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arXiv · 1908.01653

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

Giorgio Cipolloni·László Erdős·Dominik Schröder
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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 Br\'ezin-Hikami formulas in the real case, we rely on the superbosonization formula [Littelmann, Sommers, Zirnbauer, 2008].

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Giorgio Cipolloni, László Erdős, Dominik Schröder. 2019-08-05. Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble. https://doi.org/10.2140/pmp.2020.1.101

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