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

Isaac Pérez Castillo, Edgar Guzmán-González, Antonio Tonatiúh Ramos Sánchez, and Fernando L. Metz
Phys. Rev. E 103, 062108 – Published 2 June 2021

Abstract

We introduce a powerful analytic method to study the statistics of the number NA(γ) of eigenvalues inside any smooth Jordan curve γC for infinitely large non-Hermitian random matrices A. Our generic approach can be applied to different random matrix ensembles of a mean-field type, 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, and obtain explicit results for the diluted real Ginibre ensemble. The main outcome is an effective theory that determines the cumulant generating function of NA via a path integral along γ, with the path probability distribution following from the numerical solution of a nonlinear self-consistent equation. We derive expressions for the mean and the variance of NA as well as for the rate function governing rare fluctuations of NA(γ). All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.

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  • Received 28 July 2020
  • Revised 15 April 2021
  • Accepted 18 May 2021

DOI:https://doi.org/10.1103/PhysRevE.103.062108

©2021 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Isaac Pérez Castillo1, Edgar Guzmán-González1, Antonio Tonatiúh Ramos Sánchez2, and Fernando L. Metz3,4

  • 1Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340, Mexico
  • 2Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, Mexico
  • 3Physics Institute, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil
  • 4London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom

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Issue

Vol. 103, Iss. 6 — June 2021

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