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

Abstract We consider non‐Hermitian random matrices of the form , where is a general deterministic matrix and 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 and (ii) that the expected condition number of any bulk eigenvalue is bounded by ; both results are optimal up to the factor . The latter result complements the very recent matching lower bound obtained by Cipolloni et al. and improves the ‐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 , is of independent interest.