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].

arXiv commentAdded additional references to the supersymmetric literature. 37 pages

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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
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• Non-Hermitian spectral universality at critical points
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