Local laws and spectral properties of deformed sparse random matrices

TL;DR

This paper establishes local spectral laws and eigenvalue rigidity for deformed sparse random matrices, extending understanding of their spectral behavior under mild conditions.

Contribution

It introduces local laws for deformed sparse matrices and proves eigenvalue rigidity and normality of extremal eigenvalues, advancing spectral analysis in sparse regimes.

Findings

Proved local laws for deformed sparse matrices

Established eigenvalue rigidity

Showed asymptotic normality of extremal eigenvalues

Abstract

We consider deformed sparse random matrices of the form $H= W+ \lambda V$, where $W$ is a real symmetric sparse random matrix, $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$, and $\lambda = O(1) $ is a coupling constant. Under mild assumptions on the matrix entries of $W$ and $V$, we prove local laws for $H$ that compare the empirical spectral measure of it with a refined version of the deformed semicircle law. By applying the local laws, we also prove several spectral properties of $H$, including the rigidity of the eigenvalues and the asymptotic normality of the extremal eigenvalues.