Local law for eigenvalues of random Hermitian matrices with external source

Abstract

We prove a local law for eigenvalues of the random Hermitian matrices with external source $W_n=\frac{1}{n}{X}_n+A_n$ where $X_n$ is Wigner matrix and $A_n$ is diagonal matrix with only two values $a,-a$ on the diagonal. The local law is an essential step to prove the universality conjecture for this random matrix model.