Busemann functions and infinite geodesics in two-dimensional first-passage percolation

TL;DR

This paper advances the understanding of infinite geodesics in two-dimensional first-passage percolation by developing a new framework for Busemann functions, proving coalescence, and exploring conditions for geodesic existence under minimal assumptions.

Contribution

It introduces a framework for distributional limits of Busemann functions and establishes new results on geodesic coalescence and existence with minimal assumptions.

Findings

Proves coalescence of long finite geodesics in any deterministic direction.

Introduces a directional condition implying existence of directional geodesics.

Shows almost-sure coalescence and nonexistence of infinite backward paths in geodesic graphs.

Abstract

We study first-passage percolation on Z2, where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of infinite geodesics. Some of these were studied in the late 90's by C. Newman and collaborators under strong assumptions on the limiting shape and weight distribution. In this paper we develop a framework for working with distributional limits of Busemann functions and use it to prove forms of Newman's results under minimal assumptions. For instance, we show a form of coalescence of long finite geodesics in any deterministic direction. We also introduce a purely directional condition which replaces Newman's global curvature condition and whose assumption we show implies the existence of directional geodesics. Without this condition, we prove existence of infinite geodesics which are directed in sectors. Last, we analyze distributional limits of geodesic graphs, proving almost-sure coalescence and nonexistence of infinite backward paths. This result relates to the conjecture of nonexistence of "bigeodesics."