Tracy–Widom asymptotics for $q$-TASEP

Ferrari, Patrik L.; Vető, Bálint

doi:10.1214/14-AIHP614

Tracy–Widom asymptotics for

q

-TASEP

Ferrari, Patrik L. ; Vető, Bálint

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1465-1485.

Résumé
Abstract

On considère le modèle du $q$ -TASEP, qui est une modification du processus d’exclusion totalement asymétrique (TASEP) sur $ℤ$ . Le taux des sauts d’une particule dépend de la distance avec la particule à sa droite et d’un paramètre $q \in [0, 1)$ . Dans cet article on considère la condition initiale où $ℤ_{-}$ est completement occupé par les particules. On montre que les fluctuations du courant au temps $τ$ sont d’ordre $τ^{1 / 3}$ et la distribution asymptotique est la distribution de Tracy–Widom GUE. Ce résultat confirme la conjecture KPZ.

We consider the $q$ -TASEP that is a $q$ -deformation of the totally asymmetric simple exclusion process (TASEP) on $ℤ$ for $q \in [0, 1)$ where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of $q$ -TASEP at time $τ$ is of order $τ^{1 / 3}$ and asymptotically distributed as the GUE Tracy–Widom distribution, which confirms the KPZ scaling theory conjecture.

MR Zbl

DOI : 10.1214/14-AIHP614

Mots-clés : interacting particle systems, kpz universality class, $q$-TASEP, current fluctuation, Tracy–Widom distribution

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     author = {Ferrari, Patrik L. and Vet\H{o}, B\'alint},
     title = {Tracy{\textendash}Widom asymptotics for $q${-TASEP}},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1465--1485},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {4},
     year = {2015},
     doi = {10.1214/14-AIHP614},
     mrnumber = {3414454},
     zbl = {1376.60080},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/14-AIHP614/}
}

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Ferrari, Patrik L.; Vető, Bálint. Tracy–Widom asymptotics for $q$-TASEP. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1465-1485. doi : 10.1214/14-AIHP614. http://archive.numdam.org/articles/10.1214/14-AIHP614/

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