Fluctuations of eigenvalues of patterned random matrices

TL;DR

This paper investigates the eigenvalue fluctuations of various patterned random matrices, demonstrating Gaussian convergence of linear spectral statistics and calculating their limiting variances.

Contribution

It provides new results on the Gaussian fluctuation and variance of eigenvalues for circulant, symmetric circulant, reverse circulant, and Hankel matrices with i.i.d. normal entries.

Findings

Linear spectral statistics converge to Gaussian distribution in total variation norm.

Limiting variances are explicitly calculated for several matrix patterns.

Eigenvalue fluctuations follow a universal Gaussian behavior under specified conditions.

Abstract

In this article we study the fluctuation of linear statistics of eigenvalues of circulant, symmetric circulant, reverse circulant and Hankel matrices. We show that the linear spectral statistics of these matrices converges to the Gaussian distribution in total variation norm when the matrices are constructed using i.i.d. normal random variables. We also calculate the limiting variance of the linear spectral statistics for circulant, symmetric circulant and reverse circulant matrices.