## 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

Original language | English |
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Article number | 14 |

Number of pages | 8 |

Journal | Electronic Communications in Probability |

Volume | 22 |

DOIs | |

State | Published - 2017 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty