On pseudospectrum of inhomogeneous non-Hermitian random matrices

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 $\mathrm{\mid }z\mathrm{\mid }=\mathrm{\Omega }(\sigma )$. We prove that $$s_{\min }(A-z\mathrm{I}\mathrm{d})\ge \mathrm{\mid }z\mathrm{\mid }\exp (-n^{o(1)}(\frac{\sqrt{n}{\sigma }^{\ast }}{\sigma }{)}^2)$$ 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.