Process convergence of Fluctuations of linear eigenvalue statistics of random circulant matrices

TL;DR

This paper studies how the fluctuations of linear eigenvalue statistics of large random circulant matrices evolve over time, demonstrating process convergence using trace formulas, moments, and combinatorics.

Contribution

It establishes the process convergence of eigenvalue fluctuations for large random circulant matrices with Brownian motion entries, extending understanding of their spectral behavior.

Findings

Eigenvalue fluctuations converge as matrix size increases

Methodology combines trace formulas, moments, and combinatorics

Results applicable to matrices with stochastic entries

Abstract

In this paper we discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to $\infty $. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.