Universality for Weakly Non-Hermitian Matrices: Bulk Limit

We consider complex, weakly non-Hermitian matrices $A = W_1 +i\sqrt{{\tau}_N}W_2$ , where $W_1$ and $W_2$ are Hermitian matrices and $\tau_N = O(N^{-1})$. We first show that for pairs of Hermitian matrices $(W_1 , W_2)$ such that $W_1$ satisfies a multi-resolvent local law and $W_2$ is bounded in norm, the bulk correlation functions of the weakly non-Hermitian Gauss-divisible matrix $A + \sqrt{t}B$ converge pointwise to a universal limit for $t = O(N^{-1+\epsilon})$. Using this and the reverse heat flow we deduce bulk universality in the case when $W_1$ and $W_2$ are independent Wigner matrices with sufficiently smooth density.