@article{15378,
abstract = {We consider N×N non-Hermitian random matrices of the form X+A, where A is a general deterministic matrix and 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 N1+o(1) and (ii) that the expected condition number of any bulk eigenvalue is bounded by N1+o(1); both results are optimal up to the factor No(1). The latter result complements the very recent matching lower bound obtained in [15] (arXiv:2301.03549) and improves the N-dependence of the upper bounds in [5,6,32] (arXiv:1906.11819, arXiv:2005.08930, arXiv:2005.08908). Our main ingredient, a near-optimal lower tail estimate for the small singular values of X+A−z, is of independent interest.},
author = {Erdös, László and Ji, Hong Chang},
issn = {1097-0312},
journal = {Communications on Pure and Applied Mathematics},
publisher = {Wiley},
title = {{Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices}},
doi = {10.1002/cpa.22201},
year = {2024},
}