Busemann functions, geodesics, and the competition interface for directed last-passage percolation

TL;DR

This survey explores the structure of directed last-passage percolation on planar lattices, focusing on Busemann functions, geodesics, and competition interfaces, especially in non-exactly solvable models, using stationary cocycles derived from queueing theory.

Contribution

It introduces a framework for analyzing non-solvable last-passage percolation models through stationary cocycles and characterizes key geometric features like geodesics and competition interfaces.

Findings

Construction of stationary cocycles from queueing fixed points.

Characterization of the limit shape and Busemann functions.

Results on existence, uniqueness, and coalescence of geodesics.

Abstract

In this survey article we consider 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. We show how stationary cocycles are constructed from queueing fixed points and how these cocycles characterize the limit shape, yield existence of Busemann functions in directions where the shape has some regularity, describe the direction of the competition interface, and answer questions on existence, uniqueness, and coalescence of directional semi-infinite geodesics, and on nonexistence of doubly infinite geodesics.