Family of solvable generalized randommatrix ensembles with unitary symmetry
K. A. Muttalib
Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 326118440, USA
J. R. Klauder
Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 326118440, USA
and Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 326118105, 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, onevariable 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 wellknown 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, twodimensional 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 nearestneighbor spacing distribution or the spectral
rigidity
s
number variance of eigenvalues in a given range
d
known generally as the WignerDyson 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
RMEs
f
5–7
g
. Another example is the attempt to find a one
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 Spring '08
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 Physics, mechanics, Trigraph, American physical Society, physical review, random matrix

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