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.

On considère le modèle du $q$-TASEP, qui est une modification du processus d’exclusion totalement asymétrique (TASEP) sur $\mathbb{Z}$. 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 ${\mathbb{Z}}_{-}$ est completement occupé par les particules. On montre que les fluctuations du courant au temps $\tau$ sont d’ordre ${\tau }^{1/3}$ 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 $\mathbb{Z}$ 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 $\tau$ is of order ${\tau }^{1/3}$ and asymptotically distributed as the GUE Tracy–Widom distribution, which confirms the KPZ scaling theory conjecture.

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