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

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.

Infinite geodesics, asymptotic directions, and Busemann functions in first-passage percolation

“…This is an important distinction; in fact the analogue of Theorem 3.3 with uniqueness simultaneously holding for all θ P r0, 2πq should not hold. This is because two parallel geodesics should run along directions where there is a "competition interface" (see Theorem 4.7.1 in [14]). On the other hand, it is believed that the set D is a purely technical condition -i.e., that the statement of the theorem should hold with D " r0, 2πq.…”