First Passage percolation on a hyperbolic graph admits bi-infinite geodesics
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 is whether there exists a two-sided infinite geodesic in first passage percolation on Z^2, and more generally on Z^n for n>1. 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 exponential moment for 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.
Cite
@article{arxiv.1606.02449,
title = {First Passage percolation on a hyperbolic graph admits bi-infinite geodesics},
author = {Itai Benjamini and Romain Tessera},
journal= {arXiv preprint arXiv:1606.02449},
year = {2025}
}
Comments
We corrected an error in the earlier (published) version: the main theorem now requires an exponential moment condition on the edge distribution (instead of a finite first moment). The proof is mostly unchanged though