Edge universality for non-Hermitian random matrices.
Probab Theory Relat Fields 2021;
179:1-28. [PMID:
33707804 PMCID:
PMC7906960 DOI:
10.1007/s00440-020-01003-7]
[Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2019] [Revised: 09/09/2020] [Accepted: 09/13/2020] [Indexed: 11/06/2022]
Abstract
We consider large non-Hermitian real or complex random matrices \documentclass[12pt]{minimal}
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\begin{document}$$X$$\end{document}X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of \documentclass[12pt]{minimal}
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\begin{document}$$X$$\end{document}X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble.
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