Coexistence in two-type first-passage percolation models

• 成果类型: Article
• 署名作者: Garet, O; Marchand, R
• 署名单位: Universite de Orleans; Universite de Lorraine
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000503
• 发表日期: 2005
• 页码: 298-330
• 关键词: 1st passage percolation

摘要

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Z(d) or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x, y is an element of Z(d), there is a strictly positive probability that {z is an element of Z(d); d(y, z) < d(x, z)} and {z is an element of Z(d); d(y, z) > d(x, z)} are both infinite sets. We also show that there is a strictly positive probability that the graph of time-minimizing path from the origin in first-passage percolation has at least two topological ends. This generalizes results obtained by Haggstrom and Pemantle for independent exponential times on the square lattice.