Equivalence of some subcritical properties in continuum percolation

Abstract

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 $n\ge1$.