Theory of Lévy matrices

P. Cizeau and J. P. Bouchaud
Phys. Rev. E 50, 1810 – Published 1 September 1994
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Abstract

We investigate the statistical properties of the spectrum of large symmetrical matrices with each element Hij chosen according to a broad distribution ρ(H) decaying for large H as H1μ. For μ>2, 〈H2〉 is finite and the well known Gaussian orthogonal ensemble (GOE) results are recovered. When μ<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.

  • Received 15 March 1994

DOI:https://doi.org/10.1103/PhysRevE.50.1810

©1994 American Physical Society

Authors & Affiliations

P. Cizeau

  • Centre d’Etudes de Limeil Valenton, 94 195 Villeneuve St. Georges Cedex 05, France

J. P. Bouchaud

  • Service de Physique de l’Etat Condensé, Commissariat à l’Energie Atomique de Saclay, Orme des Merisiers, 91 191 Gif-sur-Yvette Cedex, France

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Vol. 50, Iss. 3 — September 1994

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