RATIO CONVERGENCE RATES FOR EUCLIDEAN FIRST-PASSAGE PERCOLATION: APPLICATIONS TO THE GRAPH INFINITY LAPLACIAN

• 成果类型: Article
• 署名作者: Bungert, Leon; Calder, Jeff; Roith, Tim
• 署名单位: University of Wurzburg; University of Minnesota System; University of Minnesota Twin Cities; Helmholtz Association; Deutsches Elektronen-Synchrotron (DESY)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2052
• 发表日期: 2024
• 页码: 3870-3910
• 关键词: hamilton-jacobi equations; random geometric graphs; p-laplacian; quantitative homogenization; Stochastic Homogenization; time constant; shortest-path; distance; limit; REGULARITY

摘要

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 semisupervised 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 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(hs)(0, se(1))/s -> sigma as s -> infinity almost surely where sigma >= 1 is a dimensional constant and h(s) greater than or similar to log(s)(1/d). A convergence rate is not available due to a lack of approximate superadditivity when h(s )-> infinity. Instead, we prove convergence rates for the ratio d(h)(0,se(1))/d(h)(0,2se(1)) -> 1/2 when h is frozen and does not depend on s. Combining this with the techniques that we developed in (IMA J. Numer. Anal. 43 (2023) 2445-2495), 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.