First-order behavior of the time constant in non-isotropic continuous first-passage percolation

Abstract

Consider $\hat{I}ž$ a homogeneous Poisson point process on ${\mathbb{R}}^d$ ($d\ge 2$) with unit intensity with respect to the Lebesgue measure. For $\epsilon \ge 0$, we define the Boolean model $\hat{I}{\pounds }_{p,\epsilon }$ as the union of the balls of volume $\epsilon$ for the $p$-norm ($p\in [1,\infty ]$) and centered at the points of $\hat{I}ž$. We define a random pseudo-metric on ${\mathbb{R}}^d$ by associating with any path a travel time equal to its $p$-length outside $\hat{I}{\pounds }_{p,\epsilon }$. This defines a continuous model of first-passage percolation, that has been studied in \cite{GT17,GT22} for $p=2$, the Euclidean norm. For $p=1$, this model is expected to share common properties with the classical first-passage percolation on the graph ${\mathbb{Z}}^d$ with a distribution of passage times of the form $\epsilon \hat{I}{´}_0+(1-\epsilon )\hat{I}{´}_1$. The exact calculation of the time constant of this model $\tilde{\hat{I}}{¼}_{p,\epsilon }(x)$ is out of reach. We investigate here the behavior of $\epsilon \mapsto \tilde{\hat{I}}{¼}_{p,\epsilon }(x)$ near $0$, and enlighten how the speed at which $\Vert x{\Vert }_p-\tilde{\hat{I}}{¼}_{p,\epsilon }(x)$ goes to $0$ depends on $x$ and $p$. For instance, for $p\in (1,\infty )$, we prove that $\Vert x{\Vert }_p-\tilde{\hat{I}}{¼}_{p,\hat{I}\mu }(x)$ is of order ${\epsilon }^{\hat{I}{º}_p(x)}$ with $$\hat{I}{º}_p(x):=\frac{1}{d-\frac{d_1(x)-1}{2}-\frac{d-d_1(x)}{p}},$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $\Vert x{\Vert }_p-\tilde{\hat{I}}{¼}_{p,\hat{I}\mu }(x)$ is also given for $p=1$ and $p=\infty$. Related results are also discussed, about properties of the geodesics, and analog properties on closely related models.