The TAZRP speed process

TL;DR

This paper introduces a speed process for the TAZRP, analogous to the TASEP case, enabling analysis of second-class particles' speeds and stationary distributions in multi-type TAZRP with reservoirs.

Contribution

It develops a new speed process framework for TAZRP, providing insights into joint distributions of second-class particles and equilibrium site contents.

Findings

Derived the joint distribution of second-class particle speeds in TAZRP.

Characterized the stationary distribution of multi-type TAZRP at a single site.

Established a connection between the speed process and stationary distributions.

Abstract

In [AAV] Amir, Angel and Valk{\'o} studied a multi-type version of the totally asymmetric simple exclusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicate questions about the joint distribution of the speed of several second-class particles in the TASEP rarefaction fan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-range process (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-class particles in the TAZRP with a reservoir. There is a close link from the speed process to questions about stationary distributions of multi-type versions of the TAZRP; for example we are able to give a precise description of the contents of a single site in equilibrium for a multi-type TAZRP with continuous labels.