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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.
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https://hal.archives-ouvertes.fr/hal-01383426
Contributor : Jean-Baptiste Gouéré <>
Submitted on : Monday, February 27, 2017 - 2:59:55 PM
Last modification on : Friday, March 27, 2020 - 3:54:26 AM

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

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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. ⟨hal-01383426v2⟩

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