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

TL;DR

We analyze a non-isotropic continuous first-passage percolation model built from a Boolean union of $p$-norm balls around a Poisson cloud in $\mathbb{R}^d$, focusing on the small-ball regime with $r=\varepsilon^{1/d}/2$ and unit intensity. By introducing a related rewards model and exploiting the local geometry of the unit ball via the cones $K_\eta(u)$ and their symmetrized variants $M_\eta(u)$, we establish sharp first-order asymptotics for the time constant: for directions $u$ with $\|u\|_p=1$, the deficit $1-\tilde{\mu}_{p,\varepsilon}(u)$ scales like $\varepsilon^{\kappa_p(u)}$, where $\kappa_p(u)$ depends on the count of nonzero coordinates and the norm index $p$. The exponent is explicit: for $p=1$, $\kappa_1(u)=1/d_1(u)$; for $p\in(1,\infty)$, $\kappa_p(u)=1\big/\left(d- (d_1(u)-1)/2 - d_2(u)/p\right)$; and for $p=\infty$, $\kappa_\infty(u)=1/(d_4(u)+1)$. The work also treats $p=2$ (isotropy) with a universal exponent $2/(d+1)$, and provides exact limit behavior in several directions, plus a comparison with the ball-model and a detailed treatment of geodesics. Overall, the results illuminate how anisotropy and unit-ball geometry govern first-order growth in continuous FPP, with implications for related percolation and geodesic properties. The methods combine geometric cone arguments, greedy constructions, and subadditive ergodic techniques, yielding precise, direction-dependent asymptotics that extend lattice-FPP intuition to non-isotropic continuous settings.

Abstract

Consider $Ξ$ 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 $Σ_{p, \varepsilon}$ as the union of the balls of volume $\varepsilon$ for the $p$-norm ($p\in [1,\infty]$) and centered at the points of $Ξ$. We define a random pseudo-metric on $\mathbb{R}^d$ by associating with any path a travel time equal to its $p$-length outside $Σ_{p,\varepsilon}$. 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 $\varepsilon δ_0 + (1-\varepsilon) δ_1$. The exact calculation of the time constant of this model $\tilde μ_{p,\varepsilon} (x)$ is out of reach. We investigate here the behavior of $\varepsilon \mapsto \tilde μ_{p,\varepsilon} (x)$ near $0$, and enlighten how the speed at which $\| x \|_p - \tilde μ_{p,\varepsilon} (x) $ goes to $0$ depends on $x$ and $p$. For instance, for $p\in (1,\infty)$, we prove that $\| x \|_p - \tilde μ_{p,ε} (x)$ is of order $\varepsilon ^{κ_p(x)}$ with $$κ_p(x): = \frac{1}{d- \frac{d_1(x)-1}{2} - \frac{d-d_1 (x)}{p}}\,,$$where $d_1(x)$ is the number of non null coordinates of $x$. The exact order of $\| x \|_p - \tilde μ_{p,ε} (x)$ 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.

Figures (5)

• Figure 1: Illustration of the sets $K_\eta(u)$ and $M_\eta(u)$ (note that $M_\eta(u)$ is here strictly included in $K_\eta(u)$).
• Figure 1: Illustration of the sets $K_\eta(u)$ and $M_\eta(u)$ (note that $M_\eta(u)$ is here strictly included in $K_\eta(u)$).
• Figure 2: Illustration of the greedy algorithm. The path constructed is drawn in blue. At each step, the path goes through the first point of $\Xi$ in the green cone in front of it.
• Figure 2: Illustration of the greedy algorithm. The path constructed is drawn in blue. At each step, the path goes through the first point of $\Xi$ in the green cone in front of it.
• Figure 3: The path $\pi$ (in red) is a generalized path from $x$ to $y$. In blue, a polygonal path $\pi'$ from $x$ to $y$ such that $\tilde{T}_\varepsilon(\pi')\le \tilde{T}_\varepsilon(\pi)$.

Theorems & Definitions (59)

• Theorem 1
• Theorem 2
• Remark 3
• Definition 4
• Lemma 5
• proof
• Theorem 6
• Proposition 7
• Lemma 8
• Lemma 9