Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Tracy–Widom asymptotics for $q$-TASEP

Patrik L. Ferrari and Bálint Vető

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1465-1485.

Dates
Received: 20 November 2013
Revised: 14 March 2014
Accepted: 17 March 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432049

Digital Object Identifier
doi:10.1214/14-AIHP614

Mathematical Reviews number (MathSciNet)
MR3414454

Zentralblatt MATH identifier
1376.60080

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Interacting particle systems KPZ universality class $q$-TASEP Current fluctuation Tracy–Widom distribution

Citation

Ferrari, Patrik L.; Vető, Bálint. Tracy–Widom asymptotics for $q$-TASEP. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1465--1485. doi:10.1214/14-AIHP614. https://projecteuclid.org/euclid.aihp/1445432049


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