Universality in the fluctuation of eigenvalues of random circulant matrices

TL;DR

This paper demonstrates that the eigenvalue fluctuations of circulant matrices follow a Gaussian distribution for a wide range of input sequences, highlighting a universal behavior in their spectral statistics.

Contribution

It establishes the Gaussian central limit theorem for eigenvalue linear statistics of circulant matrices with diverse input sequences, revealing universality.

Findings

Eigenvalue fluctuations follow Gaussian distribution

Universal behavior across various input sequences

Central limit theorem proven for circulant matrices

Abstract

We show that the linear statistics of eigenvalues of circulant matrix obey the Gaussian central limit theorem for a large class of input sequences.