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

翻译：我们得出了一个精确的低尾量估计,根据实际Gaussian(Ginibre)随机基质最低单值$sigma_1(X-z)美元(Ginibre) 随机基质美元(XX美元), 由复杂参数转折的美元美元美元美元美元。 这种转换有效地改变了条件值$kappa(X-z) 的上尾端行为, 由较慢的 $\mathbf{P}(\ kappa(X-z)\ge t)\lessm sim 1/t美元(实际基尼布雷基质典型的基质典型)和较快的1美元/ t ⁇ 2美元(美元) 的衰变率得出。 主要的技术工具是来自我们先前工作[Xiv:2005.08. 930] 的严格超声波分析。 [Xiv3]