Busemann Functions and Equilibrium Measures in Last Passage Percolation

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has always been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium measures for the related (multi-class) interacting particle process. As we shall see, in the classical Hammersley model where each point has weight one, this approach brings a new and rather geometrical solution of the longest increasing subsequence problem, as well as a detailed description of the scaling behavior of the Busemann function along different directions.