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February 2015 Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^{2}$
Antonio Auffinger, Michael Damron, Jack Hanson
Ann. Appl. Probab. 25(1): 373-405 (February 2015). DOI: 10.1214/13-AAP999

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.

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Antonio Auffinger. Michael Damron. Jack Hanson. "Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^{2}$." Ann. Appl. Probab. 25 (1) 373 - 405, February 2015. https://doi.org/10.1214/13-AAP999

Information

Published: February 2015
First available in Project Euclid: 16 December 2014

zbMATH: 1308.60106
MathSciNet: MR3297776
Digital Object Identifier: 10.1214/13-AAP999

Subjects:
Primary: 60K35
Secondary: 82B43

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 1 • February 2015
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