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Unformatted text preview: Family of solvable generalized random-matrix ensembles with unitary symmetry K. A. Muttalib Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611-8440, USA J. R. Klauder Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611-8440, USA and Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105, USA s Received 14 December 2004; published 4 May 2005 d We construct a very general family of characteristic functions describing random matrix ensembles s RME d having a global unitary invariance, and containing an arbitrary, one-variable probability measure, which we characterize by a spread function. Various choices of the spread function lead to a variety of possible generalized RMEs, which show deviations from the well-known Gaussian RME originally proposed by Wigner. We obtain the correlation functions of such generalized ensembles exactly and show examples of how particular choices of the spread function can describe ensembles with arbitrary eigenvalue densities as well as critical ensembles with multifractality. DOI: 10.1103/PhysRevE.71.055101 PACS number s s d : 02.50. 2 r, 05.40. 2 a, 05.60. 2 k, 05.90. 1 m The concept of random matrix ensembles s RME d pro- posed by Wigner to describe statistical properties of the ei- genvalues and eigenfunctions of complex nuclei f 1 g , has proved to be a very useful idea in the studies of a wide variety of physical systems including the equilibrium and transport properties of disordered quantum systems, quantum chaos, two-dimensional quantum gravity, conformal field theory, and chiral phase transitions in quantum chromody- namics, as well as financial correlations and wireless com- munications f 2,3 g . The underlying reason for such a wide range of applications is the universality of the correlations between the eigenvalues of given classes of RMEs. For ex- ample, once appropriate variables are chosen in which the mean spacing between eigenvalues is unity, the Gaussian en- sembles of random N 3 N matrices s with given symmetries d have, in the N limit, a universal s zero parameter d form for the nearest-neighbor spacing distribution or the spectral rigidity s number variance of eigenvalues in a given range d known generally as the Wigner-Dyson distribution f 1 g . The appropriately scaled energy levels of complex nuclei and transmission levels of weakly disordered mesoscopic metals both follow the above universal distributions even though the physical sizes of the systems differ by about nine orders of magnitude f 4 g . More recently, much interest has been generated in find- ing RMEs that deviate from the universal properties of the Gaussian ensembles in specific ways. One particular example is the attempt to find critical ensembles relevant for sys- tems at the critical point of, e.g., the Anderson transition in disordered conductors where the spacing distribution or the spectral rigidity is known to deviate from those of Gaussian...
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