The TASEP speed process

In a multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with a 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. By relating this form to the known stationary measures for multi-type 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 ASEP with the above initial configuration has a certain symmetry. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.