Gaussian asymptotics of discrete $\beta $-ensembles

Borodin, Alexei; Gorin, Vadim; Guionnet, Alice

doi:10.1007/s10240-016-0085-5

Gaussian asymptotics of discrete

β

-ensembles

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

Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 1-78.

Résumé

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.

MR Zbl

DOI : 10.1007/s10240-016-0085-5

Affiliations des auteurs :

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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     year = {2017},
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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. http://www.numdam.org/articles/10.1007/s10240-016-0085-5/

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