@article{12179,
  abstract     = {We derive an accurate lower tail estimate on the lowest singular value σ1(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 κ(X−z) from the slower (κ(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].},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1095-7162},
  journal      = {SIAM Journal on Matrix Analysis and Applications},
  keywords     = {Analysis},
  number       = {3},
  pages        = {1469--1487},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{On the condition number of the shifted real Ginibre ensemble}},
  doi          = {10.1137/21m1424408},
  volume       = {43},
  year         = {2022},
}