Abstract: We introduce a family of irreversible growth processes which can be seen as Markov chains 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, and "irreversible" means that the height function has nonzero average drift. The dynamics arise naturally from the Yang--Baxter equation for the six vertex model, namely from a construction called "bijectivisation".​

These dynamics preserve the KPZ phase translation invariant Gibbs measures for the stochastic six vertex model, and we compute the current (the average drift) in each KPZ phase pure state with horizontal slope s. Using this, we analyze the hydrodynamic limit of a non-stationary version of the dynamics acting on quarter plane six vertex configurations.