## Electronic Journal of Probability

### 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.

#### Article information

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

Dates
Accepted: 7 May 2017
First available in Project Euclid: 31 May 2017

https://projecteuclid.org/euclid.ejp/1496196077

Digital Object Identifier
doi:10.1214/17-EJP67

Mathematical Reviews number (MathSciNet)
MR3661663

Zentralblatt MATH identifier
1364.60132

#### Citation

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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