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

2023/01/12 by Erdős, László, Ji, Hong Chang · 1 citation
#15A12 #15B52 #60B20 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2301.04981

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 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] (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.

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