Limiting Geodesics For First-Passage Percolation On Subsets Of Z2

Antonio Auffinger, Michael Damron, Jack Hanson

Research output: Contribution to journalArticle

4 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)373-405
Number of pages33
JournalAnnals of Applied Probability
Volume25
Issue number1
DOIs
StatePublished - Feb 1 2015

Keywords

  • Busemann function
  • First-passage percolation
  • Geodesics

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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