Stationary coalescing walks on the lattice

TL;DR

This paper studies the behavior of coalescing semi-infinite walks on a lattice, classifying their collective dynamics, and applying the results to geodesics in percolation models, revealing new structural properties.

Contribution

It provides a classification of the collective behavior of coalescing walks in 2D and applies this to understand geodesics in stationary percolation models, including new examples.

Findings

Walks either coalesce almost surely or form bi-infinite trajectories.

Bi-infinite trajectories have measure-preserving properties and a common asymptotic direction.

Constructed examples show coalescence without asymptotic direction or average weight.

Abstract

We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In $2d$, we classify the collective behavior of these walks under mild assumptions: they either coalesce almost surely or form bi-infinite trajectories. Bi-infinite trajectories form measure-preserving dynamical systems, have a common asymptotic direction in $2d$, and possess other nice properties. We use our theory to classify the behavior of compatible families of semi-infinite geodesics in stationary first- and last-passage percolation. We also partially answer a question raised by C. Hoffman about the limiting empirical measure of weights seen by geodesics. We construct several examples: our main example is a standard first-passage percolation model where geodesics coalesce almost surely, but have no asymptotic direction or average weight.