SHORTEST PATH THROUGH RANDOM POINTS

• 成果类型: Article
• 署名作者: Hwang, Sung Jin; Damelin, Steven B.; Hero, Alfred O., III
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1162
• 发表日期: 2016
• 页码: 2791-2823
• 关键词: first-passage percolation; 1st-passage percolation; spanning-trees

摘要

Let (M, g(1)) be a complete d-dimensional Riemannian manifold for d > 1. Let X-n be a set of n sample points in M drawn randomly from a smooth Lebesgue density f supported in M. Let x, y be two points in M. We prove that the normalized length of the power-weighted shortest path between x, y through X-n converges almost surely to a constant multiple of the Riemannian distance between x, y under the metric tensor g(p) = f(2)(1-P)/d g(1), where p > 1 is the power parameter.