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Unformatted text preview: Power-law eigenvalue density, scaling, and critical random-matrix ensembles K. A. Muttalib * Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611-8440, 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 power-law densities s different from the scaling required in Gaussian random matrix ensembles d , we calculate exactly the two-level 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 two-level 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 nearest-neighbor 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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