Equivalence of some subcritical properties in continuum percolation

Jean-Baptiste Gouéré (1) , Marie Théret (2)

1 IDP - Institut Denis Poisson
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We consider the Boolean model on $\R^d$. 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 $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $\P(\ell \ge n) \le Ae^{-Bn}$ for all $n \ge 1$.

Mots clés: Boolean model; Continuum percolation; Critical point

Domaines: Probabilités [math.PR]

Jean-Baptiste Gouéré, Marie Théret. Equivalence of some subcritical properties in continuum percolation. 2018. ⟨hal-01721189⟩