Collision probabilities in the rarefaction fan of asymmetric exclusion processes

Ferrari, Pablo A., Gonçalves, Patricia, and Martin, James B.. "Collision probabilities in the rarefaction fan of asymmetric exclusion processes." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 1048-1064. <http://eudml.org/doc/78052>.

@article{Ferrari2009,
abstract = {We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate p∈(1/2, 1] and to the left at rate 1−p, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the initial state has first-class particles to the left of the origin, second-class particles at sites 0 and 1, and holes to the right of site 1. We show that the probability that the two second-class particles eventually collide is (1+p)/(3p), where a collision occurs when one of the particles attempts to jump over the other. This also corresponds to the probability that two ASEP processes, started from appropriate initial states and coupled using the so-called “basic coupling,” eventually reach the same state. We give various other results about the behaviour of second-class particles in the ASEP. In the totally asymmetric case (p=1) we explain a further representation in terms of a multi-type particle system, and also use the collision result to derive the probability of coexistence of both clusters in a two-type version of the corner growth model.},
author = {Ferrari, Pablo A., Gonçalves, Patricia, Martin, James B.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {asymmetric simple exclusion process; rarefaction fan; second-class particle; growth model},
language = {eng},
number = {4},
pages = {1048-1064},
publisher = {Gauthier-Villars},
title = {Collision probabilities in the rarefaction fan of asymmetric exclusion processes},
url = {http://eudml.org/doc/78052},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Ferrari, Pablo A.
AU - Gonçalves, Patricia
AU - Martin, James B.
TI - Collision probabilities in the rarefaction fan of asymmetric exclusion processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 1048
EP - 1064
AB - We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate p∈(1/2, 1] and to the left at rate 1−p, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the initial state has first-class particles to the left of the origin, second-class particles at sites 0 and 1, and holes to the right of site 1. We show that the probability that the two second-class particles eventually collide is (1+p)/(3p), where a collision occurs when one of the particles attempts to jump over the other. This also corresponds to the probability that two ASEP processes, started from appropriate initial states and coupled using the so-called “basic coupling,” eventually reach the same state. We give various other results about the behaviour of second-class particles in the ASEP. In the totally asymmetric case (p=1) we explain a further representation in terms of a multi-type particle system, and also use the collision result to derive the probability of coexistence of both clusters in a two-type version of the corner growth model.
LA - eng
KW - asymmetric simple exclusion process; rarefaction fan; second-class particle; growth model
UR - http://eudml.org/doc/78052
ER -