Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Coexistence probability in the last passage percolation model is $6-8\log2$

David Coupier and Philippe Heinrich

Full-text: Open access

Abstract

A competition model on $\mathbb{N}^{2}$ between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability $6-8\log2$. When this happens, we also prove that the central cluster almost surely has a positive density on $\mathbb{N}^{2}$. Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP, and on recent results about collision in the multi-TASEP.

Résumé

On étudie un modèle de compétition sur $\mathbb{N}^{2}$ entre trois clusters et gouverné par la percolation dirigée de dernier passage. On montre que la coexistence, c’est à dire que les trois clusters sont infinis simultanément, a lieu avec probabilité $6-8\log2$. Dans ce cas, le cluster central admet une densité positive sur $\mathbb{N}^{2}$. Nos résultats reposent sur trois couplages qui permettent de relier les interfaces de compétitions (qui représentent les frontières entres les clusters) à certaines particules du multi-TASEP, ainsi qu’à des résultats récents sur la collision dans le multi-TASEP.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 4 (2012), 973-988.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098436

Digital Object Identifier
doi:10.1214/11-AIHP438

Mathematical Reviews number (MathSciNet)
MR3052401

Zentralblatt MATH identifier
1261.60091

Subjects
Primary: 60k35 82B43: Percolation [See also 60K35]

Keywords
Last passage percolation Totally asymmetric simple exclusion process Competition interface Second class particle Coupling

Citation

Coupier, David; Heinrich, Philippe. Coexistence probability in the last passage percolation model is $6-8\log2$. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 973--988. doi:10.1214/11-AIHP438. https://projecteuclid.org/euclid.aihp/1353098436


Export citation

References

  • [1] G. Amir, O. Angel and B. Valko. The tasep speed process. Available at arXiv:0811.3706, 2008.
  • [2] D. Coupier and P. Heinrich. Stochastic domination for the last passage percolation model. Markov Process. Related Fields 17 (2011) 37–48.
  • [3] P. A. Ferrari, P. Gonçalves and J. B. Martin. Collision probabilities in the rarefaction fan of asymmetric exclusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 1048–1064.
  • [4] P. A. Ferrari, J. B. Martin and L. P. R. Pimentel. A phase transition for competition interfaces. Ann. Appl. Probab. 19 (2009) 281–317.
  • [5] P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. Ann. Probab. 33 (2005) 1235–1254.
  • [6] O. Garet and R. Marchand. Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15 (2005) 298–330.
  • [7] O. Häggström and R. Pemantle. First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 (1998) 683–692.
  • [8] C. Hoffman. Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 (2005) 739–747.
  • [9] C. Hoffman. Geodesics in first passage percolation. Ann. Appl. Probab. 18 (2008) 1944–1969.
  • [10] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer-Verlag, Berlin, 1999.
  • [11] J. B. Martin. Last-passage percolation with general weight distribution. Markov Process. Related Fields 12 (2006) 273–299.
  • [12] T. Mountford and H. Guiol. The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15 (2005) 1227–1259.
  • [13] H. Rost. Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 (1981) 41–53.
  • [14] T. Seppäläinen. Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor $K$-exclusion processes. Trans. Amer. Math. Soc. 353 (2001) 4801–4829 (electronic).
  • [15] H. Thorisson. Coupling, Stationarity, and Regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000.