

Findings

Empirical spectral measures converge in probability to a deterministic limit.

The limit is characterized by a Cauchy transform solving a Burgers' equation.

Extends previous work to cases with eigenvalue collision, including fractional Brownian motion with H<1/2.

Abstract

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(\lambda_{1}^{(n)}(t),\dots, \lambda_{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $\mu^{(n)}=(\mu_{t}^{(n)}; t\geq0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $\mu^{(n)}$ converges in probability to a deterministic limit $\mu$, in the topology of uniform convergence over compact sets. We show that the process $\mu$ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers' equation. Our results extend those of Rogers and Shi for the free Brownian motion and Pardo et al. for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.