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We introduce a class of (2+1)-dimensional stochastic growth processes, thatcan be seen as irreversible random dynamics of discrete interfaces."Irreversible" means that the interface has an average non-zero drift.Interface configurations correspond to height functions of dimer coverings ofthe infinite hexagonal or square lattice. The model can also be viewed as aninteracting driven particle system and in the totally asymmetric case thedynamics corresponds to an infinite collection of mutually interactingHammersley processes.  When the dynamical asymmetry parameter $(p-q)$ equals zero, theinfinite-volume Gibbs measures $\pi_\rho$ (with given slope $\rho$) arestationary and reversible. When $p\ne q$, $\pi_\rho$ are not reversible anymore but, remarkably, they are still stationary. In such stationary states, wefind that the average height function at any given point $x$ grows linearlywith time $t$ with a non-zero speed: $\mathbb E Q_x(t):=\mathbbE(h_x(t)-h_x(0))= V(\rho) t$ while the typical fluctuations of $Q_x(t)$ aresmaller than any power of $t$ as $t\to\infty$.  In the totally asymmetric case of $p=0,q=1$ and on the hexagonal lattice, thedynamics coincides with the "anisotropic KPZ growth model" introduced by A.Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initialcondition (that is very far from the stationary state), they were able todetermine the hydrodynamic limit and a CLT for interface fluctuations on scale$\sqrt{\log t}$, exploiting the fact that in that case certain space-timeheight correlations can be computed exactly.

A (2+1)-dimensional growth process with explicit stationary measures