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

2016/10/19 by Jean-Baptiste Gouéré, Gouéré, Jean-Baptiste, Marie Théret +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1610.05901

openalex publication_date 2016/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a non trivial Boolean model \Σ on mathbb R^d for\nd\≥ 2. For every x,y \∈ mathbb R^d we define T(x,y) as the minimum\ntime needed to travel from x to y by a traveler that walks at speed 1\noutside \Σ and at infinite speed inside \Σ. By a standard\napplication of Kingman sub-additive theorem, one easily shows that T(0,x)\nbehaves like \μ \‖x\‖ when \‖x\‖ goes to infinity, where \μ is a\nconstant named the time constant in classical first passage percolation. In\nthis paper we investigate the positivity of \μ. More precisely, under an\nalmost optimal moment assumption on the radii of the balls of the Boolean\nmodel, we prove that \μ textgreater0 if and only if the intensity\n\λ of the Boolean model satisfies \λ textless\n widehat\λ\_c, where widehat\λ\_c is one of the classical\ncritical parameters defined in continuum percolation.\n

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