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

Consider Ξ a homogeneous Poisson point process on Rᵈ (d≥ 2) with unit intensity with respect to the Lebesgue measure. For ε≥ 0, we define the Boolean model Σp, ε as the union of the balls of volume ε for the p-norm (p∈ [1,∞]) and centered at the points of Ξ. We define a random pseudo-metric on Rᵈ by associating with any path a travel time equal to its p-length outside Σp,ε. This defines a continuous model of first-passage percolation, that has been studied in {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 Zᵈ with a distribution of passage times of the form ε δ₀ + (1-ε) δ₁. The exact calculation of the time constant of this model μp,ε (x) is out of reach. We investigate here the behavior of ε ↦ μp,ε (x) near $0$, and enlighten how the speed at which \| x \|ₚ - μp,ε (x) goes to $0$ depends on x and p. For instance, for p∈ (1,∞), we prove that \| x \|ₚ - μp,ε (x) is of order ε κₚ(x) with κₚ(x): = 1d- d₁(x)-1/2 - d-d₁ (x)/p\,,where d₁(x) is the number of non null coordinates of x. The exact order of \| x \|ₚ - μp,ε (x) is also given for $p=1$ and p=∞. Related results are also discussed, about properties of the geodesics, and analog properties on closely related models.