Local deformed semicircle law and complete delocalization for Wigner matrices with random potential

TL;DR

This paper proves a local deformed semicircle law and eigenvector delocalization for a class of Wigner matrices with a random potential, extending known results to more general deformations.

Contribution

It establishes the local deformed semicircle law and eigenvector delocalization for Wigner matrices with a broad class of diagonal random potentials.

Findings

Local deformed semicircle law holds for the considered matrices.

Eigenvectors are completely delocalized.

Eigenvalue positions are characterized with high precision.

Abstract

We consider Hermitian random matrices of the form $H = W + \lambda V$, where $W$ is a Wigner matrix and $V$ a diagonal random matrix independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $\lambda \sim 1$ so that the eigenvalues of $W$ and $\lambda V$ are of the same order in the bulk of the spectrum. In this paper, we prove for a large class of diagonal matrices $V$ that the local deformed semicircle law holds for $H$, which is an analogous result to the local semicircle law for Wigner matrices. We also prove complete delocalization of eigenvectors and other results about the positions of eigenvalues.