Absence of backward infinite paths for first-passage percolation in arbitrary dimension

Abstract

In first-passage percolation (FPP), one places nonnegative random variables (weights) $({\mathit{t}}_{\mathit{e}})$ on the edges of a graph and studies the induced weighted graph metric. We consider FPP on ${\mathbb{Z}}^{\mathit{d}}$ for $\mathit{d}\ge 2$ and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating from a question of Furstenberg and most strongly supported for $\mathit{d}=2$ that, for continuously distributed i.i.d. weights, there a.s. are no bigeodesics. We provide the first progress on this question in general dimensions under no unproven assumptions. Our main result is that geodesic graphs, introduced in a previous paper of two of the authors, constructed in any deterministic direction a.s. do not contain doubly-infinite paths. As a consequence, one can construct random graphs of subsequential limits of point-to-hyperplane geodesics, which contain no bigeodesics. This gives evidence that bigeodesics, if they exist, cannot be constructed in a translation-invariant manner as limits of point-to-hyperplane geodesics.