Cavity approach to the spectral density of sparse symmetric random matrices

Timothy 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.
Original languageEnglish
Article number031116
Number of pages7
JournalPhysical Review E
Volume78
Issue number3
DOIs
Publication statusPublished - 2008

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