Equivalence of replica and cavity methods for computing spectra of sparse random matrices

TL;DR

This paper proves the exact equivalence of replica and cavity methods for spectra of sparse random matrices, introduces a variational approach, and applies it to Erdős-Rényi graphs and sparse covariance matrices.

Contribution

It establishes the exact equivalence of replica and cavity methods for sparse matrices and develops a variational framework for spectral calculations.

Findings

Replica and cavity methods are exactly equivalent for Erdős-Rényi graphs.

The variational approach provides approximate eigenvalue densities.

Application to sparse covariance matrices demonstrates versatility.

Abstract

We show by direct calculation that the replica and cavity methods are exactly equivalent for the spectrum of Erdos-Renyi random graph. We introduce a variational formulation based on the cavity method and use it to find approximate solutions for the density of eigenvalues. We also use this variational method for calculating spectra of sparse covariance matrices.