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

TL;DR

This paper investigates the asymptotic behavior of moment processes in beta Dyson's Brownian motions and beta Laguerre processes, establishing laws of large numbers and central limit theorems in a specific parameter regime.

Contribution

It extends known static results to dynamic processes, providing new limit theorems for the evolution of moments in beta ensembles.

Findings

LLNs for moment processes in the specified regime

CLTs describing fluctuations around the limits

Convergence to measures related to Hermite and Laguerre polynomials

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 (resp.\ beta Laguerre ensembles) converges to a probability measure of associated Hermite polynomials (resp.\ associated Laguerre polynomials). 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 LLNs and CLTs for their moment processes in the same regime.