Gaussian asymptotics of discrete $\beta $-ensembles

Alexei Borodin; Vadim Gorin; Alice Guionnet

doi:10.1007/s10240-016-0085-5

Gaussian asymptotics of discrete

β

-ensembles

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

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, Volume 125 (2017), pp. 1-78

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 $N \to \infty$ . 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.

Article information
Cite this article

Received: 2015-06-10
Accepted: 2016-06-06
Online First: 2016-06-14
Published online: 2017-06-07

MR Zbl

DOI: 10.1007/s10240-016-0085-5

Author's affiliations:

Alexei Borodin ¹ ; Vadim Gorin ^{1
,

2} ; Alice Guionnet 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

Alexei Borodin; Vadim Gorin; Alice Guionnet. Gaussian asymptotics of discrete $\beta $-ensembles. Publications Mathématiques de l'IHÉS, Volume 125 (2017), pp. 1-78. doi: 10.1007/s10240-016-0085-5

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