Limiting Spectral Distributions of Families of Block Matrix Ensembles

TL;DR

This paper introduces a new matrix operation called swirl and explores its impact on spectral distributions, demonstrating convergence to the Rayleigh distribution for certain ensembles with a novel combinatorial proof.

Contribution

It presents a new matrix operation and proves almost sure convergence of specific matrix ensembles to the Rayleigh distribution using a novel combinatorial approach.

Findings

The swirl operation affects spectral distributions of matrix ensembles.

Circulant Hankel matrices converge to the Rayleigh distribution.

A new combinatorial proof method is developed for spectral convergence.

Abstract

We introduce a new matrix operation on a pair of matrices, $\text{swirl}(A,X),$ and discuss its implications on the limiting spectral distribution. In a special case, the resultant ensemble converges almost surely to the Rayleigh distribution. In proving this, we provide a novel combinatorial proof that the random matrix ensemble of circulant Hankel matrices converges almost surely to the Rayleigh distribution, using the method of moments.