59 research outputs found
Capacitive flows on a 2D random net
This paper concerns maximal flows on traveling from a convex
set to infinity, the flows being restricted by a random capacity. For every
compact convex set , we prove that the maximal flow between
and infinity is such that almost surely converges to the integral
of a deterministic function over the boundary of . The limit can also be
interpreted as the optimum of a deterministic continuous max-flow problem. We
derive some properties of the infinite cluster in supercritical Bernoulli
percolation.Comment: 20 pages, 1 figure published in The Annals of Applied Probability
http://www.imstat.org/aap
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 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
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 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
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