TY - JOUR

T1 - Limiting Geodesics For First-Passage Percolation On Subsets Of Z2

AU - Auffinger, Antonio

AU - Damron, Michael

AU - Hanson, Jack

N1 - Funding Information:
The authors thank the Courant Institute for hospitality. Antonio Auffinger thanks Princeton University for accommodation and support during visits. Michael Damron thanks C. Newman for summer funding and Jack Hanson M. Aizenman for funding and support.

PY - 2015/2/1

Y1 - 2015/2/1

N2 - 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 Z2: Those whose vertex set is infinite and connected with an infinite connected complement. This includes, for instance, slit planes, half-planes and sectors. Writing xn for the sequence of boundary vertices, we show that the sequence of geodesics from any point to xn 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.

AB - 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 Z2: Those whose vertex set is infinite and connected with an infinite connected complement. This includes, for instance, slit planes, half-planes and sectors. Writing xn for the sequence of boundary vertices, we show that the sequence of geodesics from any point to xn 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.

KW - Busemann function

KW - First-passage percolation

KW - Geodesics

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U2 - 10.1214/13-AAP999

DO - 10.1214/13-AAP999

M3 - Article

AN - SCOPUS:84983297507

VL - 25

SP - 373

EP - 405

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 1

ER -