First Passage percolation on a hyperbolic graph admits bi-infinite geodesics
Itai Benjamini, Romain Tessera
TL;DR
This work shows that first passage percolation on graphs with negative curvature, notably hyperbolic graphs that contain a Morse geodesic, almost surely yields a bi-infinite geodesic under mild edge-length assumptions ($\mathbb{E} e^{c\omega(e)}<\infty$ and $\nu(\{0\})=0$). The authors combine a quantitative Morse property for bi-infinite quasi-geodesics with concentration bounds (Talagrand) to show that, for endpoints on opposite sides of a fixed Morse geodesic, the $\omega$-geodesics stay uniformly close to the Morse geodesic, forcing the existence of a global bi-infinite $\omega$-geodesic. The result provides a positive answer to Furstenberg-type questions in a broad geometric setting and has implications for the geometry of random metrics on hyperbolic graphs and related groups. It also outlines directions for extensions to other random models and potential variance phenomena along Morse 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.