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Optimal lower bound on
the least singular value of the shifted Ginibre ensemble
Giorgio Cipolloni, László Erdős and Dominik Schröder |
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Vol. 1 (2020), No. 1, 101–146
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Abstract |
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We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant . We prove an optimal lower tail estimate on this singular value in the critical regime where is around the spectral edge, thus improving the classical bound of Sankar, Spielman and Teng (SIAM J. Matrix Anal. Appl. 28:2 (2006), 446–476) for the particular shift-perturbation in the edge regime. Lacking Brézin–Hikami formulas in the real case, we rely on the superbosonization formula (Comm. Math. Phys. 283:2 (2008), 343–395). |
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Keywords
supersymmetric formalism, superbosonization, circular law
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Mathematical Subject Classification 2010
Primary: 15B52, 60B20
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Milestones
Received: 22 October 2019
Revised: 6 March 2020
Accepted: 30 March 2020
Published: 16 November 2020
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Authors |
| Giorgio Cipolloni | |
| Institute of Science and Technology
Austria Klosterneuburg Austria |
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| László Erdős | |
| Institute of Science and Technology
Austria Klosterneuburg Austria |
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| Dominik Schröder | |
| Institute of Science and Technology
Austria Klosterneuburg Austria |
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