Irreversible Markov dynamics and hydrodynamics for KPZ states in the stochastic six vertex model

Abstract

We introduce a family of Markov growth processes on discrete height functions defined on the 2-dimensional square lattice. Each height function corresponds to a configuration of the six vertex model on the infinite square lattice. We focus on the stochastic six vertex model corresponding to a particular two-parameter family of weights within the ferroelectric (Δ>1) regime. It is believed (and partially proven, see Aggarwal [3]) that the stochastic six vertex model displays nontrivial pure (i.e., translation invariant and ergodic) Gibbs states of two types, KPZ and liquid. These phases have very different long-range correlation structures. The Markov processes we construct preserve the KPZ pure states in the full plane. We also show that the same processes put on the torus preserve arbitrary Gibbs measures for generic six vertex weights (not necessarily in the ferroelectric regime). Our dynamics arise naturally from the Yang–Baxter equation for the six vertex model. Using the bijectivisation of the Yang–Baxter equation introduced in Bufetov–Petrov [17], we first construct discrete time dynamics on six vertex configurations with a particular boundary condition, namely with the step initial condition in the quarter plane. Then we take a Poisson-type limit to obtain simpler continuous time dynamics. These dynamics are irreversible; in particular, the height function has a nonzero average drift. In each KPZ pure state, we explicitly compute the average drift (also known as the current) as a function of the slope. We use this to heuristically analyze the hydrodynamics of a non-stationary version of our process acting on quarter plane stochastic six vertex configurations.