449 research outputs found
On ASEP with Step Bernoulli Initial Condition
This paper extends results of earlier work on ASEP to the case of step
Bernoulli initial condition. The main results are a representation in terms of
a Fredholm determinant for the probability distribution of a fixed particle,
and asymptotic results which in particular establish KPZ universality for this
probability in one regime. (And, as a corollary, for the current fluctuations.)Comment: 16 pages. Revised version adds references and expands the
introductio
On Orthogonal and Symplectic Matrix Ensembles
The focus of this paper is on the probability, , that a set
consisting of a finite union of intervals contains no eigenvalues for the
finite Gaussian Orthogonal () and Gaussian Symplectic ()
Ensembles and their respective scaling limits both in the bulk and at the edge
of the spectrum. We show how these probabilities can be expressed in terms of
quantities arising in the corresponding unitary () ensembles. Our most
explicit new results concern the distribution of the largest eigenvalue in each
of these ensembles. In the edge scaling limit we show that these largest
eigenvalue distributions are given in terms of a particular Painlev\'e II
function.Comment: 34 pages. LaTeX file with one figure. To appear in Commun. Math.
Physic
Formulas for ASEP with Two-Sided Bernoulli Initial Condition
For the asymmetric simple exclusion process on the integer lattice with
two-sided Bernoulli initial condition, we derive exact formulas for the
following quantities: (1) the probability that site x is occupied at time t;
(2) a correlation function, the probability that site 0 is occupied at time 0
and site x is occupied at time t; (3) the distribution function for the total
flux across 0 at time t and its exponential generating function.Comment: 18 page
Asymptotics in ASEP with Step Initial Condition
In previous work the authors considered the asymmetric simple exclusion
process on the integer lattice in the case of step initial condition, particles
beginning at the positive integers. There it was shown that the probability
distribution for the position of an individual particle is given by an integral
whose integrand involves a Fredholm determinant. Here we use this formula to
obtain three asymptotic results for the positions of these particles. In one an
apparently new distribution function arises and in another the distribuion
function F_2 arises. The latter extends a result of Johansson on TASEP to ASEP.Comment: 29 pages. Version 2 has a new title and adds asymptotics in a third
regim
The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line
In this paper we find explicit formulas for: (1) Green's function for a
system of one-dimensional bosons interacting via a delta-function potential
with particles confined to the positive half-line; and (2) the transition
probability for the one-dimensional asymmetric simple exclusion process (ASEP)
with particles confined to the nonnegative integers. These are both for systems
with a finite number of particles. The formulas are analogous to ones obtained
earlier for the Bose gas and ASEP on the line and integers, respectively. We
use coordinate Bethe Ansatz appropriately modified to account for confinement
of the particles to the half-line. As in the earlier work, the proof for the
ASEP is less straightforward than for the Bose gas.Comment: 14 Page
A Fredholm Determinant Representation in ASEP
In previous work the authors found integral formulas for probabilities in the
asymmetric simple exclusion process (ASEP) on the integer lattice. The dynamics
are uniquely determined once the initial state is specified. In this note we
restrict our attention to the case of step initial condition with particles at
the positive integers, and consider the distribution function for the m'th
particle from the left. In the previous work an infinite series of multiple
integrals was derived for this distribution. In this note we show that the
series can be summed to give a single integral whose integrand involves a
Fredholm determinant. We use this determinant representation to derive
(non-rigorously, at this writing) a scaling limit.Comment: 12 Pages. Version 3 includes a scaling conjectur
Fredholm Determinants, Differential Equations and Matrix Models
Orthogonal polynomial random matrix models of NxN hermitian matrices lead to
Fredholm determinants of integral operators with kernel of the form (phi(x)
psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm
determinants of integral operators having kernel of this form and where the
underlying set is a union of open intervals. The emphasis is on the
determinants thought of as functions of the end-points of these intervals. We
show that these Fredholm determinants with kernels of the general form
described above are expressible in terms of solutions of systems of PDE's as
long as phi and psi satisfy a certain type of differentiation formula. There is
also an exponential variant of this analysis which includes the circular
ensembles of NxN unitary matrices.Comment: 34 pages, LaTeX using RevTeX 3.0 macros; last version changes only
the abstract and decreases length of typeset versio
Mediterranean-type diet and brain structural change from 73 to 76 years in a Scottish cohort
STUDY FUNDING The data were collected by a Research into Ageing programme grant; research continues as part of the Age UKâfunded Disconnected Mind project. The work was undertaken by The University of Edinburgh Centre for Cognitive Ageing and Cognitive Epidemiology, part of the cross-council Lifelong Health and Wellbeing Initiative (MR/K026992/1), with funding from the BBSRC and Medical Research Council. Imaging and image analysis was performed at the Brain Research Imaging Centre (sbirc.ed.ac.uk/), Edinburgh, supported by the Scottish Funding Council SINAPSE Collaboration. Derivation of mean cortical thickness measures was funded by the Scottish Funding Councilâs Postdoctoral and Early Career Researchers Exchange Fund awarded by SINAPSE to David Alexander Dickie. L.C.A.C. acknowledges funding from the Scottish Government's Rural and Environment Science and Analytical Services (RESAS) division.Peer reviewedPublisher PD
Level-Spacing Distributions and the Bessel Kernel
The level spacing distributions which arise when one rescales the Laguerre or
Jacobi ensembles of hermitian matrices is studied. These distributions are
expressible in terms of a Fredholm determinant of an integral operator whose
kernel is expressible in terms of Bessel functions of order . We derive
a system of partial differential equations associated with the logarithmic
derivative of this Fredholm determinant when the underlying domain is a union
of intervals. In the case of a single interval this Fredholm determinant is a
Painleve tau function.Comment: 18 pages, resubmitted to make postscript compatible, no changes to
manuscript conten
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