Spectra of Sparse Random Matrices

TL;DR

This paper presents a novel method to compute the spectral density of various sparse symmetric random matrices, including those on different graph structures, addressing localization issues and providing detailed vertex-level spectral contributions.

Contribution

The authors develop a versatile replica-based approach that overcomes previous difficulties and applies to matrices on diverse graph types, including regular, scale-free, and constrained Laplacians.

Findings

Effective spectral density computation for various sparse matrices

Ability to analyze localization phenomena in spectra

Decomposition of spectral density into local vertex contributions

Abstract

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by Rodgers and Bray. Due attention is payed to the issue of localization. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows to unfold the total density of states into contributions coming from vertices of different local coordination.