Stochastic $R$ matrix for $U_q(A^{(1)}_n)$

TL;DR

This paper demonstrates that the quantum R matrix for symmetric tensor representations of U_q(A^{(1)}_n) can be interpreted stochastically, leading to new integrable Markov processes with explicit eigenvalues.

Contribution

It introduces a stochastic interpretation of the quantum R matrix for U_q(A^{(1)}_n) and constructs new integrable Markov processes based on this framework.

Findings

Quantum R matrix satisfies sum rule for stochastic interpretation.

Matrix elements factorize into local transition rates extending q-Hahn process.

Eigenvalues of Markov matrices obtained via Bethe ansatz.

Abstract

We show that the quantum $R$ matrix for symmetric tensor representations of $U_q(A^{(1)}_n)$ satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky's local transition rate in the $q$-Hahn process for $n=1$. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of $n$ species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.