Two-dimensional anisotropic KPZ growth and limit shapes

Abstract

A series of recent works focused on two-dimensional (2D) 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 Wolf (1991 Phys. Rev. Lett. 67 1783–6), in all known AKPZ examples the function giving the growth velocity as a function of the slope ρ 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 2D 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 with respect to a natural complex structure.