Matrix product formula for $U_q(A^{(1)}_2)$-zero range process

TL;DR

This paper derives a matrix product formula for the steady state probabilities of a specific integrable zero range process related to the quantum group $U_q(A^{(1)}_2)$, using $q$-boson operators and algebraic methods.

Contribution

It provides the first explicit matrix product representation for the steady state of the $U_q(A^{(1)}_2)$-zero range process, expanding the understanding of integrable stochastic models.

Findings

Matrix product formula for steady state probabilities derived

Representation of Zamolodchikov-Faddeev algebra constructed

Results applicable to integrable Markov processes with quantum group symmetry

Abstract

The $U_q(A^{(1)}_n)$-zero range processes introduced recently by Mangazeev, Maruyama and the authors are integrable discrete and continuous time Markov processes associated with the stochastic $R$ matrix derived from the well-known $U_q(A_n^{(1)})$ quantum $R$ matrix. By constructing a representation of the relevant Zamolodchikov-Faddeev algebra, we present, for $n=2$, a matrix product formula for the steady state probabilities in terms of $q$-boson operators.