## The Annals of Applied Probability

### Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^{2}$

#### Abstract

It is an open problem to show that in two-dimensional first-passage percolation, the sequence of finite geodesics from any point to $(n,0)$ has a limit in $n$. In this paper, we consider this question for first-passage percolation on a wide class of subgraphs of $\mathbb{Z}^{2}$: those whose vertex set is infinite and connected with an infinite connected complement. This includes, for instance, slit planes, half-planes and sectors. Writing $x_{n}$ for the sequence of boundary vertices, we show that the sequence of geodesics from any point to $x_{n}$ has an almost sure limit assuming only existence of finite geodesics. For all passage-time configurations, we show existence of a limiting Busemann function. Specializing to the case of the half-plane, we prove that the limiting geodesic graph has one topological end; that is, all its infinite geodesics coalesce, and there are no backward infinite paths. To do this, we prove in the Appendix existence of geodesics for all product measures in our domains and remove the moment assumption of the Wehr–Woo theorem on absence of bigeodesics in the half-plane.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 1 (2015), 373-405.

Dates
First available in Project Euclid: 16 December 2014

https://projecteuclid.org/euclid.aoap/1418740189

Digital Object Identifier
doi:10.1214/13-AAP999

Mathematical Reviews number (MathSciNet)
MR3297776

Zentralblatt MATH identifier
1308.60106

#### Citation

Auffinger, Antonio; Damron, Michael; Hanson, Jack. Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^{2}$. Ann. Appl. Probab. 25 (2015), no. 1, 373--405. doi:10.1214/13-AAP999. https://projecteuclid.org/euclid.aoap/1418740189

#### References

• [1] Alm, S. E. and Wierman, J. C. (1999). Inequalities for means of restricted first-passage times in percolation theory. Combin. Probab. Comput. 8 307–315.
• [2] Auffinger, A. and Damron, M. (2013). Differentiability at the edge of the percolation cone and related results in first-passage percolation. Probab. Theory Related Fields 156 193–227.
• [3] Boivin, D. (1990). First passage percolation: The stationary case. Probab. Theory Related Fields 86 491–499.
• [4] Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge Univ. Press, New York.
• [5] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583–603.
• [6] Damron, M. and Hanson, J. (2014). Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Comm. Math. Phys. 325 917–963.
• [7] Damron, M. and Hochman, M. (2013). Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models. Ann. Appl. Probab. 23 1074–1085.
• [8] Forgacs, G., Lipowsky, R. and Nieuwenhuizen, T. M. (1991). The behavior of interfaces in ordered and disordered systems. In Phase Transitions and Critical Phenomena (C. Domb and J. Lebowitz, eds.) 14 135–363. Academic, London.
• [9] Garet, O. and Marchand, R. (2005). Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15 298–330.
• [10] Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 683–692.
• [11] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif. 61–110. Springer, New York.
• [12] Hoffman, C. (2005). Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 739–747.
• [13] Hoffman, C. (2008). Geodesics in first passage percolation. Ann. Appl. Probab. 18 1944–1969.
• [14] Huse, D. A. and Henley, C. L. (1985). Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett. 54 2708–2711.
• [15] Licea, C. and Newman, C. M. (1996). Geodesics in two-dimensional first-passage percolation. Ann. Probab. 24 399–410.
• [16] Newman, C. M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) 1017–1023. Birkhäuser, Basel.
• [17] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977–1005.
• [18] Richardson, D. (1973). Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 515–528.
• [19] Wehr, J. (1997). On the number of infinite geodesics and ground states in disordered systems. J. Stat. Phys. 87 439–447.
• [20] Wehr, J. and Woo, J. (1998). Absence of geodesics in first-passage percolation on a half-plane. Ann. Probab. 26 358–367.
• [21] Wierman, J. C. and Reh, W. (1978). On conjectures in first passage percolation theory. Ann. Probab. 6 388–397.
• [22] Zhang, Y. (1999). Double behavior of critical first-passage percolation. In Perplexing Problems in Probability (M. Bramson and R. Durrett, eds.) 143–158. Birkhäuser, Boston, MA.