Distribution of singular values of random band matrices; Marchenko-Pastur law and more

TL;DR

This paper studies the spectral distribution of band matrices with random and deterministic parts, showing convergence to a deterministic measure and deriving an integral equation, generalizing the Marchenko-Pastur law.

Contribution

It extends the Marchenko-Pastur law to a broader class of band matrices with both random and deterministic components, providing a new integral equation for the limiting spectral distribution.

Findings

Convergence of the empirical spectral distribution to a deterministic measure.

Derivation of an integral equation for the Stieltjes transform.

Special case recovers the Marchenko-Pastur law.

Abstract

We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.