The least singular value of the general deformed Ginibre ensemble

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.