# Positivity of the time constant in a continuous model of first passage percolation

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1 LMPT - Laboratoire de Mathématiques et Physique Théorique 2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 $Tx,y$ 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 $T0,x$ 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.

Keywords : Time constant Critical point Boolean model Percolation First-passage percolation

Author: ** Jean-Baptiste Gouéré - Marie Théret - **

Source: https://hal.archives-ouvertes.fr/