Random coalescing geodesics in first-passage percolation

TL;DR

This paper develops an ergodic theory for infinite geodesics in planar first-passage percolation using random coalescing geodesics, revealing their density and asymptotic properties, and applies this to solve longstanding problems.

Contribution

It introduces the concept of random coalescing geodesics, establishing their properties and density, and applies this framework to solve the midpoint problem and address bigeodesic existence.

Findings

Random coalescing geodesics have well-defined asymptotic directions.

They are dense in the space of all geodesics.

The theory solves the midpoint problem and addresses bigeodesic questions.

Abstract

We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics'. Random coalescing geodesics have a range of nice asymptotic properties, such as asymptotic directions and linear Busemann functions. We show that random coalescing geodesics are (in some sense) dense in the space of geodesics. This allows us to extrapolate properties from random coalescing geodesics to obtain statements on all infinite geodesics. As an application of this theory we solve the `midpoint problem' of Benjamini, Kalai and Schramm and address a question of Furstenberg on the existence of bigeodesics.