Central Limit Theorem for $C\beta E$ Pair Dependent Statistics in Mesoscopic Regime

TL;DR

This paper extends the understanding of fluctuations in pair counting statistics for the Circular Beta Ensemble in the mesoscopic regime, providing a central limit theorem for a broad class of beta values.

Contribution

It generalizes previous results to arbitrary beta>0 and the mesoscopic scale, establishing a central limit theorem for pair dependent statistics.

Findings

Established CLT for pair counting in mesoscopic regime

Extended results to all beta>0

Analyzed fluctuations of the Circular Beta Ensemble

Abstract

We extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble $\sum_{i\neq j}f(L_N(\theta_i-\theta_j))$ for arbitrary $\beta>0$ in the mesoscopic regime $L_N=O(N^{2/3-\epsilon})$.