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Busemann functions and equilibrium measures in last passage percolation models
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  • Published: 11 April 2011

Busemann functions and equilibrium measures in last passage percolation models

  • Eric Cator1 &
  • Leandro P. R. Pimentel2 

Probability Theory and Related Fields volume 154, pages 89–125 (2012)Cite this article

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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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References

  1. Aldous D., Diaconis P.: Hammersley’s interacting particle system and longest increasing subsequences. Probab. Theory Relat. Fields 103, 199–213 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baik J., Rains E.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 102, 1085–1132 (2001)

    Article  Google Scholar 

  3. Balázs M., Cator E.A., Seppäläinen T.: Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11, 1094–1132 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Busemann H.: The Geometry of Geodesics. Academic Press, New York (1955)

    MATH  Google Scholar 

  5. Cator E.A., Groeneboom P.: Hammersley’s process with sources and sinks. Ann. Probab. 33, 879–903 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cator E.A., Groeneboom P.: Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34, 1273–1295 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cator, E.A., Pimentel, L.P.R.: A shape theorem and semi-infinite geodesics for the Hammersley model with random weights. ALEA (2010). ArXiv:1001.4706 (to appear)

  8. Ferrari P.A., Martin J.B.: Multiclass Hammersley–Aldous–Diaconis process and multiclass-customer queues. Ann. Inst. H. Poincare 99, 305–319 (2007, to appear)

    Google Scholar 

  9. Ferrari P.A., Martin J.B., Pimentel L.P.R.: A phase transition for competition interfaces. Ann. Appl. Probab. 19, 281–317 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ferrari P.A., Pimentel L.P.R.: Competition interfaces and second class particles. Ann. Probab. 33, 1235–1254 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hammersley, J.M.: A few seedlings of research. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probabability, vol. 1, pp. 345–394. University of California Press, CA (1972)

  12. Howard C.D., Newman C.M.: Geodesics and spanning trees for euclidean first passage percolation. Ann. Probab. 29, 577–623 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Kesten H.: On the speed of convergence in first passage percolation. Ann. Appl. Probab. 3, 296–338 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Newman, C.M.: A surface view of first-passage percolation. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2, pp. 1017–1023. Birkhäuser, Basel (1995) (Zürich, 1994)

  15. Seppäläien, T.: Directed random growth models on the plane. In: Analysis and stochastics of growth processes and interfaces models, pp. 9–38. Oxford University press, NY (2009)

  16. Wüthrich M.: Asymptotic behavior of semi-infinite geodesics for maximal increasing subsequences in the plane. In: Sidoravicius, V. (eds) In and Out of Equilibrium, pp. 205–226. Birkhäuser, Basel (2002)

    Chapter  Google Scholar 

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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.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    Eric Cator

  2. Institute of Mathematics, Federal University of Rio de Janeiro, Caixa Postal 68530, 21941-909, Rio de Janeiro, RJ, Brazil

    Leandro P. R. Pimentel

Authors
  1. Eric Cator
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  2. Leandro P. R. Pimentel
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Corresponding author

Correspondence to Eric Cator.

Additional information

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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Cite this article

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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  • Received: 10 December 2010

  • Revised: 28 March 2011

  • Published: 11 April 2011

  • Issue Date: October 2012

  • DOI: https://doi.org/10.1007/s00440-011-0363-6

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Keywords

  • Hammersley process
  • Last passage percolation
  • Busemann functions
  • Equilibrium

Mathematics Subject Classification (2000)

  • Primary 60C05
  • 60K35
  • Secondary 60F05
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