Gaussian asymptotics of discrete β-ensembles

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.