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

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 ${\mu }^{(n)}=({\mu }_t^{(n)};t\ge 0)$. 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.