Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble

Abstract

We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\in \mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where $z$ is around the spectral edge thus improving the classical bound of [Sankar, Spielman, Teng, 2006] in the edge regime. Lacking Brézin-Hikami formulas in the real case, we rely on the superbosonization formula [Littelmann, Sommers, Zirnbauer, 2008].

Added additional references to the supersymmetric literature. 37 pages

References in corpus (5)

• Superbosonization of invariant random matrix ensembles
• Edge Universality for non-Hermitian Random Matrices
• Comparison of the superbosonization formula and the generalized Hubbard-Stratonovich transformation
• Central Limit Theorem for Linear Eigenvalue Statistics of non-Hermitian Random Matrices
• Universality of the least singular value for the sum of random matrices

Cited by in corpus (7)

• Edge Universality for non-Hermitian Random Matrices
• The least singular value of the general deformed Ginibre ensemble
• Density of small singular values of the shifted real Ginibre ensemble
• Universality of the least singular value for the sum of random matrices
• Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations
• Non-Hermitian spectral universality at critical points
• Limiting eigenvalue distribution of the general deformed Ginibre ensemble