Gaussian fluctuations for linear spectral statistics of Wigner beta ensembles

TL;DR

This paper establishes a central limit theorem for linear spectral statistics of Wigner beta ensembles, including quaternion matrices, advancing understanding of their Gaussian fluctuations in random matrix theory.

Contribution

It provides the first general CLT for Wigner beta ensembles, extending results to quaternion matrices and broadening the theoretical framework.

Findings

Proved a CLT for Wigner quaternion matrices.

Extended CLT results to general Wigner beta ensembles.

Enhanced understanding of spectral fluctuations in random matrices.

Abstract

As an important topic in Mathematical Physics and statistics, random matrices theory has found uses in many aspects of modern physics and multivariate analysis. This paper is to investigate the Gaussian fluctuations for linear spectral statistics (LSS) of Wigner beta ensembles. We first establish a central limit theorem (CLT) for LSS of Wigner quaternion matrices, then give a general CLT for Wigner $\beta$ ensembles.