19,093 research outputs found
Probability density of determinants of random matrices
In this brief paper the probability density of a random real, complex and
quaternion determinant is rederived using singular values. The behaviour of
suitably rescaled random determinants is studied in the limit of infinite order
of the matrices
Random matrix models with log-singular level confinement: method of fictitious fermions
Joint distribution function of N eigenvalues of U(N) invariant random-matrix
ensemble can be interpreted as a probability density to find N fictitious
non-interacting fermions to be confined in a one-dimensional space. Within this
picture a general formalism is developed to study the eigenvalue correlations
in non-Gaussian ensembles of large random matrices possessing non-monotonic,
log-singular level confinement. An effective one-particle Schroedinger equation
for wave-functions of fictitious fermions is derived. It is shown that
eigenvalue correlations are completely determined by the Dyson's density of
states and by the parameter of the logarithmic singularity. Closed analytical
expressions for the two-point kernel in the origin, bulk, and soft-edge scaling
limits are deduced in a unified way, and novel universal correlations are
predicted near the end point of the single spectrum support.Comment: 13 pages (latex), Presented at the MINERVA Workshop on Mesoscopics,
Fractals and Neural Networks, Eilat, Israel, March 199
Heterotic free fermionic and symmetric toroidal orbifold models
Free fermionic models and symmetric heterotic toroidal orbifolds both
constitute exact backgrounds that can be used effectively for phenomenological
explorations within string theory. Even though it is widely believed that for
Z2xZ2 orbifolds the two descriptions should be equivalent, a detailed
dictionary between both formulations is still lacking. This paper aims to fill
this gap: We give a detailed account of how the input data of both descriptions
can be related to each other. In particular, we show that the generalized GSO
phases of the free fermionic model correspond to generalized torsion phases
used in orbifold model building. We illustrate our translation methods by
providing free fermionic realizations for all Z2xZ2 orbifold geometries in six
dimensions.Comment: 1+49 pages latex, minor revisions and references adde
Magnetic susceptibility of ultra-small superconductor grains
For assemblies of superconductor nanograins, the magnetic response is
analyzed as a function of both temperature and magnetic field. In order to
describe the interaction energy of electron pairs for a huge number of
many-particle states, involved in calculations, we develop a simple
approximation, based on the Richardson solution for the reduced BCS Hamiltonian
and applicable over a wide range of the grain sizes and interaction strengths
at arbitrary distributions of single-electron energy levels in a grain. Our
study is focused upon ultra-small grains, where both the mean value of the
nearest-neighbor spacing of single-electron energy levels in a grain and
variations of this spacing from grain to grain significantly exceed the
superconducting gap in bulk samples of the same material. For these ultra-small
superconductor grains, the overall profiles of the magnetic susceptibility as a
function of magnetic field and temperature are demonstrated to be qualitatively
different from those for normal grains. We show that the analyzed signatures of
pairing correlations are sufficiently stable with respect to variations of the
average value of the grain size and its dispersion over an assembly of
nanograins. The presence of these signatures does not depend on a particular
choice of statistics, obeyed by single-electron energy levels in grains.Comment: 40 pages, 12 figures, submitted to Phys. Rev. B, E-mail addresses:
[email protected], [email protected], [email protected]
Causality vs. Ward identity in disordered electron systems
We address the problem of fulfilling consistency conditions in solutions for
disordered noninteracting electrons. We prove that if we assume the existence
of the diffusion pole in an electron-hole symmetric theory we cannot achieve a
solution with a causal self-energy that would fully fit the Ward identity.
