Small gaps of circular $β$-ensemble

Renjie Feng, Dongyi Wei

Published 2018-06-05Version 1

In this article, we study the small gaps of the log-gas $\beta$-ensemble on the unit circle, where $\beta$ is any positive integer. The main result is that the $k$-th smallest gap, normalized by $n^{\frac {\beta+2}{\beta+1}}$, has the limit proportional to $x^{k(\beta+1)-1}e^{-x^{\beta+1}}$. In particular, the result applies 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.