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2017 Positivity of the time constant in a continuous model of first passage percolation
Jean-Baptiste Gouéré, Marie Théret
Electron. J. Probab. 22: 1-21 (2017). DOI: 10.1214/17-EJP67

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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Jean-Baptiste Gouéré. Marie Théret. "Positivity of the time constant in a continuous model of first passage percolation." Electron. J. Probab. 22 1 - 21, 2017. https://doi.org/10.1214/17-EJP67

Information

Received: 20 October 2016; Accepted: 7 May 2017; Published: 2017
First available in Project Euclid: 31 May 2017

zbMATH: 1364.60132
MathSciNet: MR3661663
Digital Object Identifier: 10.1214/17-EJP67

Subjects:
Primary: 60K35
Secondary: 82B43

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