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 {(λ1(n) (t),…,λn(n)(t)); t ≥ 0} as well as its corresponding process of empirical spectral measures μ(n) = (μt(n);t ≥ 0). Under some mild conditions on the covariance function associated to Y(n), we prove that the process μ(n) converges in probability to a deterministic limit μ, in the topology of uniform convergence over compact sets. We show that the process μ 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 [14] for the free Brownian motion and Pardo et al. [12] 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.