Busemann functions and equilibrium measures in last passage percolation models

• Published in: Probability theory and related fields Vol. 154; no. 1-2; pp. 89 - 125
• Main Authors: Cator, Eric, Pimentel, Leandro P. R.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2012; Springer Nature B.V
• Subjects: Applied mathematics; Central limit theorem; Computational fluid dynamics; Economics; Equilibrium; Finance; Fluid flow; Fluids; Growth models; Insurance; Joints; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical functions; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Percolation; Phase transitions; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Statistics for Business; Studies; Theorems; Theoretical; Two dimensional models; Uniqueness; Hammersley process; Primary 60C05; Busemann functions; Secondary 60F05; Last passage percolation; 60K35; Equilibrium
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-011-0363-6

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has 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 (or time-invariant) measures for the related (multi-class) interacting fluid system. 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 central limit theorem for the Busemann function.