Spectral properties of Levy matrices - Araujo, Mariela; Medina, Ernesto (Ciencia, 1970)
The statistical properties of the spectrum from large symmetric matrices are investigated. In these matrices the elements are chosen from a power-law distribution p(x) = Ø§νØ§x ν−1 with −2 ≤ ν ≤ 1. Universality classes or stable laws are explored by studying the density of states ρ(E) and the distribution of eigenvalue spacings P(s). Various regimes could be identified as a function of the disorder strength parameter ν. For 0 < ν < 1, the density of states obeys the simple semicircular law, and P(s) follows the Wigner surmise. For ν < 0, various zones separated by energy-dependent boundaries are observed. Furthermore, in the limit ν→ 0, we find a density of states that corresponds to the sparse matrix limit, with the characteristic singularity ρ(E) ï‚µ1/E . However, in this limit the spacing distribution exhibits power laws tails instead of the well known Brody form.