On pseudospectrum of inhomogeneous non-Hermitian random matrices

Abstract

Let $A$ be an $n\times n$ matrix with mutually independent centered Gaussian entries. Define \begin{align*} σ^*:=\max\limits_{i,j\leq n}\sqrt{{\mathbb E}\,|A_{i,j}|^2}, \quad σ:=\max\bigg(\max\limits_{j\leq n}\sqrt{{\mathbb E}\,\|{\rm col}_j(A)\|_2^2}, \max\limits_{i\leq n}\sqrt{{\mathbb E}\,\|{\rm row}_i(A)\|_2^2}\bigg). \end{align*} Assume that $\sigma \ge n^{\epsilon }{\sigma }^{\ast }$ for a constant $\epsilon >0$, and that a complex number $z$ satisfies $|z|=\Omega (\sigma )$. We prove that $$s_{\min }(A-z\mathrm{Id})\ge |z|\exp (-n^{o(1)}(\frac{\sqrt{n}{\sigma }^{\ast }}{\sigma }{)}^2)$$ with probability $1-o(1)$. Without extra assumptions on $A$, the bound is optimal up to the $n^{o(1)}$ multiple in the power of exponent. We discuss applications of this estimate in context of empirical spectral distributions of inhomogeneous non-Hermitian random matrices.

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