First Passage percolation on a hyperbolic graph admits bi-infinite geodesics

TL;DR

This paper proves that in hyperbolic graphs with certain properties, first passage percolation almost surely admits bi-infinite geodesics, providing a positive answer to a long-standing open question in geometric probability.

Contribution

It establishes the existence of bi-infinite geodesics in first passage percolation on hyperbolic graphs with Morse geodesics, under mild conditions.

Findings

Bi-infinite geodesics exist in hyperbolic graphs with Morse geodesics.

The result applies to graphs with bounded degree and positive exponential moment lengths.

Addresses a long-standing open question in geometric probability and percolation theory.

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.