Coexistence probability in the last passage percolation model is 6-8log2
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, p. 973-988

A competition model on 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 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.

On étudie un modèle de compétition sur 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 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.

DOI : https://doi.org/10.1214/11-AIHP438
Classification:  60k35,  82B43
Keywords: last passage percolation, totally asymmetric simple exclusion process, competition interface, second class particle, coupling
@article{AIHPB_2012__48_4_973_0,
     author = {Coupier, David and Heinrich, Philippe},
     title = {Coexistence probability in the last passage percolation model is $6-8\log 2$},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {4},
     year = {2012},
     pages = {973-988},
     doi = {10.1214/11-AIHP438},
     zbl = {1261.60091},
     mrnumber = {3052401},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_4_973_0}
}
Coupier, David; Heinrich, Philippe. Coexistence probability in the last passage percolation model is $6-8\log 2$. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 973-988. doi : 10.1214/11-AIHP438. http://www.numdam.org/item/AIHPB_2012__48_4_973_0/

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