## Electronic Communications in Probability

### 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 ([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
Accepted: 13 January 2017
First available in Project Euclid: 14 February 2017

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

Digital Object Identifier
doi:10.1214/17-ECP44

Mathematical Reviews number (MathSciNet)
MR3615665

Zentralblatt MATH identifier
1358.82017

#### 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

#### References

• [1] D. Ahlberg. Asymptotics of first-passage percolation on $1$-dimensional graphs. Advances in Applied Probability, 47 (2015), 182–209.
• [2] D. Ahlberg and C. Hoffman, Random coalescing geodesics in first-passage percolation. arXiv:1609.02447 (2016).
• [3] O. Angel and G. Ray. Classification of Half Planar Maps. Ann. Probab. 433 (2015), 1315–1349.
• [4] A. Auffinger, M. Damron and J. Hanson, 50 years of first passage percolation. arXiv:1511.03262
• [5] W. Ballmann, Lectures on spaces of nonpositive curvature. DMV Seminar 25, Basel: Birkhauser Verlag, pp. viii+112 (1995).
• [6] I. Benjamini, G. Kalai, O. Schramm. First passage percolation has sublinear distance variance. Ann. Probab. 31 (2003) 1970–1978.
• [7] I. Benjamini, E. Paquette and J. Pfeffer. Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation. arXiv:1409.4312
• [8] J. T. Cox and R. Durrett. Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 (1981) 583–603.
• [9] N. Curien. Planar stochastic hyperbolic infinite triangulations. Probab. Theory Relat. Fields 1653 (2016), 509–540.
• [10] N. Curien and J-F. Le Gall, First-passage percolation and local modifications of distances in random triangulations. arXiv:1511.04264
• [11] C. Drutu, S. Mozes, M. Sapir. Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362 (2010), 2451–2505.
• [12] G. Grimmett and H. Kesten. Percolation since Saint-Flour. Percolation theory at Saint-Flour, Probab. St.-Flour, Springer, Heidelberg, (2012).
• [13] M. Gromov, Hyperbolic groups, “Essays in group theory”, S. Gersten (ed), MSRI Publ. 8, Springer (1987), 75–265.
• [14] H. Kesten. Aspects of first passage percolation. École d’Été de probabilité de Saint-Flour XIV - 1984, Lecture Notes in Math., 1180, Springer, Berlin, (1986) 125–264.
• [15] C. Licea and C. Newman. Geodesics in two-dimensional first-passage percolation. Ann. Probab. 24 (1996) 399–410.
• [16] J. Wehr and J. Woo. Absence of geodesics in first-passage percolation on a half-plane. Ann. Probab. 26 (1998) 358–367.