文章基本信息

标题：First passage percolation on a hyperbolic graph admits bi-infinite geodesics
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作者：Itai Benjamini ; Romain Tessera
期刊名称：Electronic Communications in Probability
印刷版ISSN：1083-589X
出版年度：2017
卷号：22
DOI：10.1214/17-ECP44
语种：English
出版社：Electronic Communications in Probability
摘要：Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg ([14]) is whether there exists a bi-infinite geodesic in first passage percolation on the euclidean lattice of dimension at least 2. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite expectation of the random lengths, we prove that if a graph $X$ has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on $X$.
关键词：fifirst passage percolation;two-sided geodesics;hyperbolic graph;Morse geodesics.