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

Coupier, David; Heinrich, Philippe

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 973-988.

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 $6-8log2$. 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.

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 $6-8log2$. 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.

Classification : 60k35, 82B43
Mots clés : last passage percolation, totally asymmetric simple exclusion process, competition interface, second class particle, coupling

[1] G. Amir, O. Angel and B. Valko. The tasep speed process. Available at arXiv:0811.3706, 2008. | MR | Zbl

[2] D. Coupier and P. Heinrich. Stochastic domination for the last passage percolation model. Markov Process. Related Fields 17 (2011) 37-48. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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. | MR | Zbl

[5] P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. Ann. Probab. 33 (2005) 1235-1254. | MR | Zbl

[6] O. Garet and R. Marchand. Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15 (2005) 298-330. | MR | Zbl

[7] O. Häggström and R. Pemantle. First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 (1998) 683-692. | MR | Zbl

[8] C. Hoffman. Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 (2005) 739-747. | MR | Zbl

[9] C. Hoffman. Geodesics in first passage percolation. Ann. Appl. Probab. 18 (2008) 1944-1969. | MR | Zbl

[10] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer-Verlag, Berlin, 1999. | MR | Zbl

[11] J. B. Martin. Last-passage percolation with general weight distribution. Markov Process. Related Fields 12 (2006) 273-299. | MR | Zbl

[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. | Zbl

[13] H. Rost. Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 (1981) 41-53. | Zbl

[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). | Zbl

[15] H. Thorisson. Coupling, Stationarity, and Regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000. | Zbl