On pseudospectrum of inhomogeneous non-Hermitian random matrices

TL;DR

This paper establishes a near-optimal lower bound on the smallest singular value of inhomogeneous non-Hermitian Gaussian matrices shifted by a complex scalar, with implications for their spectral distribution.

Contribution

It provides a new probabilistic lower bound on the smallest singular value for a broad class of inhomogeneous Gaussian matrices, extending understanding of their spectral properties.

Findings

Derived a lower bound for the smallest singular value with high probability.

Bound is optimal up to subexponential factors without additional assumptions.

Discussed implications for empirical spectral distributions of such matrices.

Abstract

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.