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 $d\ge 2$. 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 $\mu \Vert x\Vert$ when $\Vert x\Vert$ 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 $μ\textgreater{}0$ if and only if the intensity $\lambda$ of the Boolean model satisfies $λ\textless{} \widehatλ\_c$, where $\hat{\lambda }\_c$ is one of the classical critical parameters defined in continuum percolation.