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 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.
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Acknowledgments
Part of this work was done during our stay at the Institute Henri Poincare, Centre Emile Borel, attending the program Interacting Particle Systems, Statistical Mechanics and Probability Theory (September 5–December 19, 2008). Both authors wish to thank the organizers and the Institute for their hospitality and support during our stay there. We would also like to thank an anonymous referee for many helpful suggestions to improve this manuscript.
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L. P. R. Pimentel was supported by grant number 613.000.605 from the Netherlands Organisation for Scientific Research (NWO).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Cator, E., Pimentel, L.P.R. Busemann functions and equilibrium measures in last passage percolation models. Probab. Theory Relat. Fields 154, 89–125 (2012). https://doi.org/10.1007/s00440-011-0363-6
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DOI: https://doi.org/10.1007/s00440-011-0363-6