A Multi-species ASEP(q,j) and q-TAZRP with Stochastic Duality

TL;DR

This paper develops a multi-species ASEP(q,j) process allowing multiple particles per site, establishes its duality properties, and shows its convergence to a multi-species q-TAZRP as j approaches infinity, extending known models.

Contribution

It introduces a multi-species ASEP(q,j) with explicit reversible measures and self-duality, generalizing previous two-species results and connecting to multi-species q-TAZRP.

Findings

Explicit reversible measures and self-duality functions for multi-species ASEP(q,j)

Duality between the process and its space-reversed version

Convergence to multi-species q-TAZRP as j approaches infinity

Abstract

This paper introduces a multi-species version of a process called ASEP(q,j). In this process, up to 2j particles are allowed to occupy a lattice site, the particles drift to the right with asymmetry 0<q^{2j}<1, and there are n-1 species of particles in which heavier particles can force lighter particles to switch places. Assuming closed boundary conditions, we explicitly write the reversible measures and a self-duality function, generalizing previously known results for two-species ASEP and single-species ASEP(q,j). Additionally, it is shown that this multi-species ASEP(q,j) is dual to its space-reversed version, in which particles drift to the left. As j goes to infinity, this multi-species ASEP(q,j) converges to a multi-species q-TAZRP and the self-duality function has a non-trivial limit, showing that this multi-species q-TAZRP satisfies a space-reversed self-duality. The construction of the process and the proofs are accomplished utilizing spin j representations of U_q(gl_n), extending the approach used for single-species ASEP(q,j).