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

TL;DR

This paper studies a continuous first passage percolation model, proving that the time constant is positive if and only if the Boolean model's intensity is below a critical threshold, linking percolation properties to travel times.

Contribution

It establishes a precise condition for the positivity of the time constant in a continuous percolation model, connecting it to the percolation phase transition.

Findings

The time constant is positive if and only if the Boolean model's intensity is below a critical value.

Under certain moment conditions, the positivity of the time constant characterizes subcritical percolation.

The results extend classical percolation theory to a dynamic travel-time setting.

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 $T(x,y)$ 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 $\mu\textgreater{}0$ if and only if the intensity $\lambda$ of the Boolean model satisfies $\lambda \textless{} \widehat{\lambda}\_c$, where $ \widehat{\lambda}\_c$ is one of the classical critical parameters defined in continuum percolation.