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.
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. 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.