Universality of a family of random matrix ensembles with logarithmic softconFnement potentials
Jinmyung Choi and K. A. Muttalib
Department of Physics, University of Florida, Gainesville, Florida 326118440, USA
s
Received 9 June 2010; revised manuscript received 23 July 2010; published 8 September 2010
d
Recently we introduced a family of
U
s
N
d
invariant random matrix ensembles which is characterized by a
parameter
l
describing logarithmic softconFnement potentials
V
s
H
d,f
ln
H
g
s
1+
l
d
s
l.
0
d
. We showed that we
can study eigenvalue correlations of these “
l
ensembles” based on the numerical construction of the corre
sponding orthogonal polynomials with respect to the weight function exp
f
±
s
ln
x
d
1+
l
g
. In this work, we expand
our previous work and show that:
s
i
d
the eigenvalue density is given by a power law of the form
r
s
x
d
~
f
ln
x
g
l
±1
/
x
and
s
ii
d
the twolevel kernel has an anomalous structure, which is characteristic of the critical
ensembles. We further show that the anomalous part, or the socalled “ghostcorrelation peak,” is controlled by
the parameter
l
; decreasing
l
increases the anomaly. We also identify the twolevel kernel of the
l
ensembles
in the semiclassical regime, which can be written in a sinhkernel form with more general argument that
reduces to that of the critical ensembles for
l
=1. ²inally, we discuss the universality of the
l
ensembles, which
includes WignerDyson universality
s
l
→
`
limit
d
, the uncorrelated Poissontype behavior
s
l
→
0 limit
d
, and a
critical behavior for all the intermediate
l
s
0
,l,`
d
in the semiclassical regime. We also comment on the
implications of our results in the context of the localizationdelocalization problems as well as the
N
depen
dence of the twolevel kernel of the fattail random matrices.
DOI:
10.1103/PhysRevB.82.104202
PACS number
s
s
d
: 72.15.Rn, 05.40.
2
a, 05.60.
2
k, 05.90.
1
m
I. INTRODUCTION
Random matrix theory
s
RMT
d
deals with the statistical
properties of eigenvalues and eigenvectors of random matri
ces drawn from a certain probability measure. The theory
was successfully applied by Wigner
1
in the 1950s to describe
the spectral properties of complex manybody nuclei where
the underlying Hamiltonians are so complicated that a useful
way to study the system turned out to be through a statistical
treatment of the Hamiltonians.
2
Since then, RMT has been
applied to a wide variety of systems in diverse areas, includ
ing, e.g., manybody atoms and nuclei, quantum chaos, me
soscopic disordered conductors, twodimensional quantum
gravity, conformal Feld theory, chiral phase transitions as
well as zeros of Riemann zeta function, scalefree networks,
biological networks, communication systems, and Fnancial
markets.
3
,
4
This broad range of applicability of RMT in
seemingly unrelated areas highlights the universal features of
the correlations of the eigenvalues in RMT. Within the clas
sical Gaussian model pioneered by Wigner, these correla
tions are known as the WignerDyson
s
WD
d
statistics of the
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 Spring '08
 CHAN
 Physics, mechanics, Normal Distribution, American physical Society, Critical phenomena, Orthogonal polynomials, Physical Review B

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