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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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