Papers › Ratio convergence rates for Euclidean first-passage percolation: Applications to the...
Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian
Leon Bungert, Jeff Calder, Tim Roith
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
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 dₕ(x,y) on a homogeneous Poisson point process on ℝᵈ, where admissible paths have step size at most h>0. Using a suitable regularization of the distance function and subadditivity we prove that d_(hₛ)(0,se₁)/ s →σ as s→∞ almost surely where σ≥1 is a dimensional constant and hₛ≳log(s)¹d. A convergence rate is not available due to a lack of approximate superadditivity when hₛ→∞. Instead, we prove convergence rates for the ratio (dₕ(0,se₁))/(dₕ(0,2se₁))→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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections