Euclidean First-Passage Percolation

• Euclidean first-passage percolation is a family of random metric models that define distances via minimal passage times over stochastic paths, typically using Poisson point processes.
• It bridges classical lattice methods and continuum percolation by employing techniques such as Kingman’s subadditive ergodic theorem and concentration inequalities to establish deterministic limit shapes.
• The framework provides actionable insights for analyzing spatial random metrics with applications in manifold learning and graph-based semi-supervised learning.

Euclidean first-passage percolation (Euclidean FPP) is a family of random metric models on Euclidean space $\mathbb{R}^d$, in which the distance between points is defined by the minimal "passage time" over admissible paths whose geometry and costs are determined by an underlying stochastic structure, typically a Poisson point process or related random geometric graph. These models interpolate between classical lattice FPP and continuum percolation, providing a canonical framework for studying random spatial metrics and associated geometric and probabilistic phenomena.

1. Core Models and Definitions

Several principal instantiations of Euclidean FPP have been rigorously analyzed:

1. Poisson Point Process Power-Weighted Model: Points are distributed according to a homogeneous (or nonhomogeneous) Poisson point process $Q \subset \mathbb{R}^d$ of intensity $\lambda > 0$. For $\alpha > 1$, the passage time along a path $r=(q_0, q_1, ..., q_k) \subset Q$ is defined as $T(r) = \sum_{i=1}^k \|q_i - q_{i-1}\|^\alpha$. The FPP value between $x, y \in \mathbb{R}^d$ is $T(x, y) := T(q(x), q(y))$, where $q(\cdot)$ denotes the nearest Poisson point to a location. The model exhibits rotational invariance and subadditivity (1901.10325).
2. Random Geometric and Proximity Graphs: On a graph such as the Poisson–Delaunay triangulation or random geometric graphs at supercritical connectivity, edge passage times are defined via i.i.d. random marks or deterministic edge lengths, and FPP between two locations becomes the minimal cost among all feasible paths (Coletti et al., 2021, Coupier et al., 2016).
3. Nonhomogeneous Manifold Models: When $M \subset \mathbb{R}^D$ is a $d$-dimensional $C^1$ submanifold with density $f : M \to [m_f, M_f]$, passage times are similarly defined, but the underlying point process is a nonhomogeneous Poisson process (or i.i.d. sample) reflecting $f$ (Groisman et al., 2018).
4. Hop-length (Cutoff) Graphs: For fixed $h > 0$, paths consist of jumps between Poisson points at most distance $h$ apart, and distances are defined as the minimal length sum over such paths connecting two locations (Bungert et al., 2022).

The term "geodesic" is used for length- or time-minimizing paths, which may be finite or semi-infinite.

2. Macroscopic Limit and Shape Theorems

The asymptotic geometry of Euclidean FPP is governed by the emergence of a deterministic metric (the time or Lyapunov norm), underpinned by spatial ergodicity and subadditivity. The generalized shape theorem for stationary, ergodic random pseudometrics $\{\tau(x, y)\}$ on $\mathbb{R}^d$ establishes that there exists a deterministic seminorm $\mu(\cdot)$ such that

$$\lim_{\|x\|_2 \to \infty} \frac{\tau(0, x) - \mu(x)}{\|x\|_2} = 0,$$

almost surely, and the rescaled "metric balls" converge in shape to $\{x : \mu(x) \le 1\}$ (Ziesche, 2016). For isotropic Poisson–based models with i.i.d. power-weighted edge costs (homogeneous), the asymptotic shape is a Euclidean ball. In settings with nonhomogeneous point intensity or density $f$, the limiting distance $d_f$ is given by a weighted path integral (the Fermat distance): $$d_f(p, q) = \inf_{\gamma} \int_0^1 f(\gamma(t))^{-\beta} \|\gamma'(t)\| dt, \quad \beta = \frac{\alpha - 1}{d},$$ characterizing distortion by both the ambient geometry and sampling density (Groisman et al., 2018).

Explicit bounds on the convergence rate have been established in models such as Poisson hyperplane FPP, where exponential error rates can be derived, and in random geometric graphs, where the shape is precisely a Euclidean ball provided connectivity conditions and moment bounds on edge weights hold (Ziesche, 2016, Coletti et al., 2021).

3. Fluctuation Theory and Concentration

The paper of fluctuations in Euclidean FPP centers on both random and non-random components:

• For passage times $T(0, x)$, the decomposition $$T(0, x) = g|x| + \left[T(0, x) - \mathbb{E} T(0, x)\right] + \left[\mathbb{E} T(0, x) - g|x|\right]$$ separates random fluctuations from nonrandom error.
• Under uniform concentration assumptions, if $\sigma(r) \sim r^\chi$ with $0 < \chi < 1$, exponential bounds of the form

$$\mathbb{P}(|T(x, y) - \mathbb{E} T(x, y)| \ge t \sigma(|x - y|)) \leq C_1 e^{-c_1 t}$$

hold for all $t > 0$. In this regime, geodesic wandering (transversal fluctuation) is of order $$\Delta_r = \sqrt{r \sigma(r)} \sim r^{(1 + \chi)/2}$$, reflecting KPZ-type scaling relations (Alexander, 2020).

