Bulk universality for deformed Wigner matrices

Ji Oon Lee, Kevin Schnelli, Ben Stetler, Horng-Tzer Yau

Published 2014-05-26, updated 2016-06-07Version 3

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 $N$.