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Rigidity and Edge Universality of Discrete β ‐Ensembles
Author(s) -
Guionnet Alice,
Huang Jiaoyang
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21818
Subject(s) - universality (dynamical systems) , mathematics , rigidity (electromagnetism) , statistical physics , particle system , probability distribution , physics , statistics , computer science , quantum mechanics , operating system
We study discrete β ‐ensembles as introduced in [17]. We obtain rigidity estimates on the particle locations; i.e., with high probability, the particles are close to their classical locations with an optimal error estimate. We prove the edge universality of the discrete β ‐ensembles; i.e., for β ≥ 1, the distribution of extreme particles converges to the Tracy‐Widom β ‐distribution. As far as we know, this is the first proof of general Tracy‐Widom β ‐distributions in the discrete setting. A special case of our main results implies that under the Jack deformation of the Plancherel measure, the distribution of the lengths of the first few rows in Young diagrams converges to the Tracy‐Widom β ‐distribution, which answers an open problem in [38]. Our proof relies on Nekrasov's (or loop) equations, a multiscale analysis, and a comparison argument with continuous β ‐ensembles. © 2019 Wiley Periodicals, Inc.