Theory of Lévy matrices.

Abstract

We investigate the statistical properties of the spectrum of large symmetrical matrices with each element ${\mathit{H}}_{\mathit{i}\mathit{j}}$ chosen according to a broad distribution \ensuremath{\rho}(H) decaying for large H as ${\mathit{H}}^{\mathrm{\ensuremath{-}}1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}}$. For \ensuremath{\mu}>2, 〈${\mathit{H}}^{2}$〉 is finite and the well known Gaussian orthogonal ensemble (GOE) results are recovered. When \ensuremath{\mu}2, the semicircular law is replaced by a density which extends over the whole energy axis. Furthermore, while all states are extended in the case of GOE matrices, we show numerically and analytically that two mobility edges appear, separating extended from localized states, with an intermediate ``mixed'' phase in between. The unusual nature of these localized states is discussed.