69 research outputs found
Convergence in distribution for subcritical 2D oriented percolation seen from its rightmost point
We study subcritical two-dimensional oriented percolation seen from its
rightmost point on the set of infinite configurations which are bounded above.
This a Feller process whose state space is not compact and has no invariant
measures. We prove that it converges in distribution to a measure which charges
only finite configurations.Comment: Published in at http://dx.doi.org/10.1214/13-AOP841 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Law of large numbers for the asymmetric simple exclusion process
We consider simple exclusion processes on Z for which the underlying random
walk has a finite first moment and a non-zero mean and whose initial
distributions are product measures with different densities to the left and to
the right of the origin. We prove a strong law of large numbers for the number
of particles present at time t in an interval growing linearly with t.Comment: 16 page
Extreme paths in oriented 2D Percolation
A useful result about leftmost and rightmost paths in two dimensional bond
percolation is proved. This result was introduced without proof in \cite{G} in
the context of the contact process in continuous time. As discussed here, it
also holds for several related models, including the discrete time contact
process and two dimensional site percolation. Among the consequences are a
natural monotonicity in the probability of percolation between different sites
and a somewhat counter-intuitive correlation inequality
On the asymmetric zero-range in the rarefaction fan
We consider the one-dimensional asymmetric zero-range process starting from a
step decreasing profile. In the hydrodynamic limit this initial condition leads
to the rarefaction fan of the associated hydrodynamic equation. Under this
initial condition and for totally asymmetric jumps, we show that the weighted
sum of joint probabilities for second class particles sharing the same site is
convergent and we compute its limit. For partially asymmetric jumps we derive
the Law of Large Numbers for the position of a second class particle under the
initial configuration in which all the positive sites are empty, all the
negative sites are occupied with infinitely many first class particles and with
a single second class particle at the origin. Moreover, we prove that among the
infinite characteristics emanating from the position of the second class
particle, this particle chooses randomly one of them. The randomness is given
in terms of the weak solution of the hydrodynamic equation through some sort of
renormalization function. By coupling the zero-range with the exclusion process
we derive some limiting laws for more general initial conditions.Comment: 22 pages, to appear in Journal of Statistical Physic
Product Measure Steady States of Generalized Zero Range Processes
We establish necessary and sufficient conditions for the existence of
factorizable steady states of the Generalized Zero Range Process. This process
allows transitions from a site to a site involving multiple particles
with rates depending on the content of the site , the direction of
movement, and the number of particles moving. We also show the sufficiency of a
similar condition for the continuous time Mass Transport Process, where the
mass at each site and the amount transferred in each transition are continuous
variables; we conjecture that this is also a necessary condition.Comment: 9 pages, LaTeX with IOP style files. v2 has minor corrections; v3 has
been rewritten for greater clarit
Nonequilibrium phase transition in a non integrable zero-range process
The present work is an endeavour to determine analytically features of the
stationary measure of a non-integrable zero-range process, and to investigate
the possible existence of phase transitions for such a nonequilibrium model.
The rates defining the model do not satisfy the constraints necessary for the
stationary measure to be a product measure. Even in the absence of a drive,
detailed balance with respect to this measure is violated. Analytical and
numerical investigations on the complete graph demonstrate the existence of a
first-order phase transition between a fluid phase and a condensed phase, where
a single site has macroscopic occupation. The transition is sudden from an
imbalanced fluid where both species have densities larger than the critical
density, to a critical neutral fluid and an imbalanced condensate
Spreading in narrow channels
We study a lattice model for the spreading of fluid films, which are a few
molecular layers thick, in narrow channels with inert lateral walls. We focus
on systems connected to two particle reservoirs at different chemical
potentials, considering an attractive substrate potential at the bottom,
confining side walls, and hard-core repulsive fluid-fluid interactions. Using
kinetic Monte Carlo simulations we find a diffusive behavior. The corresponding
diffusion coefficient depends on the density and is bounded from below by the
free one-dimensional diffusion coefficient, valid for an inert bottom wall.
These numerical results are rationalized within the corresponding continuum
limit.Comment: 16 pages, 10 figure
Structure of the stationary state of the asymmetric target process
We introduce a novel migration process, the target process. This process is
dual to the zero-range process (ZRP) in the sense that, while for the ZRP the
rate of transfer of a particle only depends on the occupation of the departure
site, it only depends on the occupation of the arrival site for the target
process. More precisely, duality associates to a given ZRP a unique target
process, and vice-versa. If the dynamics is symmetric, i.e., in the absence of
a bias, both processes have the same stationary-state product measure. In this
work we focus our interest on the situation where the latter measure exhibits a
continuous condensation transition at some finite critical density ,
irrespective of the dimensionality. The novelty comes from the case of
asymmetric dynamics, where the target process has a nontrivial fluctuating
stationary state, whose characteristics depend on the dimensionality. In one
dimension, the system remains homogeneous at any finite density. An alternating
scenario however prevails in the high-density regime: typical configurations
consist of long alternating sequences of highly occupied and less occupied
sites. The local density of the latter is equal to and their
occupation distribution is critical. In dimension two and above, the asymmetric
target process exhibits a phase transition at a threshold density much
larger than . The system is homogeneous at any density below ,
whereas for higher densities it exhibits an extended condensate elongated along
the direction of the mean current, on top of a critical background with density
.Comment: 30 pages, 16 figure
Shock Profiles for the Asymmetric Simple Exclusion Process in One Dimension
The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice
is a system of particles which jump at rates and (here ) to
adjacent empty sites on their right and left respectively. The system is
described on suitable macroscopic spatial and temporal scales by the inviscid
Burgers' equation; the latter has shock solutions with a discontinuous jump
from left density to right density , , which
travel with velocity . In the microscopic system we
may track the shock position by introducing a second class particle, which is
attracted to and travels with the shock. In this paper we obtain the time
invariant measure for this shock solution in the ASEP, as seen from such a
particle. The mean density at lattice site , measured from this particle,
approaches at an exponential rate as , with a
characteristic length which becomes independent of when
. For a special value of the
asymmetry, given by , the measure is
Bernoulli, with density on the left and on the right. In the
weakly asymmetric limit, , the microscopic width of the shock
diverges as . The stationary measure is then essentially a
superposition of Bernoulli measures, corresponding to a convolution of a
density profile described by the viscous Burgers equation with a well-defined
distribution for the location of the second class particle.Comment: 34 pages, LaTeX, 2 figures are included in the LaTeX file. Email:
[email protected], [email protected], [email protected]
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