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.