β-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems

Abstract

Abstract We study the $\beta $ analogue of the nonintersecting Poisson random walks. We derive a stochastic differential equation of the Stieltjes transform of the empirical measure process, which can be viewed as a dynamical version of the Nekrasov’s equation in [7, Section 4]. We find that the empirical measure process converges weakly in the space of cádlág measure-valued processes to a deterministic process, characterized by the quantized free convolution, as introduced in [11]. For suitable initial data, we prove that the rescaled empirical measure process converges weakly in the space of distributions acting on analytic test functions to a Gaussian process. The means and the covariances coincide with those of $\beta $-Dyson Brownian motions with the initial data constructed by the Markov–Krein correspondence. Especially, the covariance structure can be described in terms of the Gaussian free field. Our proof relies on integrable features of the generators of the $\beta $-nonintersecting Poisson random walks, the method of characteristics, and a coupling technique for Poisson random walks.