Classification of Stationary distributions for the stochastic vertex models

TL;DR

This paper characterizes the extremal stationary distributions of the stochastic six vertex model, showing they are product Bernoulli measures, and extends these results to higher spin models using coupling, current analysis, and fusion methods.

Contribution

It provides a complete classification of extremal stationary distributions for the stochastic six vertex model and its higher spin generalization, using novel coupling and fusion techniques.

Findings

Extremal stationary distributions are product Bernoulli measures.

Under a moving frame, extremal distributions include blocking measures.

Results extend to stochastic higher spin six vertex models.

Abstract

In this paper, we study the stationary distributions for the stochastic vertex models. Our main focus is the stochastic six vertex (S6V) model. We show that the extremal stationary distributions of the S6V model are given by product Bernoulli measures. Moreover, for the S6V model under a moving frame of speed $1$, we show that the extremal stationary distributions are given by product Bernoulli measures and blocking measures. Finally, we generalize our results to the stochastic higher spin six vertex model. Our proof relies on the coupling of the S6V models introduced in [Aggarwal, 2020], the analysis of current and the method of fusion.