Bigeodesics in first-passage percolation

TL;DR

This paper proves the non-existence of bigeodesics with deterministic directions in first-passage percolation under differentiability assumptions, advancing understanding of geodesic structures and related models.

Contribution

It establishes the absence of bigeodesics with deterministic directions assuming shape boundary differentiability, resolving longstanding conjectures and problems.

Findings

No bigeodesics with one end in any deterministic direction.

Rules out ground state pairs with fixed interface directions.

Addresses the Benjamini-Kalai-Schramm midpoint problem.

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.