Spectra of sparse non-Hermitian random matrices: an analytical solution

Abstract

We present the exact analytical expression for the spectrum of a sparse non-Hermitian random matrix ensemble, generalizing two classical results in random-matrix theory: this analytical expression forms a non-Hermitian version of the Kesten-Mckay law as well as a sparse realization of Girko's elliptic law. Our exact result opens new perspectives in the study of several physical problems modelled on sparse random graphs. In this context, we show analytically that the convergence rate of a transport process on a very sparse graph depends upon the degree of symmetry of the edges in a non-monotonous way.

5 pages, 5 figures, 12 pages supplemental material

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• Universality in the spectral and eigenfunction properties of random networks
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• Spacing ratio characterization of the spectra of directed random networks
• On the low dimensional dynamics of structured random networks
• Linear stability analysis for large dynamical systems on directed random graphs
• Universal transient behavior in large dynamical systems on networks
• Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations
• Eigenvalue Repulsion and Eigenfunction Localization in Sparse Non-Hermitian Random Matrices
• Chebyshev-polynomial expansion of the localization length of Hermitian and non-Hermitian random chains
• Eigenvalue spectra and stability of directed complex networks
• The spectral density of dense random networks and the breakdown of the Wigner semicircle law
• Spectral density of the non-backtracking operator
• Extended Anderson Criticality in Heavy-Tailed Neural Networks
• Large deviation theory for diluted Wishart random matrices
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Second largest Eigenpair Statistics for Sparse Graphs
• Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry
• Real spectra of large real asymmetric random matrices
• Finite size effects. The averaged eigenvalue density of Wigner random sign real symmetric matrices
• Random Antagonistic Matrices
• A path integral approach to sparse non-Hermitian random matrices
• Top eigenpair statistics of diluted Wishart matrices