Linear statistics of the circular $\beta$-ensemble, Stein's method, and circular Dyson Brownian motion

TL;DR

This paper introduces a new Stein's method approach using circular Dyson Brownian motion to analyze linear statistics of the circular $eta$-ensemble, enabling simultaneous study of multiple statistics with minimal moment estimates.

Contribution

It generalizes previous results for the CUE and provides a novel, scalable method for analyzing linear statistics of $eta$-ensembles.

Findings

Allows simultaneous analysis of multiple linear statistics.

Requires only low-order moment estimates.

Generalizes results from the CUE to broader $eta$-ensembles.

Abstract

We study the linear statistics of the circular $\beta$-ensemble with a Stein's method argument, where the exchangeable pair is generated through circular Dyson Brownian motion. This generalizes previous results obtained in such a way for the CUE and provides a novel approach for studying linear statistics of $\beta$-ensembles. This approach allows studying simultaneously a collection of linear statistics whose number grows with the dimension of the ensemble. Also this approach requires estimating only low order moments of the linear statistics.