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Equivalence of some subcritical properties in continuum percolation. (English) Zbl 1428.62425
Summary: We consider the Boolean model on $$\mathbb{R}^d$$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $$C$$ denote the connected component of the origin in the Boolean model. Let $$|C|$$ denotes its volume. Let $$\ell$$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $$\mathbb{E}(|C|)$$ is finite if and only if there exists $$A,B>0$$ such that $$\mathbb{P}(\ell\ge n)\le Ae^{-Bn}$$ for all $$n\ge1$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
Boolean model; continuum percolation; critical point
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##### References:
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