125 research outputs found
On a conjecture of Widom
We prove a conjecture of H.Widom stated in [W] (math/0108008) about the
reality of eigenvalues of certain infinite matrices arising in asymptotic
analysis of large Toeplitz determinants. As a byproduct we obtain a new proof
of A.Okounkov's formula for the (determinantal) correlation functions of the
Schur measures on partitions.Comment: 9 page
Structured matrices, continued fractions, and root localization of polynomials
We give a detailed account of various connections between several classes of
objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices,
Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems,
total positivity, and root localization of univariate polynomials. Along with a
survey of many classical facts, we provide a number of new results.Comment: 79 pages; new material added to the Introductio
Gibbs Ensembles of Nonintersecting Paths
We consider a family of determinantal random point processes on the
two-dimensional lattice and prove that members of our family can be interpreted
as a kind of Gibbs ensembles of nonintersecting paths. Examples include
probability measures on lozenge and domino tilings of the plane, some of which
are non-translation-invariant.
The correlation kernels of our processes can be viewed as extensions of the
discrete sine kernel, and we show that the Gibbs property is a consequence of
simple linear relations satisfied by these kernels. The processes depend on
infinitely many parameters, which are closely related to parametrization of
totally positive Toeplitz matrices.Comment: 6 figure
Infinite-dimensional diffusions as limits of random walks on partitions
The present paper originated from our previous study of the problem of
harmonic analysis on the infinite symmetric group. This problem leads to a
family {P_z} of probability measures, the z-measures, which depend on the
complex parameter z. The z-measures live on the Thoma simplex, an
infinite-dimensional compact space which is a kind of dual object to the
infinite symmetric group. The aim of the paper is to introduce stochastic
dynamics related to the z-measures. Namely, we construct a family of diffusion
processes in the Toma simplex indexed by the same parameter z. Our diffusions
are obtained from certain Markov chains on partitions of natural numbers n in a
scaling limit as n goes to infinity. These Markov chains arise in a natural
way, due to the approximation of the infinite symmetric group by the increasing
chain of the finite symmetric groups. Each z-measure P_z serves as a unique
invariant distribution for the corresponding diffusion process, and the process
is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so
that the process is reversible. We describe the spectrum of its generator and
compute the associated (pre)Dirichlet form.Comment: AMSTex, 33 pages. Version 2: minor changes, typos corrected, to
appear in Prob. Theor. Rel. Field
Intensity Distribution of Waves Transmitted Through a Multiple Scattering Medium
The distributions of the angular transmission coefficient and of the total
transmission are calculated for multiple scattered waves. The calculation is
based on a mapping to the distribution of eigenvalues of the transmission
matrix. The distributions depend on the profile of the incoming beam. The
distribution function of the angular transmission has a stretched exponential
decay. The total-transmission distribution grows log-normally whereas it decays
exponentially.Comment: 8 pages, revtex3.0, 3 postscript figures, NvR0
Localization in non-chiral network models for two-dimensional disordered wave mechanical systems
Scattering theoretical network models for general coherent wave mechanical
systems with quenched disorder are investigated. We focus on universality
classes for two dimensional systems with no preferred orientation: Systems of
spinless waves undergoing scattering events with broken or unbroken time
reversal symmetry and systems of spin 1/2 waves with time reversal symmetric
scattering. The phase diagram in the parameter space of scattering strengths is
determined. The model breaking time reversal symmetry contains the critical
point of quantum Hall systems but, like the model with unbroken time reversal
symmetry, only one attractive fixed point, namely that of strong localization.
Multifractal exponents and quasi-one-dimensional localization lengths are
calculated numerically and found to be related by conformal invariance.
Furthermore, they agree quantitatively with theoretical predictions. For
non-vanishing spin scattering strength the spin 1/2 systems show
localization-delocalization transitions.Comment: 4 pages, REVTeX, 4 figures (postscript
Non-perturbative calculation of the probability distribution of plane-wave transmission through a disordered waveguide
A non-perturbative random-matrix theory is applied to the transmission of a
monochromatic scalar wave through a disordered waveguide. The probability
distributions of the transmittances T_{mn} and T_n=\sum_m T_{mn} of an incident
mode n are calculated in the thick-waveguide limit, for broken time-reversal
symmetry. A crossover occurs from Rayleigh or Gaussian statistics in the
diffusive regime to lognormal statistics in the localized regime. A
qualitatively different crossover occurs if the disordered region is replaced
by a chaotic cavity. ***Submitted to Physical Review E.***Comment: 7 pages, REVTeX-3.0, 5 postscript figures appended as self-extracting
archive. A complete postscript file with figures and text (4 pages) is
available from http://rulgm4.LeidenUniv.nl/preprints.htm
Reflection and transmission of waves in surface-disordered waveguides
The reflection and transmission amplitudes of waves in disordered multimode
waveguides are studied by means of numerical simulations based on the invariant
embedding equations. In particular, we analyze the influence of surface-type
disorder on the behavior of the ensemble average and fluctuations of the
reflection and transmission coefficients, reflectance, transmittance, and
conductance. Our results show anomalous effects stemming from the combination
of mode dispersion and rough surface scattering: For a given waveguide length,
the larger the mode transverse momentum is, the more strongly is the mode
scattered. These effects manifest themselves in the mode selectivity of the
transmission coefficients, anomalous backscattering enhancement, and speckle
pattern both in reflection and transmission, reflectance and transmittance, and
also in the conductance and its universal fluctuations. It is shown that, in
contrast to volume impurities, surface scattering in quasi-one-dimensional
structures (waveguides) gives rise to the coexistence of the ballistic,
diffusive, and localized regimes within the same sample.Comment: LaTeX (REVTeX), 12 pages with 14 EPS figures (epsf macro), minor
change
Intensity Distribution of Modes in Surface Corrugated Waveguides
Exact calculations of transmission and reflection coefficients in surface
randomly corrugated optical waveguides are presented. As the length of the
corrugated part of the waveguide increases, there is a strong preference to
forward coupling through the lowest mode. An oscillating behavior of the
enhanced backscattering as a function of the wavelength is predicted. Although
the transport is strongly non isotropic, the analysis of the probability
distributions of the transmitted waves confirms in this configuration
distributions predicted by Random Matrix Theory for volume disorder
Spectral Compressibility at the Metal-Insulator Transition of the Quantum Hall Effect
The spectral properties of a disordered electronic system at the
metal-insulator transition point are investigated numerically. A recently
derived relation between the anomalous diffusion exponent and the
spectral compressibility at the mobility edge, , is
confirmed for the integer quantum Hall delocalization transition. Our
calculations are performed within the framework of an unitary network-model and
represent a new method to investigate spectral properties of disordered
systems.Comment: 5 pages, RevTeX, 3 figures, Postscript, strongly revised version to
be published in PR
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