# 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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https://hal.archives-ouvertes.fr/hal-00113319
Contributor : Jean-Baptiste Gouéré <>
Submitted on : Monday, November 13, 2006 - 1:48:18 AM
Last modification on : Monday, May 31, 2021 - 11:14:15 PM
Long-term archiving on: : Tuesday, April 6, 2010 - 10:19:50 PM

### Citation

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

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