Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian

TL;DR

This work connects Euclidean first-passage percolation on Poisson clouds to graph-based PDEs, deriving the first quantitative convergence rates for the graph infinity Laplacian at the connectivity (percolation) length scale. By introducing an enriched Poisson process and exploiting near-subadditivity and concentration of measure, the authors prove almost-sure convergence and tail bounds for regularized graph distances, along with explicit ratio convergence to 1/2. These percolation-based results feed into a homogenization-style framework that yields uniform convergence rates for solutions to the graph infinity Laplace equation (Lipschitz learning) on random geometric graphs with ε_n at connectivity threshold, including a concrete rate at ε_n ∼ (log n / n)^{1/d}. The findings bridge stochastic geometry and nonlinear PDEs on graphs, with direct implications for graph-based semi-supervised learning and the design of percolation-aware numerical schemes. Overall, the paper provides the first fully quantitative (probabilistic) guarantees linking percolation geometry to continuum limits of nonlinear graph PDEs and demonstrates their utility in Lipschitz learning.

Abstract

In this paper we prove the first quantitative convergence rates for the graph infinity Laplace equation for length scales at the connectivity threshold. In the graph-based semi-supervised learning community this equation is also known as Lipschitz learning. The graph infinity Laplace equation is characterized by the metric on the underlying space, and convergence rates follow from convergence rates for graph distances. At the connectivity threshold, this problem is related to Euclidean first passage percolation, which is concerned with the Euclidean distance function $d_{h}(x,y)$ on a homogeneous Poisson point process on $\mathbb{R}^d$, where admissible paths have step size at most $h>0$. Using a suitable regularization of the distance function and subadditivity we prove that ${d_{h_s}(0,se_1)}/ s \to σ$ as $s\to\infty$ almost surely where $σ\geq 1$ is a dimensional constant and $h_s\gtrsim \log(s)^\frac{1}{d}$. A convergence rate is not available due to a lack of approximate superadditivity when $h_s\to \infty$. Instead, we prove convergence rates for the ratio $\frac{d_{h}(0,se_1)}{d_{h}(0,2se_1)}\to \frac{1}{2}$ when $h$ is frozen and does not depend on $s$. Combining this with the techniques that we developed in (Bungert, Calder, Roith, IMA Journal of Numerical Analysis, 2022), we show that this notion of ratio convergence is sufficient to establish uniform convergence rates for solutions of the graph infinity Laplace equation at percolation length scales.

Figures (6)

• Figure 1: The covering of the Poisson process $X$ on the left yields empty boxes in red. We define the enriched process $\mathcal{X}_s$ on the right by adding the points in blue.
• Figure 2: Boxes covering the line segment between $0$ and $se_1$.
• Figure 3: Construction in the proof of \ref{['prop:superadditivity']}. The blue points on the spheres constitute the deterministic coverings $x_i$ and $x_i^\prime$.
• Figure 4: Visualizations of optimal paths for $d=2$.
• Figure 6: The averaged distance from $0$ to $s$ as defined in \ref{['eq:sample_ratio']}. Each curve corresponds to a different choice of the factor $a>0$ in front of the log scaling \ref{['eq:logscaling']}. The left plots have standard axes, the right ones are log-log plots of the error to the last iterate.

Theorems & Definitions (75)

• Theorem 2.1: Euclidean first-passage percolation
• proof
• Remark 2.2
• Theorem 2.3: Convergence rates
• Corollary 2.4
• proof
• Remark 2.5
• Corollary 2.6
• proof
• Corollary 2.7