# Positivity of the time constant in a continuous model of first passage percolation

Abstract : We consider a non trivial Boolean model $\Sigma$ on ${\mathbb R}^d$ for $d\geq 2$. For every $x,y \in {\mathbb R}^d$ we define $T(x,y)$ as the minimum time needed to travel from $x$ to $y$ by a traveler that walks at speed $1$ outside $\Sigma$ and at infinite speed inside $\Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that $T(0,x)$ behaves like $\mu \|x\|$ when $\|x\|$ goes to infinity, where $\mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the positivity of $\mu$. More precisely, under an almost optimal moment assumption on the radii of the balls of the Boolean model, we prove that $\mu>0$ if and only if the intensity $\lambda$ of the Boolean model satisfies $\lambda < \widehat{\lambda}_c$, where $\widehat{\lambda}_c$ is one of the classical critical parameters defined in continuum percolation.
Keywords :
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Article dans une revue
Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2017, 〈https://projecteuclid.org/euclid.ejp/1496196077〉
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https://hal.archives-ouvertes.fr/hal-01383426
Contributeur : Jean-Baptiste Gouéré <>
Soumis le : lundi 27 février 2017 - 14:59:55
Dernière modification le : vendredi 4 janvier 2019 - 17:32:34

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• HAL Id : hal-01383426, version 2
• ARXIV : 1610.05901

### Citation

Jean-Baptiste Gouéré, Marie Théret. Positivity of the time constant in a continuous model of first passage percolation. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2017, 〈https://projecteuclid.org/euclid.ejp/1496196077〉. 〈hal-01383426v2〉

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