Equivalence of some subcritical properties in continuum percolation
Résumé
We consider the Boolean model on Rd. 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 ℓ 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(ℓ ≥ n) ≤ Ae−Bn for all n ≥ 1.
Domaines
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relationship_isVariantFormatOf hal-01721189 Preprint Jean-Baptiste Gouéré, Marie Théret. Equivalence of some subcritical properties in continuum percolation. 2018. ⟨hal-01721189⟩