Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices (2301.04981v2)

Abstract: We consider $N\times N$ non-Hermitian random matrices of the form $X+A$, where $A$ is a general deterministic matrix and $\sqrt{N}X$ consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, i.e. that the local density of eigenvalues is bounded by $N^{1+o(1)}$ and (ii) that the expected condition number of any bulk eigenvalue is bounded by $N^{1+o(1)}$; both results are optimal up to the factor $N^{o(1)}$. The latter result complements the very recent matching lower bound obtained in 15 and improves the $N$-dependence of the upper bounds in 5,6,32. Our main ingredient, a near-optimal lower tail estimate for the small singular values of $X+A-z$, is of independent interest.