Busemann functions and equilibrium measures in last passage percolation models

Journal article (2011)
Department
Delft Institute of Applied Mathematics (EEMCS) (TU Delft)
Copyright: This article is published with open access at Springerlink.com
DOI: https://doi.org/doi:10.1007/s00440-011-0363-6
Equilibrium Hammersley process Last passage percolation Busemann functions
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Published Date

11-04-2011

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Source:

Probability Theory and Related Fields, 2011

ISSN:

0178-8051

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Abstract

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 timeinvariant) 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.

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