Bigeodesics in first-passage percolation

Abstract

In first-passage percolation, we place i.i.d. continuous weights at the edges of Z^2 and consider the weighted graph metric. A distance-minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics. In the '90s, Licea-Newman showed that, under a curvature assumption on the "asymptotic shape", all infinite geodesics have an asymptotic direction, and there is a full-measure set D of [0, 2 pi) such that for any theta in D, there are no bigeodesics with one end directed in direction theta. In this paper, we show that there are no bigeodesics with one end directed in any deterministic direction, assuming the shape boundary is differentiable. This rules out existence of ground state pairs for the related disordered ferromagnet whose interface has a deterministic direction. Furthermore, it resolves the Benjamini-Kalai-Schramm "midpoint problem" under the extra assumption that the limit shape boundary is differentiable.

28 pages, 6 figures, addressed minor typographical errors and clarified measurability issues

References in corpus (2)

• Competition interfaces and second class particles
• Semi-infinite paths of the 2d-Radial Spanning Tree

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• Non-existence of bi-infinite geodesics in the exponential corner growth model
• Busemann functions and Gibbs measures in directed polymer models on ${\mathbb{Z}}^2$
• Optimal Exponent for Coalescence of Finite Geodesics in Exponential Last Passage Percolation
• Geodesics Toward Corners in First Passage Percolation
• Non-constant ground configurations in the disordered ferromagnet
• Geodesic trees in last passage percolation and some related problems