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 (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.

References in corpus (5)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Radial Dunkl Processes : Existence and uniqueness, Hitting time, Beta Processes and Random Matrices

Cited by in corpus (3)

• Matrix addition and the Dunkl transform at high temperature
• Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform
• Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases