Exclusion processes and Zero-range processes are two important examples of particle systems on **Z**. In this talk I will focus on the totally asymmetric exclusion process (TASEP) and the constant-rate totally asymmetric zero-range process (TAZRP). I will define the two models and discuss their basic properties such as their stationary measures. We will then introduce multi-type version of these processes (where particles of several types walk on **Z** according to the dynamics of the model, with lower type particles having priority over higher type ones). In a paper with Angel and Valko, results on multi-type stationary measures for the TASEP and about the (random) speed of a second class particle in the "step" initial condition were used to define a speed process for the TASEP which turned out to have many interesting properties. I will briefly survey some results concerning that process, and then go on to describe a new speed process for the TAZRP. We will show how to generalize the multi-type "step" initial condition to the Zero-Range process and how to define a bijection between the models that allows to transfer some results in 2nd class particles to the TAZRP. We will then use these to define a speed process and analyze it. An important step is finding the stationary measures for the multi-type TAZRP, which will be analyzed as a system of queues in tandem. We will also show how to use the speed process to make simple calculations of the speed of a 2nd class particle in the rarefacation fan for TAZRP under certain initial conditions.

The talk is based on joint work with O. Angel and B. Valko (TASEP) and current work in progress with P. Goncalves and J. Martin.

No previous knowledge of the models involved will be assumed.

Back to seminar page