an:07110153
Zbl 1428.62425
Gou??r??, Jean-Baptiste; Th??ret, Marie
Equivalence of some subcritical properties in continuum percolation
EN
Bernoulli 25, No. 4B, 3714-3733 (2019).
00438827
2019
j
60K35
Boolean model; continuum percolation; critical point
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 \(n\ge1\).