Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble.

Abstract

We study the eigensolution statistics of large N\ifmmode\times\else\texttimes\fi{}N real and symmetric sparse random matrices as a function of the average number p of nonzero matrix elements per row. In the very sparse matrix limit (small p) the averaged density of states deviates from the Wigner semicircle law with the appearance of a singularity 〈\ensuremath{\rho}(E)〉\ensuremath{\propto}1/\ensuremath{\Vert}E\ensuremath{\Vert} as E\ensuremath{\rightarrow}0. A localization threshold is identified at ${\mathit{p}}_{\mathit{q}}$\ensuremath{\simeq}1.4 via a simple criterion based on the density fluctuations, and the nearest-level-spacing function P(S) is shown to obey the Wigner surmise law in the delocalized phase (pg${\mathit{p}}_{\mathit{q}}$). Our findings are in agreement with previous supersymmetric and replica theories and studies of the Anderson transition in dilute Bethe lattices.