78 research outputs found
Microscopic concavity and fluctuation bounds in a class of deposition processes
We prove fluctuation bounds for the particle current in totally asymmetric
zero range processes in one dimension with nondecreasing, concave jump rates
whose slope decays exponentially. Fluctuations in the characteristic directions
have order of magnitude . This is in agreement with the expectation
that these systems lie in the same KPZ universality class as the asymmetric
simple exclusion process. The result is via a robust argument formulated for a
broad class of deposition-type processes. Besides this class of zero range
processes, hypotheses of this argument have also been verified in the authors'
earlier papers for the asymmetric simple exclusion and the constant rate zero
range processes, and are currently under development for a bricklayers process
with exponentially increasing jump rates.Comment: Improved after Referee's comments: we added explanations and changed
some parts of the text. 50 pages, 1 figur
There are No Nice Interfaces in 2+1 Dimensional SOS-Models in Random Media
We prove that in dimension translation covariant Gibbs states
describing rigid interfaces in a disordered solid-on-solid (SOS) cannot exist
for any value of the temperature, in contrast to the situation in .
The prove relies on an adaptation of a theorem of Aizenman and Wehr.
Keywords: Disordered systems, interfaces, SOS-modelComment: 8 pages, gz-compressed Postscrip
Selected recollections of my relationship with Leo Breiman
During the period 1962--1964, I had a tenure track Assistant Professorship in
Mathematics at Cornell University in Ithaca, New York, where I did research in
probability theory, especially on linear diffusion processes. Being somewhat
lonely there and not liking the cold winter weather, I decided around the
beginning of 1964 to try to get a job in the Mathematics Department at UCLA, in
the city in which I was born and raised. At that time, Leo Breiman was an
Associate Professor in that department. Presumably, he liked my research on
linear diffusion processes and other research as well, since the department
offered me a tenure track Assistant Professorship, which I happily accepted.
During the Summer of 1965, I worked on various projects with Sidney Port, then
at RAND Corporation, especially on random walks and related material. I was
promoted to Associate Professor, effective in Fall, 1966, presumably thanks in
part to Leo. Early in 1966, I~was surprised to be asked by Leo to participate
in a department meeting called to discuss the possible hiring of Sidney. The
conclusion was that Sidney was hired as Associate Professor in the department,
as of Fall, 1966. Leo communicated to me his view that he thought that Sidney
and I worked well together, which is why he had urged the department to hire
Sidney. Anyhow, Sidney and I had a very fruitful and enjoyable collaboration in
probability and, to a much lesser extent, in theoretical statistics, for a
number of years thereafter.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS431 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition
The one-dimensional totally asymmetric simple exclusion process (TASEP) is
considered. We study the time evolution property of a tagged particle in TASEP
with the step-type initial condition. Calculated is the multi-time joint
distribution function of its position. Using the relation of the dynamics of
TASEP to the Schur process, we show that the function is represented as the
Fredholm determinant. We also study the scaling limit. The universality of the
largest eigenvalue in the random matrix theory is realized in the limit. When
the hopping rates of all particles are the same, it is found that the joint
distribution function converges to that of the Airy process after the time at
which the particle begins to move. On the other hand, when there are several
particles with small hopping rate in front of a tagged particle, the limiting
process changes at a certain time from the Airy process to the process of the
largest eigenvalue in the Hermitian multi-matrix model with external sources.Comment: 48 pages, 8 figure
A multi-layer extension of the stochastic heat equation
Motivated by recent developments on solvable directed polymer models, we
define a 'multi-layer' extension of the stochastic heat equation involving
non-intersecting Brownian motions.Comment: v4: substantially extended and revised versio
Locating the minimum : Approach to equilibrium in a disordered, symmetric zero range process
We consider the dynamics of the disordered, one-dimensional, symmetric zero
range process in which a particle from an occupied site hops to its nearest
neighbour with a quenched rate . These rates are chosen randomly from the
probability distribution , where is the lower cutoff.
For , this model is known to exhibit a phase transition in the steady
state from a low density phase with a finite number of particles at each site
to a high density aggregate phase in which the site with the lowest hopping
rate supports an infinite number of particles. In the latter case, it is
interesting to ask how the system locates the site with globally minimum rate.
We use an argument based on local equilibrium, supported by Monte Carlo
simulations, to describe the approach to the steady state. We find that at
large enough time, the mass transport in the regions with a smooth density
profile is described by a diffusion equation with site-dependent rates, while
the isolated points where the mass distribution is singular act as the
boundaries of these regions. Our argument implies that the relaxation time
scales with the system size as with for and
suggests a different behaviour for .Comment: Revtex, 7 pages including 3 figures. Submitted to Pramana -- special
issue on mesoscopic and disordered system
A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation
We estimate the variance of the value function for a random optimal control
problem. The value function is the solution of a Hamilton-Jacobi
equation with random Hamiltonian
in dimension . It is known that homogenization occurs as , but little is known about the statistical fluctuations of .
Our main result shows that the variance of the solution is bounded
by . The proof relies on a modified Poincar\'e
inequality of Talagrand
Occupation times of exclusion processes
In this paper we consider exclusion processes evolving on the one-dimensional lattice , under the diffusive time scale and starting from the invariant state - the Bernoulli product measure of parameter . Our goal consists in establishing the scaling
limits of the additive functional - {\em{ the occupation time of the origin}}. We present a method, recently introduced in \cite{G.J.}, from which a
{\em{local Boltzmann-Gibbs Principle}} can be derived for a general class of exclusion processes. In this case, this
principle says that is very well approximated to the additive functional of the density of particles. As a consequence, the scaling limits of
follow from the scaling limits of the density of particles. As examples we present the mean-zero exclusion, the symmetric simple exclusion and
the weakly asymmetric simple exclusion. For the latter under a strong asymmetry regime, the limit of is given in terms of the solution of the KPZ equation.FC
Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure
In [arXiv:0804.3035] we studied an interacting particle system which can be
also interpreted as a stochastic growth model. This model belongs to the
anisotropic KPZ class in 2+1 dimensions. In this paper we present the results
that are relevant from the perspective of stochastic growth models, in
particular: (a) the surface fluctuations are asymptotically Gaussian on a
sqrt(ln(t)) scale and (b) the correlation structure of the surface is
asymptotically given by the massless field.Comment: 13 pages, 4 figure
Generalized Green Functions and current correlations in the TASEP
We study correlation functions of the totally asymmetric simple exclusion
process (TASEP) in discrete time with backward sequential update. We prove a
determinantal formula for the generalized Green function which describes
transitions between positions of particles at different individual time
moments. In particular, the generalized Green function defines a probability
measure at staircase lines on the space-time plane. The marginals of this
measure are the TASEP correlation functions in the space-time region not
covered by the standard Green function approach. As an example, we calculate
the current correlation function that is the joint probability distribution of
times taken by selected particles to travel given distance. An asymptotic
analysis shows that current fluctuations converge to the process.Comment: 46 pages, 3 figure
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