Coexistence probability in the last passage percolation model is $6-8\log 2$

Coupier, David ; Heinrich, Philippe

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 973-988 / Harvested from Numdam

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 $6-8\log 2$. 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.

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 $6-8\log 2$. 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.

Détail

Publié le : 2012-01-01

MR 3052401 | Zbl 1261.60091

DOI : https://doi.org/10.1214/11-AIHP438
Classification:  60k35,  82B43

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