1,631 research outputs found
Moderate deviations for the chemical distance in Bernoulli percolation
In this paper, we establish moderate deviations for the chemical distance in
Bernoulli percolation. The chemical distance between two points is the length
of the shortest open path between these two points. Thus, we study the size of
random fluctuations around the mean value, and also the asymptotic behavior of
this mean value. The estimates we obtain improve our knowledge of the
convergence to the asymptotic shape. Our proofs rely on concentration
inequalities proved by Boucheron, Lugosi and Massart, and also on the
approximation theory of subadditive functions initiated by Alexander.Comment: 19 pages, in english. A french version, entitled "D\'eviations
mod\'er\'ees de la distance chimique" is also availabl
Asymptotic shape for the chemical distance and first-passage percolation in random environment
The aim of this paper is to generalize the well-known asymptotic shape result
for first-passage percolation on \Zd to first-passage percolation on a random
environment given by the infinite cluster of a supercritical Bernoulli
percolation model. We prove the convergence of the renormalized set of wet
points to a deterministic shape that does not depend on the random environment.
As a special case of the previous result, we obtain an asymptotic shape theorem
for the chemical distance in supercritical Bernoulli percolation. We also prove
a flat edge result. Some various examples are also given.Comment: redaction du 10 avril 200
Competition between growths governed by Bernoulli Percolation
We study a competition model on where the two infections are
driven by supercritical Bernoulli percolations with distinct parameters and
. We prove that, for any , there exist at most countably many values of
such that coexistence can occur.Comment: 30 pages with figure
Asymptotic shape for the contact process in random environment
The aim of this article is to prove asymptotic shape theorems for the contact
process in stationary random environment. These theorems generalize known
results for the classical contact process. In particular, if H_t denotes the
set of already occupied sites at time t, we show that for almost every
environment, when the contact process survives, the set H_t/t almost surely
converges to a compact set that only depends on the law of the environment. To
this aim, we prove a new almost subadditive ergodic theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AAP796 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Continuity of the asymptotic shape of the supercritical contact process
We prove the continuity of the shape governing the asymptotic growth of the
supercritical contact process in Z^d , with respect to the infection parameter.
The proof is valid in any dimension d 1
Does Eulerian percolation on percolate ?
Eulerian percolation on Z 2 with parameter p is the classical Bernoulli bond
percolation with parameter p conditioned on the fact that every site has an
even degree. We first explain why Eulerian percolation with parameter p
coincides with the contours of the Ising model for a well-chosen parameter
(p). Then we study the percolation properties of Eulerian percolation.Comment: This improves the previous version, only the status of one value for
p is unknow
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