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

Abstract

We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_n+1}(R+X)(R+X{)}^{\ast }$, 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.

Many typos are corrected. Statement and the proofs of the main theorems are written in a more elaborated way

References in corpus (1)

• Matrix concentration inequalities via the method of exchangeable pairs

Cited by in corpus (2)

• Modelling equilibration of local many-body quantum systems by random graph ensembles
• Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth