Stationary cocycles and Busemann functions for the corner growth model

TL;DR

This paper constructs stationary cocycles for a general directed last-passage percolation model, enabling analysis of limit shapes and Busemann functions without relying on exact solvability.

Contribution

It introduces a novel method to construct stationary cocycles from queueing fixed points for non-solvable models, advancing understanding of limit shapes and Busemann functions.

Findings

Existence of stationary cocycles for general i.i.d. weights

Solution of variational formulas for limit shapes

Existence of Busemann functions in regular directions

Abstract

We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles will be used to prove results about semi-infinite geodesics and the competition interface.