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.

在参数 beta 与系统规模的倒数成正比的情况下，已知高斯 beta 系综（分别为 beta Laguerre 系综）的经验分布弱收敛于相关 Hermite 多项式（分别为相关 Laguerre 系综）的概率测度多项式），几乎可以肯定。极限附近的高斯波动也是已知的。本文旨在研究这些结果的动态版本。更准确地说，我们研究了β戴森布朗运动和β拉盖尔过程，并为它们在同一体系中的矩过程建立了大数定律（LLN）和中心极限定理（CLT）。