Gaussian asymptotics of discrete $\beta $ -ensembles

Abstract : 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.
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Alexei Borodin, Vadim Gorin, Alice Guionnet. Gaussian asymptotics of discrete $\beta $ -ensembles. Publications Mathématiques de L'IHÉS, Springer Verlag, 2017, 125, pp.1--78. ⟨10.1007/s10240-016-0085-5⟩. ⟨ensl-01400956⟩

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