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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
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