Cavity approach to the spectral density of sparse symmetric random matrices

2008/03/31 by Tim Rogers, Koujin Takeda, Isaac Pérez Castillo +1 · 2 citations
Mathematics · Physics and Astronomy · #Quantum chaos and dynamical systems #Random Matrices and Applications #Stochastic processes and statistical mechanics #cond-mat.dis-nn

paper · pdf · doi:10.1103/physreve.78.031116

published as Phys. Rev. E. 78, 031116 (2008) · 7 pages, 6 figures

arxiv created 2008/09/10 · openalex publication_date 2008/09/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 treelike, 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, showing excellent agreement.

Citations

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• Euclidean random matrices: solved and open problems
• Random-matrix theories in quantum physics: common concepts
• Mean-field theory for a spin-glass model of neural networks: TAP free energy and paramagnetic to spin-glass transition
• On the Distribution of the Roots of Certain Symmetric Matrices

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