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Positivity of the time constant in a continuous model of first passage percolation. (English) Zbl 1364.60132
Summary: 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 $$\lambda < \hat{\lambda}_c$$, where $$\hat{\lambda}_c$$ is one of the classical critical parameters defined in continuum percolation.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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