A Short Note on Markov Duality in Multi-species Higher Spin Stochastic Vertex Models

TL;DR

This paper demonstrates a duality property in multi-species higher spin stochastic vertex models, extending known dualities and applying charge reversal, with implications for boundary conditions and related models.

Contribution

It introduces a new duality for the multi-species higher spin stochastic vertex model using charge reversal, connecting to open boundary conditions and recovering recent results.

Findings

Established a duality involving indicator functions for multi-species models.

Applied charge reversal to derive duality functions with open boundary conditions.

Reproduced recent duality results for the stochastic six vertex model.

Abstract

We show that the multi-species higher spin stochastic vertex model, also called the U_q(A_n^{(1)}) vertex model, satisfies a duality where the indicator function has the form {\eta^x_{[1,n]} \geq \xi^x_{[1,n]} }. In other words, for every particle in the \xi configuration of species i at vertex x, there must be a particle of species j > i at vertex x in the \eta configuration. The proof follows by applying charge reversal to previously discovered duality functions, which also results in open boundary conditions. As a corollary, we recover the duality for the stochastic six vertex model recently found by Y. Lin.