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*} \sigma^*:=\max\limits_{i,j\leq n}\sqrt{{\mathbb E}\,|A_{i,j}|^2}, \quad \sigma:=\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 $\sigma\geq n^\varepsilon\,\sigma^*$ for a constant $\varepsilon>0$, and that a complex number $z$ satisfies $|z|=\Omega(\sigma)$. We prove that $$ s_{\min}(A-z\,{\rm Id}) \geq |z|\,\exp\bigg(-n^{o(1)}\,\Big(\frac{\sqrt{n}\,\sigma^*}{\sigma}\Big)^2\bigg) $$ 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.