For the classical Poisson model, the variance of passage time admits a sublinear upper bound $\text{Var}[T(0, n e_1)] \leq C n / \log n$, and concentration on the scale $\sqrt{n / \log n}$ is conjectured to be sharp (1901.10325). Iterative entropy reduction arguments further show that the nonrandom error $\mathbb{E} T_n - n\mu$ can be controlled on the scale of random fluctuations up to polylog factors, lending support to the assertion that random and nonrandom errors are of the same order (Damron et al., 2016).

A major insight is the logarithmic divergence of the nonrandom fluctuation term, which cannot remain bounded—i.e., $\mathbb{E} T(0, x) - g|x| \gtrsim \log\phi(|x|)$—reflecting inherent roughness of the asymptotic geometry (Nakajima, 2020).

4. Geodesic Structure and Semi-Infinite Behavior

Geodesic properties are central to the global geometry and scaling limits:

• Uniqueness and Coalescence: For each fixed asymptotic direction $\hat{u}$ in planar models (including Poisson–Delaunay triangulation and lattice FPP), there exists a unique semi-infinite geodesic from each point, and any two such geodesics with the same direction coalesce almost surely (Coupier et al., 2016). Coalescence underlies the absence of bi-infinite geodesics and sublinearity in the number of distinct infinite routes.
• "Highways and Byways" Phenomenon: In planar first-passage percolation, the probability that a distant site is traversed by an infinite geodesic from the origin decays to zero, and the expected number of infinite geodesics crossing a circle of radius $r$ is $o(r)$ as $r \to \infty$ (Ahlberg et al., 2022). In the Poisson–Delaunay model, a similar result holds for the expected number of semi-infinite branches leaving a disk of radius $r$, confirming that the global geodesic structure is sparse in the scaling limit (Coupier et al., 2016).

5. Extensions to Nonhomogeneous, Random Tessellation, and Graph-based Models

Significant generalizations of Euclidean FPP include:

• Nonhomogeneous Manifold FPP: For i.i.d. samples or Poisson processes on manifolds $M \subset \mathbb{R}^D$ with density $f$, the graph-based passage times converge (after scaling) to the macroscopic Fermat distance. Discrete minimizers (polygonal geodesics on the point cloud) converge uniformly to continuum minimizers, affirming the use of FPP distances in data-driven manifold learning (Groisman et al., 2018).
• Random Tessellations and Poisson Hyperplane Models: Passage times on tessellations (e.g., cell-boundary models with i.i.d. face times) fit into the ergodic pseudometric framework, with explicit limit shapes determined by zonoids for Poisson hyperplane tessellations. Exponential convergence rates and precise analytic expressions for the time constant are available in these cases (Ziesche, 2016).
• Graph-based Learning Applications: The metric induced by Euclidean FPP, especially in k-nearest neighbor graphs at percolation threshold, underpins consistency and convergence rates for solutions of the discrete infinity Laplacian (Lipschitz learning) equation. Ratio convergence results for discrete distances translate into uniform rates for graph-based semi-supervised learning (Bungert et al., 2022).

6. Proof Strategies and Technical Tools

The analytic backbone of Euclidean FPP includes:

• Kingman’s Subadditive Ergodic Theorem: Ensures the existence of deterministic time constants and limit shapes.
• Concentration Inequalities and Entropy Methods: Stretched exponential and polynomial tail bounds for passage times follow from Talagrand-type inequalities, Falik–Samorodnitsky inequalities, and greedy lattice animal arguments (1901.10325, Alexander, 2020).
• Coupling and Geometric Decomposition: Local couplings between homogeneous and nonhomogeneous processes, subdivision into tubes or blocks, and modular path constructions facilitate proofs of convergence and wandering bounds (Groisman et al., 2018, Alexander, 2020).
• Coalescence and Planarity Arguments: Adaptations of the Burton–Keane method, together with planarity and absence of cycles or bi-infinite geodesics, are central in establishing the asymptotic sparsity of geodesic forests (Coupier et al., 2016, Ahlberg et al., 2022).
• Near-Subadditivity and Ratio Lemmas: Quantitative limiting theorems for hop-length models and distance functions rely critically on de Bruijn–Erdős lemmas for nearly subadditive sequences and Pólya-type ratio estimates (Bungert et al., 2022).

7. Open Problems and Ongoing Directions

Several fundamental questions remain:

• Determination of the true variance and transverse fluctuation exponents for passage times and geodesics in Euclidean and lattice FPP.
• Characterization of the sharpness and universality of concentration phenomena in the continuum, especially with inhomogeneous intensities, non-Euclidean norms, or random environment features (Alexander, 2020, 1901.10325).
• Rigorous establishment of KPZ scaling relations for fluctuation and wandering, and their implications for geodesic geometry in higher dimensions.
• Extension of explicit limit shape results and convergence rates to broader classes of random graphs and stochastic spatial models.
• Deeper connections to high-dimensional statistical and learning frameworks, especially the use of FPP-type distances as consistent estimators of intrinsic manifold geometry (Groisman et al., 2018, Bungert et al., 2022).

Euclidean FPP thus constitutes a unifying and highly active domain, interlinking spatial stochastic geometry, scaling limits, random metric theory, and applied probabilistic methodologies.