  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.
