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Analytic approach for the number statistics of non-Hermitian random matrices

Authors :
Antonio Tonatiúh Ramos Sánchez
Isaac Pérez Castillo
Edgar Guzmán-González
Fernando L. Metz
Source :
Physical Review E. 103
Publication Year :
2021
Publisher :
American Physical Society (APS), 2021.

Abstract

We introduce a powerful analytic method to study the statistics of the number $\mathcal{N}_{\textbf{A}}(\gamma)$ of eigenvalues inside any contour $\gamma \in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf A}$. Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable. The main outcome is an effective theory that determines the cumulant generating function of $\mathcal{N}_{\textbf{A}}$ via a path integral along $\gamma$, with the path probability distribution following from the solution of a self-consistent equation. We derive the expressions for the mean and the variance of $\mathcal{N}_{\textbf{A}}$ as well as for the rate function governing rare fluctuations of ${\mathcal{N}}_{\textbf{A}}{(\gamma)}$. All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.<br />Comment: 6 pages, 2 figures. SI as ancillary file

Details

ISSN :
24700053 and 24700045
Volume :
103
Database :
OpenAIRE
Journal :
Physical Review E
Accession number :
edsair.doi.dedup.....10349675169e50b776649831ee15fe5f