151 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
Long-range exclusion processes, generator and invariant measures
We show that if is an invariant measure for the long range exclusion
process putting no mass on the full configuration, is the formal generator
of that process and is a cylinder function, then
and . This result is then applied to determine (i) the set of
invariant and translation-invariant measures of the long range exclusion
process on when the underlying random walk is irreducible; (ii)
the set of invariant measures of the long range exclusion process on
when the underlying random walk is irreducible and either has zero
mean or allows jumps only to the nearest-neighbors.Comment: Published at http://dx.doi.org/10.1214/009117905000000486 in 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
A shape theorem for an epidemic model in dimension
We prove a shape theorem for the set of infected individuals in a spatial
epidemic model with 3 states (susceptible-infected-recovered) on , when there is no extinction of the infection. For this, we derive
percolation estimates (using dynamic renormalization techniques) for a locally
dependent random graph in correspondence with the epidemic model.Comment: 39 pages; soumi
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
Tightness for the interface of the one-dimensional contact process
We consider a symmetric, finite-range contact process with two types of
infection; both have the same (supercritical) infection rate and heal at rate
1, but sites infected by Infection 1 are immune to Infection 2. We take the
initial configuration where sites in have Infection 1 and sites
in have Infection 2, then consider the process defined as
the size of the interface area between the two infections at time . We show
that the distribution of is tight, thus proving a conjecture posed by
Cox and Durrett in [Bernoulli 1 (1995) 343--370].Comment: Published in at http://dx.doi.org/10.3150/09-BEJ236 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Convergence to the maximal invariant measure for a zero-range process with random rates
We consider a one-dimensional totally asymmetric nearest-neighbor zero-range
process with site-dependent jump-rates - an environment. For each environment p
we prove that the set of all invariant measures is the convex hull of a set of
product measures with geometric marginals. As a consequence we show that for
environments p satisfying certain asymptotic property, there are no invariant
measures concentrating on configurations with critical density bigger than
, a critical value. If is finite we say that there is
phase-transition on the density. In this case we prove that if the initial
configuration has asymptotic density strictly above , then the
process converges to the maximal invariant measure.Comment: 19 pages, Revised versio
Zero-range processes with rapidly growing rates
We provide two methods to construct zero-range processes with superlinear
rates on . In the first method these rates can grow very fast,
if either the dynamics and the initial distribution are translation invariant
or if only nearest neigbour translation invariant jumps are permitted, in the
one-dimensional lattice. In the second method the rates cannot grow as fast but
more general dynamics are allowed.Comment: 33 page
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