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

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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.

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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     title = {Gaussian asymptotics of discrete $\beta $-ensembles},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--78},
     publisher = {Springer Berlin Heidelberg},
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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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