Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices

TL;DR

This paper studies how the eigenvalue statistics of certain patterned matrices with Brownian motion entries fluctuate over time, establishing their process convergence as matrix size grows, using combinatorial methods.

Contribution

It introduces a detailed analysis of time-dependent eigenvalue fluctuations for reverse and symmetric circulant matrices with Brownian entries, focusing on polynomial test functions.

Findings

Eigenvalue statistics converge as processes when matrix size increases.

The fluctuations are characterized for polynomial test functions.

The proofs rely on combinatorial trace formulas and the method of moments.

Abstract

Consider the $n \times n$ reverse circulant $RC_n(t)$ and symmetric circulant $SC_n(t)$ matrices with independent Brownian motion entries. We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of these matrices as $n \tends \infty$, when the test functions of the statistics are polynomials. The proofs are mainly combinatorial, based on the trace formula, method of moments and some results on process convergence.