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

TL;DR

This paper studies how the time constant in a continuous first passage percolation model varies with the underlying measure, focusing on its regularity and continuity properties.

Contribution

It investigates the continuity of the time constant as a function of the measure in a continuous percolation model, extending classical results to a more general setting.

Findings

Established conditions for the continuity of the time constant

Analyzed the dependence of the time constant on the measure ν

Provided insights into the regularity of the time constant function

Abstract

For a given dimension d $\ge$ 2 and a finite measure $\nu$ on (0, +$\infty$), we consider $\xi$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $\nu$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $\Sigma$ = $\cup$ (c,r)$\in$$\xi$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ 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 regularity of $\mu$ as a function of the measure $\nu$ associated with the underlying Boolean model.