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.

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