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

We study the eigensolution statistics of large N×{}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 〈{ρ}(E)〉{∝}1/{}E{} as E{→}0. A localization threshold is identified at pq{}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 (p>pq). Our findings are in agreement with previous supersymmetric and replica theories and studies of the Anderson transition in dilute Bethe lattices.