$\beta$-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems

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 c\'adl\'ag 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 are universal, and 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.