On étudie un modèle de compétition sur 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 . Dans ce cas, le cluster central admet une densité positive sur . 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 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 . When this happens, we also prove that the central cluster almost surely has a positive density on . 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.
Mots clés : 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}, pages = {973--988}, publisher = {Gauthier-Villars}, volume = {48}, number = {4}, year = {2012}, doi = {10.1214/11-AIHP438}, mrnumber = {3052401}, zbl = {1261.60091}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP438/} }
TY - JOUR AU - Coupier, David AU - Heinrich, Philippe TI - Coexistence probability in the last passage percolation model is $6-8\log 2$ JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 973 EP - 988 VL - 48 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP438/ DO - 10.1214/11-AIHP438 LA - en ID - AIHPB_2012__48_4_973_0 ER -
%0 Journal Article %A Coupier, David %A Heinrich, Philippe %T Coexistence probability in the last passage percolation model is $6-8\log 2$ %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 973-988 %V 48 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP438/ %R 10.1214/11-AIHP438 %G en %F 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, Tome 48 (2012) no. 4, pp. 973-988. doi : 10.1214/11-AIHP438. http://www.numdam.org/articles/10.1214/11-AIHP438/
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