The Annals of Probability

The TASEP speed process

Gideon Amir, Omer Angel, and Benedek Valkó

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Abstract

In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of ℤ is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [−1, 1]. We study the joint distribution of these speeds: the TASEP speed process.

We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes.

By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed.

We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of ℤ, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.

Article information

Source
Ann. Probab., Volume 39, Number 4 (2011), 1205-1242.

Dates
First available in Project Euclid: 5 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1312555796

Digital Object Identifier
doi:10.1214/10-AOP561

Mathematical Reviews number (MathSciNet)
MR2857238

Zentralblatt MATH identifier
1225.82039

Subjects
Primary: 82C22: Interacting particle systems [See also 60K35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Exclusion process TASEP ASEP multi-type second class stationary measure

Citation

Amir, Gideon; Angel, Omer; Valkó, Benedek. The TASEP speed process. Ann. Probab. 39 (2011), no. 4, 1205--1242. doi:10.1214/10-AOP561. https://projecteuclid.org/euclid.aop/1312555796


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