Gaussian asymptotics of discrete β-ensembles
Borodin, Alexei1 idref ; Gorin, Vadim1, 2 ; Guionnet, Alice1 idref
1Department of Mathematics, Massachusetts Institute of Technology Cambridge MA USA
2Institute for Information Transmission Problems of Russian Academy of Sciences Moscow Russia
Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 1-78

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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DOI : 10.1007/s10240-016-0085-5

Borodin, Alexei  1   ; Gorin, Vadim  1 , 2   ; Guionnet, Alice  1

1 Department of Mathematics, Massachusetts Institute of Technology Cambridge MA USA
2 Institute for Information Transmission Problems of Russian Academy of Sciences Moscow Russia 
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Borodin, Alexei; Gorin, Vadim; Guionnet, Alice. Gaussian asymptotics of discrete $\beta $-ensembles. Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 1-78. doi: 10.1007/s10240-016-0085-5

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