Gaussian Fluctuation for Smoothed Local Correlations in CUE

TL;DR

This paper investigates the limiting distribution of smoothed local correlations in the Circular Unitary Ensemble, extending the understanding of eigenvalue statistics and their fluctuations.

Contribution

It provides a Gaussian fluctuation result for smoothed local correlations in CUE, inspired by the Rudnick-Sarnak theorem, for sufficiently smooth test functions.

Findings

Establishes Gaussian fluctuation for smoothed correlations in CUE

Extends Rudnick-Sarnak theorem to local correlation functions

Provides asymptotic distribution results for eigenvalue statistics

Abstract

Motivated by the Rudnick-Sarnak theorem we study limiting distribution of smoothed local correlations of the form $$ \sum_{j_1, j_2, \ldots, j_n} f(N\*(\theta_{j_2}-\theta_{j_1}), N\*(\theta_{j_3}-\theta_{j_1}), \ldots, N\*(\theta_{j_n}-\theta_{j_1}))$$ for the Circular United Ensemble of random matrices for sufficiently smooth test functions.