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

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.

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Cited by in corpus (13)

• Cavity and replica methods for the spectral density of sparse symmetric random matrices
• Universality in the spectral and eigenfunction properties of random networks
• Localization of eigenvectors in random graphs
• Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations
• A unifying model for random matrix theory in arbitrary space dimensions
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
• Second largest Eigenpair Statistics for Sparse Graphs
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model
• Localization in random bipartite graphs: numerical and empirical study
• Equivalence between algorithmic instability and transition to replica symmetry breaking in perceptron learning systems
• Sparse Random Block Matrices
• A path integral approach to sparse non-Hermitian random matrices