The TASEP speed process

TL;DR

This paper investigates the joint distribution of particle speeds in the multi-type TASEP, proving stationarity of the speed process, computing marginals, and extending results to ASEP, revealing surprising properties like positive probability of equal speeds.

Contribution

It introduces the TASEP speed process, proves its stationarity, and generalizes previous finite-class results to an infinite-class setting, also extending findings to ASEP.

Findings

The speed process is stationary under TASEP dynamics.

Two speeds are equal with positive probability.

Infinite particles can share the same speed.

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.