# Equivalence of some subcritical properties in continuum percolation

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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Cited literature [17 references]

https://hal.archives-ouvertes.fr/hal-01721189
Contributor : Jean-Baptiste Gouéré <>
Submitted on : Thursday, March 1, 2018 - 5:47:54 PM
Last modification on : Saturday, April 3, 2021 - 3:29:39 AM
Long-term archiving on: : Wednesday, May 30, 2018 - 3:34:47 PM

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### Identifiers

• HAL Id : hal-01721189, version 1
• ARXIV : 1803.00793

### Citation

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

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