A NUMERICAL STUDY OF SPARSE RANDOM MATRICES

A numerical study is presented for the eigensolution statistics of large N x N real and symmetric sparse random matrices as a function of the mean number p of nonzero elements per row. The model shows classical percolation and quantum localization transitions at p(c) = 1 and p(q) > 1, respectively. In the rigid limit p = N we demonstrate that the averaged density of states follows the Wigner semicircle law and the corresponding nearest energy-level-spacing distribution function P(S) obeys the Wigner surmise. In the very sparse matrix limit p much less than N, with p > p(q), a singularity <rho(E)) is-proportional-to 1/Absolute value of E is found as Absolute value of E --> 0 and exponential tails develop in the high-Absolute value of E regions, but the P(S) distribution remains consistent with level repulsion. The localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations- The value p(q) congruent-to 1.4 is roughly estimated, in agreement with previous studies of the Anderson transition in dilute Bethe lattices.