Supersymmetric approach to the deformed Ginibre ensemble

We consider non-Hermitian random matrices of the form $H=A+H_0$, where $H_0$ is a standard Ginibre matrix, and $A$ is a rather general $n\times n$ matrix (Hermitian or non-Hermitian) independent of $H_0$. It is known that under some reasonable conditions the limiting spectrum of $H$ lies in some domain $D$ with a smooth boundary $\Gamma$. We apply the supersymmetric approach to study the asymptotic behavior of the smallest singular value of the matrix $(H-z)$ if $z\in D$ but $\mathrm{dist}(z,\Gamma)\sim n^{-1/2}$.