Bulk universality for deformed Wigner matrices

TL;DR

This paper proves that the local spectral statistics in the bulk of large deformed Wigner matrices are universal, regardless of the specific diagonal perturbation, under certain decay and size conditions.

Contribution

It establishes bulk universality for a broad class of deformed Wigner matrices with general diagonal perturbations.

Findings

Bulk eigenvalue statistics are universal for large N.

Universality holds for both real symmetric and complex Hermitian matrices.

Results apply to matrices with subexponential decay and comparable eigenvalue scales.

Abstract

We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$, and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$, we show that the local statistics in the bulk of the spectrum are universal in the limit of large $N$.