Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices

We consider NxN$N\times N$ non-Hermitian random matrices of the form X+A$X+A$, where A$A$ is a general deterministic matrix and NX$\sqrt {N}X$ consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by N1+o(1)$N<^>{1+o(1)}$ and (ii) that the expected condition number of any bulk eigenvalue is bounded by N1+o(1)$N<^>{1+o(1)}$; both results are optimal up to the factor No(1)$N<^>{o(1)}$. The latter result complements the very recent matching lower bound obtained by Cipolloni et al. and improves the N$N$-dependence of the upper bounds by Banks et al. and Jain et al. Our main ingredient, a near-optimal lower tail estimate for the small singular values of X+A-z$X+A-z$, is of independent interest.