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.
Apart from the above references, that deal with specific models, let us mention @cite_11 , that gives a sufficient condition for a @math -dimensional growth model to belong to the AKPZ class. In simple terms, [Th. 2.1] borodin2018two states that if the hydrodynamic equation @math preserves solutions of the Euler-Lagrange equations associated to strictly convex surface tension function @math , then @math . This condition can be verified on several growth models, e.g. the one defined in @cite_19 , and it is related to the fact that these stochastic processes preserve a certain local Gibbs property'' ( @math is then the surface tension corresponding with such Gibbs potential).
@cite_19 @cite_11
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A series of recent works focused on two-dimensional interface growth models in the so-called Anisotropic KPZ (AKPZ) universality class, that have a large-scale behavior similar to that of the Edwards-Wilkinson equation. In agreement with the scenario conjectured by D. Wolf (1991), in all known AKPZ examples the function @math giving the growth velocity as a function of the slope @math has a Hessian with negative determinant ("AKPZ signature"). While up to now negativity was verified model by model via explicit computations, in this work we show that it actually has a simple geometric origin in the fact that the hydrodynamic PDEs associated to these non-equilibrium growth models preserves the Euler-Lagrange equations determining the macroscopic shapes of certain equilibrium two-dimensional interface models. In the case of growth processes defined via dynamics of dimer models on planar lattices, we further prove that the preservation of the Euler-Lagrange equations is equivalent to harmonicity of @math with respect to a natural complex structure. 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). This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. 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.