The domino-shuffling algorithm can be seen as a stochastic process describing the irreversible growth of a @math -dimensional discrete interface. Its stationary speed of growth @math depends on the average interface slope @math , as well as on the edge weights @math , that are assumed to be periodic in space. We show that this growth model belongs to the Anisotropic KPZ class: one has @math and the height fluctuations grow at most logarithmically in time. Moreover, we prove that @math is discontinuous at each of the (finitely many) smooth (or gaseous'') slopes @math ; at these slopes, fluctuations do not diverge as time grows. For a special case of spatially @math periodic weights, analogous results have been recently proven in Chhita-Toninelli (2018) via an explicit computation of @math . In the general case, such a computation is out of reach; instead, our proof goes through a relation between the speed of growth and the limit shape of domino tilings of the Aztec diamond.
Historically, the first rigorous result we are aware of, on a @math -dimensional growth model in the AKPZ class, is @cite_23 , that computed the speed of growth of the Gates-Westcott model @cite_12 , verified that @math and proved that stationary states are only logarithmically rough, in agreement with the above conjecture (growth of flucutations in time was not studied there). More recently, a growth model that is a @math -dimensional, discrete, analog of Hammersley's process has been introduced in @cite_19 . Besides the computation of the speed of growth and the verification of @math , rigorous results on this model include the proof that height fluctuations grow at most logarithmically in space time @cite_19 @cite_6 , the study of stationary states @cite_6 , hydrodynamic limits for the height profile @cite_19 @cite_0 , determinantal formulas for certain space-time correlations @cite_19 and a CLT on scale @math for height flucutations under special initial conditions @cite_19 . Some of these results have been extended to an AKPZ growth process defined in terms of the dimer model on the square grid, see @cite_2 .
@cite_6 @cite_0 @cite_19 @cite_23 @cite_2 @cite_12
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We introduce a class of (2+1)(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)(p−q) equals zero, the infinite-volume Gibbs measures πρπρ (with given slope ρρ) are stationary and reversible. When p≠qp≠q, πρπρ 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 xx grows linearly with time tt with a nonzero speed: EQx(t):=E(hx(t)−hx(0))=V(ρ)tEQx(t):=E(hx(t)−hx(0))=V(ρ)t while the typical fluctuations of Qx(t)Qx(t) are smaller than any power of tt as t→∞t→∞. In the totally asymmetric case of p=0,q=1p=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 logt−−−−√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→∞t→∞, the local statistics of height fluctuations tends to that of a Gibbs state (which led to the prediction that Gibbs states should be stationary). We construct a family of stochastic growth models in 2 + 1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. We generalize the surface growth model of Gates and Westcott to arbitrary inclination. The exact steady growth velocity is of saddle type with principal curvatures of opposite sign. According to Wolf, this implies logarithmic height correlations, which we prove by mapping the steady state of the surface to world lines of free fermions with chiral boundary conditions. We consider driven dimer models on the square and honeycomb graphs, starting from a stationary Gibbs measure. Each model can be thought of as a two dimensional stochastic growth model of an interface, belonging to the anisotropic KPZ universality class. We use a combinatorial approach to determine the speed of growth and show logarithmic growth in time of the variance of the height function fluctuations. A new Markov process describing crystal growth in three dimensions is introduced. States of the process are configurations of the crystal surface, which has a terrace-edge-kink structure. The states are continuous along edges but discrete across edges, in accordance with the very different rates for the two types of captures of particles. Stationary distributions, describing steady crystal growth, are found in general. To our knowledge, these are the first examples of stationary distributions for layered crystal growth in three dimensions. The steady growth rate and other quantities are obtained explicitly for two interacting edges. For many interacting edges, growth behavior is determined (a) in various asymptotic regimes including thermodynamic limits, (b) via simulations, and (c) using series (cluster) expansions in the slope of the surface, the first three coefficients being computed. The theoretical growth rates show a marked dependence on surface dimensions. This may contribute to the size dependence and dispersion in the observed growth rate of small crystals.