Two-dimensional Anisotropic KPZ growth and limit shapes

TL;DR

This paper investigates the geometric origin of the AKPZ universality class in 2D interface growth models, showing that the negative Hessian of the growth velocity function stems from the preservation of Euler-Lagrange equations by hydrodynamic PDEs.

Contribution

It reveals the geometric reason behind the AKPZ signature and establishes the link between hydrodynamic PDEs and equilibrium shape equations in 2D growth models.

Findings

The Hessian of the growth velocity function has a negative determinant due to geometric preservation properties.

Hydrodynamic PDEs preserve Euler-Lagrange equations, explaining the AKPZ signature.

In dimer model dynamics, preservation of Euler-Lagrange equations is equivalent to harmonicity of the growth velocity.

Abstract

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 $v(\rho)$ giving the growth velocity as a function of the slope $\rho$ 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 $v$ with respect to a natural complex structure.