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 ${\mathit{p}}_{\mathit{q}}$≃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>${\mathit{p}}_{\mathit{q}}$). Our findings are in agreement with previous supersymmetric and replica theories and studies of the Anderson transition in dilute Bethe lattices.

• Received 4 September 1991

DOI:https://doi.org/10.1103/PhysRevLett.68.361