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

Let N be a norm on ℝ^d with d≥ 2 and consider χ a homogeneous Poisson point process on ℝ^d with intensity ε∈ [0,∞). We define the Boolean model Σ_N, ε as the union of the balls of diameter 1 for the norm N and centered at the points of χ. For every x,y ∈ℝ^d, Let T_N, ε (x,y) be the minimum time needed to travel from x to y if one travels at speed 1 outside Σ_N,ε and at infinite speed inside Σ_N,ε: this defines a continuous model of first-passage percolation, that has been studied in for N=·_2, the Euclidean norm. The exact calculation of the time constant of this model μ_N,ε (x):=lim_n→∞T_N,ε (0,n x) / n is out of reach. We investigate here the behavior of ε↦μ_N,ε (x) near 0, and enlight how the speed at which N(x) - μ_N,ε (x) goes to 0 depends on x and N. For instance, if N is the p-norm for p∈ (1,∞), we prove that N(x) - μ_·_p,ϵ (x) is of order ε ^κ_p(x) with κ_p(x): = 1/d- d_1(x)-1/2 - d-d_1 (x)/p , where d_1(x) is the number of non null coordinates of x.Together with the study of the time constant, we also prove a control on the N-length of the geodesics, and get some informations on the number of points of χ really useful to those geodesics. The results are in fact more natural to prove in a slightly different setting, where instead of centering a ball of diameter 1 at each point of χ and traveling at infinite speed inside Σ_Nε, we put a reward of one unit of time at those points.