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J. Phys. A: Math. Theor.
42
(2009) 152001 (8pp)
doi:10.1088/17518113/42/15/152001
FAST TRACK COMMUNICATION
Rotationally invariant family of L´evylike random
matrix ensembles
Jinmyung Choi and K A Muttalib
Department of Physics, University of Florida, Gainesville, FL 326118440, USA
Email:
[email protected]
and
[email protected]
Received 29 January 2009, in final form 4 March 2009
Published 20 March 2009
Online at
stacks.iop.org/JPhysA/42/152001
Abstract
We introduce a family of rotationally invariant random matrix ensembles
characterized by a parameter
λ
.
While
λ
=
1 corresponds to wellknown
critical ensembles, we show that
λ
=
1 describes ‘L´evylike’ ensembles,
characterized by powerlaw 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´evylike 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
manybody 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, twodimensional gravity, conformal field
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 ‘qrandom matrix
ensembles’ (qRMEs) [
5
,
6
]. These were later shown to be models of ‘critical ensembles’,
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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 qRMEs 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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 Spring '08
 CHAN
 Physics, mechanics, Eigenvalue, eigenvector and eigenspace, J. Phys, Orthogonal matrix, random matrix, Orthogonal polynomials

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