Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit law of a second class particle in TASEP with non-random initial condition

P. L. Ferrari, P. Ghosal, and P. Nejjar

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We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition and density $\lambda$ on $\mathbb{Z}_{-}$ and $\rho$ on $\mathbb{Z}_{+}$, and a second class particle initially at the origin. For $\lambda<\rho$, there is a shock and the second class particle moves with speed $1-\lambda-\rho$. For large time $t$, we show that the position of the second class particle fluctuates on the $t^{1/3}$ scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time $t$.


On considère le processus d’exclusion simple totalement asymétrique avec des condition initiales déterministes, densité $\lambda$ sur $\mathbb{Z}_{-}$ et $\rho$ sur $\mathbb{Z}_{+}$. Initialement on place une particule de deuxième classe à l’origine. Si $\lambda<\rho$, un choc est créé et la particule de deuxième classe le suit avec vitesse $1-\lambda-\rho$. Dans la limite $t\to\infty$, on démontre que les fluctuations de la position de la particule de deuxième classe sont de l’ordre $t^{1/3}$ et on obtient sa loi limite. On détermine aussi la loi limite du nombre de sauts faits par la particule de deuxième classe jusqu’à l’instant $t$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1203-1225.

Received: 26 October 2017
Revised: 4 May 2018
Accepted: 21 May 2018
First available in Project Euclid: 25 September 2019

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Mathematical Reviews number (MathSciNet)

Primary: 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Exclusion process Second class particle Shock Scaling limit KPZ universality class


Ferrari, P. L.; Ghosal, P.; Nejjar, P. Limit law of a second class particle in TASEP with non-random initial condition. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1203--1225. doi:10.1214/18-AIHP916.

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