Bigeodesics in First-Passage Percolation

Abstract

In first-passage percolation, we place i.i.d. continuous weights at the edges of ${\mathbb{Z}^2}$ 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 1990s, 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 ${D \subset [0,2\pi)}$ 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” (Benjamini et al. in Ann Probab 31, p. 1976, 2003). under the extra assumption that the limit shape boundary is differentiable.