On the Condition Number of the Shifted Real Ginibre Ensemble

We derive an accurate lower tail estimate on the lowest singular value $\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by complex parameter $z$. Such shift effectively changes upper behaviour condition number $\kappa(X-z)$ from slower $\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t$ decay typical for Ginibre matrices to faster $1/t^2$ seen as long $z$ is away axis. This sharpens and resolves recent conjecture in [arXiv:2005.08930] regularizing effect ensemble with genuinely shift. As consequence we obtain improved bound eigenvalue numbers (known also eigenvector overlaps) matrices. The main technical tool rigorous supersymmetric analysis our earlier work [arXiv:1908.01653].