Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices

TL;DR

This paper applies the cavity method to analyze the spectral density of sparse symmetric random matrices, deriving equations that recover known laws and match numerical results.

Contribution

It introduces a cavity-based approach to efficiently compute spectral densities of sparse matrices, including derivations of classical laws like Wigner and Marcenko-Pastur.

Findings

Accurately recovers Wigner semicircle law for Gaussian matrices.

Derives Marcenko-Pastur law for covariance matrices.

Shows excellent agreement with numerical diagonalization.

Abstract

The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally tree-like, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, finding excellent agreement.