Cavity approach to the spectral density of non-Hermitian sparse matrices

TL;DR

This paper introduces a cavity method-based approach to efficiently compute the spectral density of non-Hermitian sparse random matrices, recovering known laws and validating results against numerical diagonalization.

Contribution

The paper develops a set of equations using the cavity method for exact spectral density calculation of non-Hermitian sparse matrices, including a derivation of the generalized Girko's law.

Findings

Exact spectral densities match numerical diagonalization results

The approach recovers the generalized Girko's law

Efficient computation for various matrix ensembles

Abstract

The spectral densities of ensembles of non-Hermitian sparse random matrices are analysed using the cavity method. We present a set of equations from which the spectral density of a given ensemble can be efficiently and exactly calculated. Within this approach, the generalised Girko's law is recovered easily. We compare our results with direct diagonalisation for a number of random matrix ensembles, finding excellent agreement.