Geodesics and the competition interface for the corner growth model

TL;DR

This paper investigates the geometric properties of the corner growth model with general weights, focusing on geodesics and the competition interface, extending understanding beyond exactly solvable cases.

Contribution

It introduces new results on the competition interface and geodesic structures using stationary cocycles and Busemann functions in a non-solvable setting.

Findings

Characterization of the competition interface

Existence and uniqueness of directional semi-infinite geodesics

Nonexistence of doubly infinite geodesics

Abstract

We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. In a previous paper we constructed stationary cocycles and Busemann functions for this model. Using these objects, we prove new results on the competition interface, on existence, uniqueness, and coalescence of directional semi-infinite geodesics, and on nonexistence of doubly infinite geodesics.