Electronic Journal of Probability

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

Jean-Baptiste Gouéré and Marie Théret

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 49, 21 pp.

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

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Boolean model continuum percolation first passage percolation critical point time constant

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Gouéré, Jean-Baptiste; Théret, Marie. Positivity of the time constant in a continuous model of first passage percolation. Electron. J. Probab. 22 (2017), paper no. 49, 21 pp. doi:10.1214/17-EJP67. https://projecteuclid.org/euclid.ejp/1496196077

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