Since the self-energy must be causal, we conclude that the Ward identity is
partly violated in the diffusive transport regime of disordered electrons. We
explain this violation in physical terms and discuss its consequences.Comment: 4 pages, REVTeX, 6 EPS figure
Minimizing Higgs Potentials via Numerical Polynomial Homotopy Continuation
The study of models with extended Higgs sectors requires to minimize the
corresponding Higgs potentials, which is in general very difficult. Here, we
apply a recently developed method, called numerical polynomial homotopy
continuation (NPHC), which guarantees to find all the stationary points of the
Higgs potentials with polynomial-like nonlinearity. The detection of all
stationary points reveals the structure of the potential with maxima,
metastable minima, saddle points besides the global minimum. We apply the NPHC
method to the most general Higgs potential having two complex Higgs-boson
doublets and up to five real Higgs-boson singlets. Moreover the method is
applicable to even more involved potentials. Hence the NPHC method allows to go
far beyond the limits of the Gr\"obner basis approach.Comment: 9 pages, 4 figure
2*2 random matrix ensembles with reduced symmetry: From Hermitian to PT-symmetric matrices
A possibly fruitful extension of conventional random matrix ensembles is
proposed by imposing symmetry constraints on conventional Hermitian matrices or
parity-time- (PT-) symmetric matrices. To illustrate the main idea, we first
study 2*2 complex Hermitian matrix ensembles with O(2) invariant constraints,
yielding novel level-spacing statistics such as singular distributions,
half-Gaussian distribution, distributions interpolating between GOE (Gaussian
Orthogonal Ensemble) distribution and half Gaussian distributions, as well as
gapped-GOE distribution. Such a symmetry-reduction strategy is then used to
explore 2*2 PT-symmetric matrix ensembles with real eigenvalues. In particular,
PT-symmetric random matrix ensembles with U(2) invariance can be constructed,
with the conventional complex Hermitian random matrix ensemble being a special
case. In two examples of PT-symmetric random matrix ensembles, the
level-spacing distributions are found to be the standard GUE (Gaussian Unitary
Ensemble) statistics or "truncated-GUE" statistics
Competition and cooperation:aspects of dynamics in sandpiles
In this article, we review some of our approaches to granular dynamics, now
well known to consist of both fast and slow relaxational processes. In the
first case, grains typically compete with each other, while in the second, they
cooperate. A typical result of {\it cooperation} is the formation of stable
bridges, signatures of spatiotemporal inhomogeneities; we review their
geometrical characteristics and compare theoretical results with those of
independent simulations. {\it Cooperative} excitations due to local density
fluctuations are also responsible for relaxation at the angle of repose; the
{\it competition} between these fluctuations and external driving forces, can,
on the other hand, result in a (rare) collapse of the sandpile to the
horizontal. Both these features are present in a theory reviewed here. An arena
where the effects of cooperation versus competition are felt most keenly is
granular compaction; we review here a random graph model, where three-spin
interactions are used to model compaction under tapping. The compaction curve
shows distinct regions where 'fast' and 'slow' dynamics apply, separated by
what we have called the {\it single-particle relaxation threshold}. In the
final section of this paper, we explore the effect of shape -- jagged vs.
regular -- on the compaction of packings near their jamming limit. One of our
major results is an entropic landscape that, while microscopically rough,
manifests {\it Edwards' flatness} at a macroscopic level. Another major result
is that of surface intermittency under low-intensity shaking.Comment: 36 pages, 23 figures, minor correction
Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
We exploit a relation between the mean number of minima of random
Gaussian surfaces and extreme eigenvalues of random matrices to understand the
critical behaviour of in the simplest glass-like transition occuring in
a toy model of a single particle in -dimensional random environment, with
. Varying the control parameter through the critical value
we analyse in detail how drops from being exponentially
large in the glassy phase to on the other side of the
transition. We also extract a subleading behaviour of in both
glassy and simple phases. The width of the critical region
is found to scale as and inside that region converges
to a limiting shape expressed in terms of the Tracy-Widom distribution
Chern-Simons theory, exactly solvable models and free fermions at finite temperature
We show that matrix models in Chern-Simons theory admit an interpretation as
1D exactly solvable models, paralleling the relationship between the Gaussian
matrix model and the Calogero model. We compute the corresponding Hamiltonians,
ground-state wavefunctions and ground-state energies and point out that the
models can be interpreted as quasi-1D Coulomb plasmas. We also study the
relationship between Chern-Simons theory on and a system of N
one-dimensional fermions at finite temperature with harmonic confinement. In
particular we show that the Chern-Simons partition function can be described by
the density matrix of the free fermions in a very particular, crystalline,
configuration. For this, we both use the Brownian motion and the matrix model
description of Chern-Simons theory and find several common features with c=1
theory at finite temperature. Finally, using the exactly solvable model result,
we show that the finite temperature effect can be described with a specific
two-body interaction term in the Hamiltonian, with 1D Coulombic behavior at
large separations.Comment: 19 pages, v2: references adde
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