SMALL GAPS OF CIRCULAR β-ENSEMBLE
成果类型：
Article
署名作者：
Feng, Renjie; Wei, Dongyi
署名单位：
Chinese Academy of Sciences; University of Science & Technology of China, CAS; Peking University
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/20-AOP1468
发表日期：
2021
页码：
997-1032
关键词：
摘要：
In this article, we study the smallest gaps of the circular beta-ensemble (C beta E) on the unit circle, where beta is any positive integer. The main result is that the smallest gaps, after being normalized by n(beta+2/beta+1), will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the kth smallest gap, which is proportional to x(k(beta+1)-1)e(-x beta+1). In particular, the results apply to the classical COE, CUE and CSE in random matrix theory. The essential part of the proof is to derive several identities and inequalities regarding the Selberg integral, which should have their own interest.