Infinite geodesics, asymptotic directions, and Busemann functions in first-passage percolation

TL;DR

This paper investigates the properties of infinite geodesics in first-passage percolation, establishing existence, uniqueness, and directedness results with and without unproven curvature assumptions, using novel proof techniques.

Contribution

It provides new proofs for the existence of multiple infinite geodesics and advances understanding of their directedness and uniqueness without relying on unproven assumptions.

Findings

Existence of at least two infinite geodesics without unproven assumptions

Directedness properties of infinite geodesics established

Generalized uniqueness results for infinite geodesics obtained

Abstract

We show existence, uniqueness, and directedness properties for infinite geodesics in the FPP model. After giving the fundamental definitions, we describe results by Newman and collaborators giving existence and uniqueness of directed geodesics under an unproven curvature assumption. We then give two proofs of the existence of at least two infinite geodesics under no unproven assumptions. In the final two sections, we give proofs of directedness statements for infinite geodesics using more recent methods which give information even under no unproven assumptions and prove a generalized uniqueness statement for infinite geodesics.