Preprints, Working Papers, ... Year : 2018

Equivalence of some subcritical properties in continuum percolation

Jean-Baptiste Gouéré

Abstract

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 $n \ge 1$.
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Is original form of hal-03158712 Journal article Jean-Baptiste Gouéré, Marie Théret. Equivalence of some subcritical properties in continuum percolation. Bernoulli, 2019, 25 (4B), pp.3714-3733. ⟨10.3150/19-BEJ1108⟩. ⟨hal-03158712⟩

Dates and versions

hal-01721189 , version 1 (01-03-2018)

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