Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices

Tim Rogers; Koujin Takeda; Isaac Pérez Castillo; Reimer Kühn

doi:10.1103/physreve.78.031116

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arXiv · 0803.1553

Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices

Tim Rogers·Koujin Takeda·Isaac Pérez Castillo·Reimer Kühn

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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.

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Tim Rogers, Koujin Takeda, Isaac Pérez Castillo, Reimer Kühn. 2008-09-10. Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices. https://doi.org/10.1103/physreve.78.031116

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