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2017 First passage percolation on a hyperbolic graph admits bi-infinite geodesics
Itai Benjamini, Romain Tessera
Electron. Commun. Probab. 22(none): 1-8 (2017). DOI: 10.1214/17-ECP44

Abstract

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 $X$.

Citation

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Itai Benjamini. Romain Tessera. "First passage percolation on a hyperbolic graph admits bi-infinite geodesics." Electron. Commun. Probab. 22 1 - 8, 2017. https://doi.org/10.1214/17-ECP44

Information

Received: 8 June 2016; Accepted: 13 January 2017; Published: 2017
First available in Project Euclid: 14 February 2017

zbMATH: 1358.82017
MathSciNet: MR3615665
Digital Object Identifier: 10.1214/17-ECP44

Subjects:
Primary: 51F99, 82B43, 97K50

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