Asymptotics of discrete $\beta$-corners processes via two-level discrete loop equations

Abstract

We introduce and study stochastic particle ensembles which are natural discretizations of general $\beta$-corners processes. We prove that under technical assumptions on a general analytic potential the global fluctuations for the difference between two adjacent levels are asymptotically Gaussian. The covariance is universal and remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that are two-level analogues of the discrete loop equations.Comment: 102 pages, 2 figures. Fixed a typo in v1 that propagated all the way to the main result -- Theorem 1.7 (Theorem 1.9 in v1). Assumption 6 in Section 3 has been removed and Assumption 5 relaxed. A new appendix has been added. Several applications have been added in Section 6 (Section 7 in v1). In Section 7 (Section 6 in v1) the restriction $\theta \geq 1$ has been relaxed to $\theta > 0