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

Abstract

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

added results on bipartite statistics in Section 4