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IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 42 (2009) 152001 (8pp) doi:10.1088/1751-8113/42/15/152001 FAST TRACK COMMUNICATION Rotationally invariant family of L´evy-like random matrix ensembles Jinmyung Choi and K A Muttalib Department of Physics, University of Florida, Gainesville, FL 32611-8440, USA E-mail: jmchoi@phys.u±.edu and muttalib@phys.u±.edu Received 29 January 2009, in ²nal form 4 March 2009 Published 20 March 2009 Online at Abstract We introduce a family of rotationally invariant random matrix ensembles characterized by a parameter λ . While λ = 1 corresponds to well-known critical ensembles, we show that λ 6= 1 describes ‘L´evy-like’ ensembles, characterized by power-law eigenvalue densities. For λ> 1 the density is bounded, as in Gaussian ensembles, but λ< 1 describes ensembles characterized by densities with long tails. In particular, the model allows us to evaluate, in terms of a novel family of orthogonal polynomials, the eigenvalue correlations for L´evy-like ensembles. These correlations differ qualitatively from those in either the Gaussian or the critical ensembles. PACS numbers: 05.40. a, 05.60. k, 05.90.+m 1. Introduction Gaussian random matrix ensembles (RMEs) were proposed by Wigner about half a century ago to describe the statistical properties of the eigenvalues and eigenfunctions of complex many-body quantum systems in which the Hamiltonians are considered only in a probabilistic way [ 1 , 2 ]. Over the past several decades, they proved to be a very useful tool in the studies of equilibrium and transport properties of disordered quantum systems, classically chaotic systems with a few degrees of freedom, two-dimensional gravity, conformal ²eld theory and chiral phase transition in quantum chromodynamics [ 3 ]. This wide applicability results from certain universal properties of the correlation of the eigenvalues known as the Wigner distributions, as opposed to the Poisson distributions that result from random sets of eigenvalues. More recently, attempts have been made to construct generalized ensembles that show a crossover from a Wigner to a Poisson distribution as a function of a parameter, as seen in many physical systems [ 4 ]. One such generalization is the family of ‘q-random matrix ensembles’ (q-RMEs) [ 5 , 6 ]. These were later shown to be models of ‘critical ensembles’, 1751-8113/09/152001+08$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1
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J. Phys. A: Math. Theor. 42 (2009) 152001 Fast Track Communication with statistical properties different from the Gaussian RMEs and relevant for systems near a metal–insulator transition [ 7 ]. While there are other models that also describe critical statistics [ 8 ], one advantage of the q-RMEs is that they are rotationally invariant, and therefore can be analytically studied in great detail by the powerful method of orthogonal polynomials [ 2 ].
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