Eigenvalues of large chiral non-Hermitian random matrices

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Abstract

We study a non-Hermitian chiral random matrix of which the eigenvalues are complex random variables. The empirical distributions and the radius of the eigenvalues are investigated. The limit of the empirical distributions is a new probability distribution defined on the complex plane. The graphs of the density functions are plotted; the surfaces formed by the density functions are understood through their convexity and their Gaussian curvatures. The limit of the radius is a Gumbel distribution. The main observation is that the joint density function of the eigenvalues of the chiral ensemble, after a transformation, becomes a rotation-invariant determinantal point process on the complex plane. Then, the eigenvalues are studied by the tools developed by Jiang and Qi [J. Theor. Probab. 30, 326 (2017); 32, 353 (2019)]. Most efforts are devoted to deriving the central limit theorems for distributions defined by the Bessel functions via the method of steepest descent and the estimates of the zero of a non-Trivial equation as the saddle point.

Original languageEnglish (US)
Article number013508
JournalJournal of Mathematical Physics
Volume61
Issue number1
DOIs
StatePublished - Jan 1 2020

Bibliographical note

Funding Information:
The authors would like to thank the referee for his/her careful reading of the manuscript and valuable comments that have led to improvements on the paper. Shuhua Chang’s research was supported in part by the NSFC (Grant Nos. 91430108 and 11771322).

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