The least singular value of the general deformed Ginibre ensemble

TL;DR

This paper establishes optimal tail estimates for the least singular value of deformed Ginibre matrices, extending previous results to more general deterministic matrices and improving classical bounds.

Contribution

It generalizes recent results on the least singular value of deformed Ginibre matrices to include non-zero deterministic matrices, providing optimal tail estimates.

Findings

Proves optimal tail bounds for the least singular value near the spectral edge.

Extends previous results to matrices with a general deterministic component.

Improves classical bounds by Sankar, Spielman, and Teng.

Abstract

We study the least singular value of the $n\times n$ matrix $H-z$ with $H=A_0+H_0$, where $H_0$ is drawn from the complex Ginibre ensemble of matrices with iid Gaussian entries, and $A_0$ is some general $n\times n$ matrix with complex entries (it can be random and in this case it is independent of $H_0$). Assuming some rather general assumptions on $A_0$, we prove an optimal tail estimate on the least singular value in the regime where $z$ is around the spectral edge of $H$ thus generalize the recent result of Cipolloni, Erd\H{o}s, Schr\"{o}der arxiv:1908.01653 to the case $A_0\ne 0$. The result improves the classical bound by Sankar, Spielman and Teng.