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Busemann Functions and Infinite Geodesics in Two-Dimensional First-Passage Percolation

Abstract

We study first-passage percolation on \({\mathbb{Z}^2}\), where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of infinite geodesics. Some of these were studied in the late 1990s by C. Newman and collaborators under strong assumptions on the limiting shape and weight distribution. In this paper we develop a framework for working with distributional limits of Busemann functions and use it to prove forms of Newman’s results under minimal assumptions. For instance, we show a form of coalescence of long finite geodesics in any deterministic direction. We also introduce a purely directional condition which replaces Newman’s global curvature condition and whose assumption we show implies the existence of directional geodesics. Without this condition, we prove existence of infinite geodesics which are directed in sectors. Last, we analyze distributional limits of geodesic graphs, proving almost-sure coalescence and nonexistence of infinite backward paths. This result relates to the conjecture of nonexistence of “bigeodesics.”

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Correspondence to Jack Hanson.

Additional information

M. D. is supported by an NSF postdoctoral fellowship and NSF Grants DMS-0901534 and DMS-1007626.

J. H. is supported by an NSF graduate fellowship and NSF Grant PHY-1104596.

Communicated by F. Toninelli

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Damron, M., Hanson, J. Busemann Functions and Infinite Geodesics in Two-Dimensional First-Passage Percolation. Commun. Math. Phys. 325, 917–963 (2014). https://doi.org/10.1007/s00220-013-1875-y

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Keywords

  • Passage Time
  • Ergodic Theorem
  • Translation Invariance
  • Supporting Line
  • Lattice Path