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November 2012 Coexistence probability in the last passage percolation model is $6-8\log2$
David Coupier, Philippe Heinrich
Ann. Inst. H. Poincaré Probab. Statist. 48(4): 973-988 (November 2012). DOI: 10.1214/11-AIHP438

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.

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.

Citation

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David Coupier. Philippe Heinrich. "Coexistence probability in the last passage percolation model is $6-8\log2$." Ann. Inst. H. Poincaré Probab. Statist. 48 (4) 973 - 988, November 2012. https://doi.org/10.1214/11-AIHP438

Information

Published: November 2012
First available in Project Euclid: 16 November 2012

zbMATH: 1261.60091
MathSciNet: MR3052401
Digital Object Identifier: 10.1214/11-AIHP438

Subjects:
Primary: 60K35 , 82B43

Keywords: competition interface , coupling , Last passage percolation , Second class particle , Totally asymmetric simple exclusion process

Rights: Copyright © 2012 Institut Henri Poincaré

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Vol.48 • No. 4 • November 2012
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