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Unformatted text preview: Powerlaw eigenvalue density, scaling, and critical randommatrix ensembles K. A. Muttalib * Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 326118440, USA Mourad E. H. Ismail † Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA s Received 18 July 2007; published 8 November 2007 d We consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law. Under a scaling appropriate for such powerlaw densities s different from the scaling required in Gaussian random matrix ensembles d , we calculate exactly the twolevel kernel that deter mines all eigenvalue correlations. We show that such ensembles belong to the class of critical ensembles. DOI: 10.1103/PhysRevE.76.051105 PACS number s s d : 05.40. 2 a, 05.60. 2 k, 05.90. 1 m I. INTRODUCTION Gaussian random matrix ensembles, introduced by Wigner and Dyson f 1 g , have been studied extensively over half a century in the context of nuclear physics, atomic and molecular physics, condensed matter physics, as well as par ticle physics f 2 g . The wide applicability results from the uni versal properties of the correlation functions of the eigenval ues, once they are appropriately scaled. Thus, e.g., the correlation functions of the eigenvalues x of an ensemble of N 3 N Hermitian matrices become, after proper scaling, in dependent of the size of the matrix or of the details of the microscopic distribution from which the matrix elements are drawn. For a Gaussian distribution, the density is a semi circle given by s s x d = Î 2 N x 2 / p ; this requires, for univer sality, the double scaling limit N → ` , x → 0, u ; x Î N finite, s 1 d such that the density s ¯ s u d =const near the origin. This par ticular double scaling is required for all Gaussian random matrix ensembles of different symmetry classes f 3 g in order to have a universal large N limit. In particular, for Gaussian unitary ensembles, after a second trivial scaling z = Î 2 u / p such that s ˜ s z d =1, it leads to the universal twolevel kernel K G s z , h d = sin f p s z h dg p s z h d s 2 d independent of N , which gives rise to universal eigenvalue correlations like the nearestneighbor spacing distribution or the number variance, commonly known as the Wigner distri butions. Note that in the above double scaling limit, one is always restricted to the eigenvalues far from the tails of the density. This is a highly nontrivial feature of Gaussian random matrix ensembles. Indeed, the eigenvalue correlations for Gaussian ensembles are deeply related to the properties of Hermite polynomials f 3 g , and the above scaling is dictated by the asymptotic properties of Hermite polynomials of order N and argument x in the large N limit for small x . Since Hermite polynomials have a different asymptotic behavior for large x in the large N limit f 4 g , a different scaling is dictated for the...
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This note was uploaded on 12/15/2011 for the course PHY 3221 taught by Professor Chan during the Spring '08 term at University of Florida.
 Spring '08
 CHAN
 mechanics, Power

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