Small gaps of circular $\beta$-ensemble

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.