# Existence of subcritical regimes in the Poisson Boolean model of continuum percolation

Abstract : We consider the so-called Poisson Boolean model of continuum percolation. At each point of an homogeneous Poisson point process on the Euclidean space $\R^d$, we center a ball with random radius. We assume that the radii of the balls are independent, identically distributed and independent of the point process. We denote by $\Sigma$ the union of the balls and by $S$ the connected component of $\Sigma$ that contains the origin. We show that $S$ is almost surely bounded for small enough density $\lambda$ of the point process if and only if the mean volume of the balls is finite. Let us denote by $D$ the diameter of $S$ and by $R$ one of the random radii. We also show that, for all positive real number $s$, $D^s$ is integrable for small enough $\lambda$ if and only if $R^{d+s}$ is integrable.
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Annals of Probability, 2008, pp.36(4):1209-1220
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https://hal.archives-ouvertes.fr/hal-00113319
Contributeur : Jean-Baptiste Gouéré <>
Soumis le : lundi 13 novembre 2006 - 01:48:18
Dernière modification le : jeudi 3 mai 2018 - 15:32:06
Document(s) archivé(s) le : mardi 6 avril 2010 - 22:19:50

### Citation

Jean-Baptiste Gouéré. Existence of subcritical regimes in the Poisson Boolean model of continuum percolation. Annals of Probability, 2008, pp.36(4):1209-1220. 〈hal-00113319〉

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