On the condition number of the shifted real Ginibre ensemble
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We derive an accurate lower tail estimate on the lowest singular value σ_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 κ(X-z) from the slower 𝐏(κ(X-z)≥ t)≲ 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].
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