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

Abstract

We consider Hermitian random matrices of the form H=W + lambda V, where W is a Wigner matrix and V is a diagonal random matrix independent of W. We assume subexponential decay for the matrix entries of W and we choose lambda similar to 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.