Limit theorems for moment processes of beta Dyson’s Brownian motions and beta Laguerre processes

Abstract

In the regime where the parameter beta is proportional to the reciprocal of the system size, it is known that the empirical distribution of Gaussian beta ensembles (respectively, beta Laguerre ensembles) converges weakly to a probability measure of associated Hermite polynomials (respectively, associated Laguerre polynomials), almost surely. Gaussian fluctuations around the limit have been known as well. This paper aims to study a dynamical version of those results. More precisely, we study beta Dyson’s Brownian motions and beta Laguerre processes and establish law of large numbers (LLNs) and central limit theorems (CLTs) for their moment processes in the same regime.