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Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

Erhard, Dirk; Martinez, Julian; Poisat, Julien (2017), Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster, Journal of Theoretical Probability, 30, 3, p. 784-812. 10.1007/s10959-015-0661-5

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-00903727
Date
2017
Journal name
Journal of Theoretical Probability
Volume
30
Number
3
Publisher
Springer
Pages
784-812
Publication identifier
10.1007/s10959-015-0661-5
Metadata
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Author(s)
Erhard, Dirk
Mathematical institute
Martinez, Julian
Instituto de Investigaciones Matemáticas "Luis A. Santaló" [Buenos Aires] [IMAS]
Poisat, Julien
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We consider a continuum percolation model on $\R^d$, $d\geq 1$.For $t,\lambda\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $\lambda>0$.When $d\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r>0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\geq 2$, there is a non-trivial percolation transitionin $t$, provided $\lambda$ and $r$ are chosen properly.The last statement means that $\lambda$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero(which is infinite when $d\in\{2,3\}$, but finite and dependent on $r$ when $d\geq 4$).We further show that for all $d\geq 2$, the unbounded cluster in the supercritical phase is unique.Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest.The present paper settles the basic properties of the model and should be viewed as a jumpboard for finer results.
Subjects / Keywords
Poisson point process; phase transition; Boolean percolation.; Continuum percolation; Brownian motion

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