## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 22 (2017), paper no. 14, 8 pp.

### First passage percolation on a hyperbolic graph admits bi-infinite geodesics

Itai Benjamini and Romain Tessera

#### 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 $X$.

#### Article information

**Source**

Electron. Commun. Probab., Volume 22 (2017), paper no. 14, 8 pp.

**Dates**

Received: 8 June 2016

Accepted: 13 January 2017

First available in Project Euclid: 14 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1487062816

**Digital Object Identifier**

doi:10.1214/17-ECP44

**Mathematical Reviews number (MathSciNet)**

MR3615665

**Zentralblatt MATH identifier**

1358.82017

**Subjects**

Primary: 82B43: Percolation [See also 60K35] 51F99: None of the above, but in this section 97K50: Probability theory

**Keywords**

first passage percolation two-sided geodesics hyperbolic graph Morse geodesics

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Benjamini, Itai; Tessera, Romain. First passage percolation on a hyperbolic graph admits bi-infinite geodesics. Electron. Commun. Probab. 22 (2017), paper no. 14, 8 pp. doi:10.1214/17-ECP44. https://projecteuclid.org/euclid.ecp/1487062816