Pair Dependent Linear Statistics for Circular Beta Ensemble

TL;DR

This paper investigates the limiting distribution of pair counting statistics in the circular beta ensemble of random matrices, extending classical results on pair correlations to a broader class of ensembles.

Contribution

It introduces new limiting distribution results for pair statistics in the circular beta ensemble, generalizing Montgomery's classical pair correlation results.

Findings

Derived limiting distributions for pair counting statistics

Extended Montgomery's pair correlation results to circular beta ensembles

Applicable for smooth test functions with L_N=O(N)

Abstract

We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N\*(\theta_i-\theta_j))$ for the circular $\beta$-ensemble (C$\beta$E) of random matrices for sufficiently smooth test function $f$ and $L_N=O(N).$ For $\beta=2$ and $L_N=N$ our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.