Density of states of a sparse random matrix

Abstract

The density of states \ensuremath{\rho}(\ensuremath{\mu}) of an N\ifmmode\times\else\texttimes\fi{}N real, symmetric, random matrix with elements 0,\ifmmode\pm\else\textpm\fi{}1 is calculated in the limit N\ensuremath{\rightarrow}\ensuremath{\infty} as a function of the average ``connectivity'' p, i.e., of the mean number of nonzero elements per row. For p\ensuremath{\rightarrow}\ensuremath{\infty}, the Wigner semicircular distribution is recovered. For finite p the distribution has tails extending beyond the semicircle, with for ${\ensuremath{\mu}}^{2}$\ensuremath{\rightarrow}\ensuremath{\infty}. Applications to the theory of ``Griffiths singularities'' in dilute magnets are discussed.