Open Access
November 2019 Equivalence of some subcritical properties in continuum percolation
Jean-Baptiste Gouéré, Marie Théret
Bernoulli 25(4B): 3714-3733 (November 2019). DOI: 10.3150/19-BEJ1108


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 $n\ge1$.


Download Citation

Jean-Baptiste Gouéré. Marie Théret. "Equivalence of some subcritical properties in continuum percolation." Bernoulli 25 (4B) 3714 - 3733, November 2019.


Received: 1 March 2018; Revised: 1 December 2018; Published: November 2019
First available in Project Euclid: 25 September 2019

zbMATH: 07110153
MathSciNet: MR4010970
Digital Object Identifier: 10.3150/19-BEJ1108

Keywords: Boolean model , continuum percolation , critical point

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4B • November 2019
Back to Top