385 research outputs found
An Extension of Level-spacing Universality
Dyson's short-distance universality of the correlation functions implies the
universality of P(s), the level-spacing distribution. We first briefly review
how this property is understood for unitary invariant ensembles and consider
next a Hamiltonian H = H_0+ V , in which H_0 is a given, non-random, N by N
matrix, and V is an Hermitian random matrix with a Gaussian probability
distribution. n-point correlation function may still be expressed as a
determinant of an n by n matrix, whose elements are given by a kernel
as in the H_0=0 case. From this representation we can show
that Dyson's short-distance universality still holds. We then conclude that
P(s) is independent of H_0.Comment: 12 pages, Revte
Universal correlations for deterministic plus random Hamiltonians
We consider the (smoothed) average correlation between the density of energy
levels of a disordered system, in which the Hamiltonian is equal to the sum of
a deterministic H0 and of a random potential . Remarkably, this
correlation function may be explicitly determined in the limit of large
matrices, for any unperturbed H0 and for a class of probability distribution
P of the random potential. We find a compact representation of the
correlation function. From this representation one obtains readily the short
distance behavior, which has been conjectured in various contexts to be
universal. Indeed we find that it is totally independent of both H0 and
P().Comment: 26P, (+5 figures not included
Characteristic polynomials of random matrices at edge singularities
We have discussed earlier the correlation functions of the random variables
\det(\la-X) in which is a random matrix. In particular the moments of the
distribution of these random variables are universal functions, when measured
in the appropriate units of the level spacing. When the \la's, instead of
belonging to the bulk of the spectrum, approach the edge, a cross-over takes
place to an Airy or to a Bessel problem, and we consider here these modified
classes of universality.
Furthermore, when an external matrix source is added to the probability
distribution of , various new phenomenons may occur and one can tune the
spectrum of this source matrix to new critical points. Again there are
remarkably simple formulae for arbitrary source matrices, which allow us to
compute the moments of the characteristic polynomials in these cases as well.Comment: 22 pages, late
Universal singularity at the closure of a gap in a random matrix theory
We consider a Hamiltonian , in which is a given
non-random Hermitian matrix,and is an Hermitian random matrix
with a Gaussian probability distribution.We had shown before that Dyson's
universality of the short-range correlations between energy levels holds at
generic points of the spectrum independently of . We consider here the
case in which the spectrum of is such that there is a gap in the
average density of eigenvalues of which is thus split into two pieces. When
the spectrum of is tuned so that the gap closes, a new class of
universality appears for the energy correlations in the vicinity of this
singular point.Comment: 20pages, Revtex, to be published in Phys. Rev.
Spectral form factor in a random matrix theory
In the theory of disordered systems the spectral form factor , the
Fourier transform of the two-level correlation function with respect to the
difference of energies, is linear for and constant for
. Near zero and near its exhibits oscillations which have
been discussed in several recent papers. In the problems of mesoscopic
fluctuations and quantum chaos a comparison is often made with random matrix
theory. It turns out that, even in the simplest Gaussian unitary ensemble,
these oscilllations have not yet been studied there. For random matrices, the
two-level correlation function exhibits several
well-known universal properties in the large N limit. Its Fourier transform is
linear as a consequence of the short distance universality of
. However the cross-over near zero and
requires to study these correlations for finite N. For this purpose we use an
exact contour-integral representation of the two-level correlation function
which allows us to characterize these cross-over oscillatory properties. The
method is also extended to the time-dependent case.Comment: 36P, (+5 figures not included
A simple model of dimensional collapse
We consider a simple model of d families of scalar field interacting with
geometry in two dimensions. The geometry is locally flat and has only global
degrees of freedom. When d<0 the universe is locally two dimensional but for
d>0 it collapses to a one dimensional manifold. The model has some, but not
all, of the characteristics believed to be features of the full theory of
conformal matter interacting with quantum gravity which has local geometric
degrees of freedom.Comment: 10 pages, plain Late
Critical behavior of the Random-Field Ising model at and beyond the Upper Critical Dimension
The disorder-driven phase transition of the RFIM is observed using exact
ground-state computer simulations for hyper cubic lattices in d=5,6,7
dimensions. Finite-size scaling analyses are used to calculate the critical
point and the critical exponents of the specific heat, magnetization,
susceptibility and of the correlation length. For dimensions d=6,7 which are
larger or equal to the assumed upper critical dimension, d_u=6, mean-field
behaviour is found, i.e. alpha=0, beta=1/2, gamma=1, nu=1/2. For the analysis
of the numerical data, it appears to be necessary to include recently proposed
corrections to scaling at and beyond the upper critical dimension.Comment: 8 pages and 13 figures; A consise summary of this work can be found
in the papercore database at http://www.papercore.org/Ahrens201
Scaling Relations for Logarithmic Corrections
Multiplicative logarithmic corrections to scaling are frequently encountered
in the critical behavior of certain statistical-mechanical systems. Here, a
Lee-Yang zero approach is used to systematically analyse the exponents of such
logarithms and to propose scaling relations between them. These proposed
relations are then confronted with a variety of results from the literature.Comment: 4 page
"Single Ring Theorem" and the Disk-Annulus Phase Transition
Recently, an analytic method was developed to study in the large limit
non-hermitean random matrices that are drawn from a large class of circularly
symmetric non-Gaussian probability distributions, thus extending the existing
Gaussian non-hermitean literature. One obtains an explicit algebraic equation
for the integrated density of eigenvalues from which the Green's function and
averaged density of eigenvalues could be calculated in a simple manner. Thus,
that formalism may be thought of as the non-hermitean analog of the method due
to Br\'ezin, Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian
random matrices. A somewhat surprising result is the so called "Single Ring"
theorem, namely, that the domain of the eigenvalue distribution in the complex
plane is either a disk or an annulus. In this paper we extend previous results
and provide simple new explicit expressions for the radii of the eigenvalue
distiobution and for the value of the eigenvalue density at the edges of the
eigenvalue distribution of the non-hermitean matrix in terms of moments of the
eigenvalue distribution of the associated hermitean matrix. We then present
several numerical verifications of the previously obtained analytic results for
the quartic ensemble and its phase transition from a disk shaped eigenvalue
distribution to an annular distribution. Finally, we demonstrate numerically
the "Single Ring" theorem for the sextic potential, namely, the potential of
lowest degree for which the "Single Ring" theorem has non-trivial consequences.Comment: latex, 5 eps figures, 41 page
The 3-D O(4) universality class and the phase transition in two-flavor QCD
We determine the critical equation of state of the three-dimensional O(4)
universality class. We first consider the small-field expansion of the
effective potential (Helmholtz free energy). Then, we apply a systematic
approximation scheme based on polynomial parametric representations that are
valid in the whole critical regime, satisfy the correct analytic properties
(Griffiths' analyticity), take into account the Goldstone singularities at the
coexistence curve, and match the small-field expansion of the effective
potential. From the approximate representations of the equation of state, we
obtain estimates of several universal amplitude ratios.
The three-dimensional O(4) universality class is expected to describe the
finite-temperature chiral transition of quantum chromodynamics with two light
flavors. Within this picture, the O(4) critical equation of state relates the
reduced temperature, the quark masses, and the condensates around T_c in the
limit of vanishing quark masses.Comment: 19 pages, 5 fig
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