On the Condition Number of the Shifted Real Ginibre Ensemble

We derive an accurate lower tail estimate on the lowest singular value ₁ (X-z) of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z. Such shift effectively changes the upper tail behavior of the condition number (X-z) from the slower ( (X-z) t) 1/t decay typical for real Ginibre matrices to the faster 1/t² decay seen for complex Ginibre matrices as long as z is away from the real axis. This sharpens and resolves a recent conjecture in J. Banks et al. , https: //arxiv. org/abs/2005. 08930, 2020 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 Probab. Math. Phys. , 1 (2020), pp. 101--146.