Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices

Details Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices

2008-03-11

arxiv.org/abs/0803.1553v2

Phys. Rev. E. 78, 031116 (2008)

Physics

Condensed Matter

Disordered Systems and Neural Networks

7 pages, 6 figures

Scientific paper

10.1103/PhysRevE.78.031116

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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