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.

中文翻译：

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非厄米特随机矩阵特征值条件数的韦格纳估计和上限

我们考虑 形式的非厄米随机矩阵，其中 是一般确定性矩阵，由均值为零、单位方差和有界密度的独立项组成。对于这个系综，我们证明 (i) Wegner 估计，即特征值的局部密度受 限制，并且 (ii) 任何体特征值的预期条件数受 限制；两个结果都是最优的。后一个结果补充了 Cipolloni 等人最近获得的匹配下限。并改善了 Banks 等人对上限的依赖性。和贾恩等人。我们的主要成分，对 的小奇异值的近乎最优的下尾估计，是具有独立意义的。