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

Abstract: Consider $\Xi$ a homogeneous Poisson point process on $\mathbb{R}^d$ ($d\geq 2$) with unit intensity with respect to the Lebesgue measure. For $\varepsilon\geq 0$, we define the Boolean model $\Sigma_{p, \varepsilon}$ as the union of the balls of volume $\varepsilon$ for the $p$-norm ($p\in [1,\infty]$) and centered at the points of $\Xi$. We define a random pseudo-metric on $\mathbb{R}^d$ by associating with any path a travel time equal to its $p$-length outside $\Sigma_{p,\varepsilon}$. 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 $\varepsilon \delta_0 + (1-\varepsilon) \delta_1$. The exact calculation of the time constant of this model $\tilde \mu_{p,\varepsilon} (x)$ is out of reach. We investigate here the behavior of $\varepsilon \mapsto \tilde \mu_{p,\varepsilon} (x)$ near $0$, and enlighten how the speed at which $| x |p - \tilde \mu{p,\varepsilon} (x) $ goes to $0$ depends on $x$ and $p$. For instance, for $p\in (1,\infty)$, we prove that $| x |p - \tilde \mu{p,\epsilon} (x)$ is of order $\varepsilon ^{\kappa_p(x)}$ with $$\kappa_p(x): = \frac{1}{d- \frac{d_1(x)-1}{2} - \frac{d-d_1 (x)}{p}}\,,$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $| x |p - \tilde \mu{p,\epsilon} (x)$ 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.