Gaussian asymptotics of discrete β β -ensembles

Abstract

© 2016, IHES and Springer-Verlag Berlin Heidelberg. We introduce and study stochastic N-particle ensembles which are discretizations for general-β log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, (z, w) -measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as N→ ∞. The covariance is universal and coincides with its counterpart in random matrix theory. Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.