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

TL;DR

This paper derives an exact analytical spectrum for sparse non-Hermitian random matrices, extending classical laws and providing insights into physical processes on sparse graphs.

Contribution

It presents a novel analytical expression for the spectrum of sparse non-Hermitian matrices, generalizing classical random matrix laws.

Findings

Spectrum follows a non-Hermitian Kesten-Mckay law

Spectrum also exhibits a sparse Girko's elliptic law

Transport convergence rate depends non-monotonously on edge symmetry

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.