Edge Universality for Deformed Wigner Matrices

TL;DR

This paper proves that for a broad class of deformed Wigner matrices, the distribution of the largest eigenvalues converges to the Tracy-Widom law, extending universality results to these deformed models.

Contribution

It establishes edge universality for deformed Wigner matrices with general diagonal perturbations, including complex Hermitian cases.

Findings

Largest eigenvalues follow Tracy-Widom distribution

Universality holds for a wide class of diagonal matrices

Results extend to complex Hermitian matrices

Abstract

We consider $N\times N$ random matrices of the form $H = W + V$ where $W$ is a real symmetric Wigner matrix and $V$ 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 rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution $F_1$ in the limit of large $N$. Our proofs also apply to the complex Hermitian setting, i.e., when $W$ is a complex Hermitian Wigner matrix.