Multispecies totally asymmetric zero range process: I. Multiline process and combinatorial $R$

TL;DR

This paper introduces an $n$-species totally asymmetric zero range process ($n$-TAZRP) on a periodic lattice, constructs a related multiline process using combinatorial $R$, and derives a matrix product formula for steady state probabilities, linking it to quantum affine algebra representations.

Contribution

The paper develops a new $n$-species TAZRP model, constructs a multiline process via combinatorial $R$, and provides a matrix product formula for steady states, connecting to quantum affine algebra structures.

Findings

Established a projection from the multiline process to $n$-TAZRP.

Derived a matrix product formula for steady state probabilities.

Linked $n$-TAZRP to quantum affine algebra representations.

Abstract

We introduce an $n$-species totally asymmetric zero range process ($n$-TAZRP) on one-dimensional periodic lattice with $L$ sites. It is a continuous time Markov process in which $n$ species of particles hop to the adjacent site only in one direction under the condition that smaller species ones have the priority to do so. Also introduced is an $n$-line process, a companion stochastic system having the uniform steady state from which the $n$-TAZRP is derived as the image by a certain projection $\pi$. We construct the $\pi$ by a combinatorial $R$ of the quantum affine algebra $U_q(\hat{sl}_L)$ and establish a matrix product formula of the steady state probability of the $n$-TAZRP in terms of corner transfer matrices of a $q=0$-oscillator valued vertex model. These results parallel the recent reformulation of the $n$-species totally asymmetric simple exclusion process ($n$-TASEP) by the authors, demonstrating that $n$-TAZRP and $n$-TASEP are the canonical sister models associated with the symmetric and the antisymmetric tensor representations of $U_q(\hat{sl}_L)$ at $q=0$, respectively.