Small gaps of circular $\beta$-ensemble

TL;DR

This paper investigates the smallest gaps in the log-gas $eta$-ensemble on the unit circle, showing they converge to a Poisson process after normalization, with explicit density formulas for the $k$-th smallest gap.

Contribution

It provides the first rigorous analysis of the smallest gaps in the circular $eta$-ensemble for any positive integer $eta$, including explicit limiting distributions.

Findings

Smallest gaps, after normalization, converge to a Poisson process.

Derived explicit density for the $k$-th smallest gap.

Results apply to classical random matrix ensembles like COE, CUE, and CSE.

Abstract

In this article, we study the smallest gaps of the log-gas $\beta$-ensemble on the unit circle (C$\beta$E), where $\beta$ is any positive integer. The main result is that the smallest gaps, after being normalized by $n^{\frac {\beta+2}{\beta+1}}$, will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the $k$-th smallest gap, which is 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.