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Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices

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arxiv 0803.1553 v2 pith:WHYBDAT3 submitted 2008-03-11 cond-mat.dis-nn

Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices

Tim Rogers , Koujin Takeda , Isaac P\'erez Castillo , Reimer K\"uhn This is my paper
classification cond-mat.dis-nn
keywords matricesdensitysparseapproachcavitycovariancerandomspectral
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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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