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

Abstract

We introduce a class of $(2+1)$-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. “Irreversible” means that the interface has an average nonzero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes.

When the dynamical asymmetry parameter $(p-q)$ equals zero, the infinite-volume Gibbs measures $\pi_{\rho}$ (with given slope $\rho$) are stationary and reversible. When $p\ne q$, $\pi_{\rho}$ are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point $x$ grows linearly with time $t$ with a nonzero speed: $\mathbb{E}Q_{x}(t):=\mathbb{E}(h_{x}(t)-h_{x}(0))=V(\rho)t$ while the typical fluctuations of $Q_{x}(t)$ are smaller than any power of $t$ as $t\to\infty$.

In the totally asymmetric case of $p=0,q=1$ and on the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” introduced by A. Borodin and P. L. Ferrari in [J. Stat. Mech. Theory Exp. 2009 (2009) P02009, Comm. Math. Phys. 325 603–684]. For a suitably chosen, “integrable”, initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale $\sqrt{\log t}$, exploiting the fact that in that case certain space-time height correlations can be computed exactly. In the same setting, they proved that, asymptotically for $t\to\infty$, the local statistics of height fluctuations tends to that of a Gibbs state (which led to the prediction that Gibbs states should be stationary).