A (2+1)-dimensional Anisotropic KPZ growth model with a smooth phase

TL;DR

This paper introduces a (2+1)-dimensional anisotropic KPZ growth model with a smooth phase at zero slope, demonstrating a transition from rough to smooth stationary states and analyzing the associated fluctuation behaviors and Hessian properties.

Contribution

The paper constructs a specific growth model exhibiting a smooth stationary phase and analyzes its fluctuation properties and the non-smoothness of the growth velocity function at zero slope.

Findings

At non-zero slope, fluctuations grow logarithmically in space and time.

At zero slope, the stationary state is smooth with bounded correlations.

The model belongs to the AKPZ class with negative Hessian determinant for non-zero slopes.

Abstract

Stochastic growth processes in dimension $(2+1)$ were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes, according to whether the Hessian $H_\rho$ of the speed of growth $v(\rho)$ as a function of the average slope $\rho$ satisfies $\det H_\rho>0$ ("isotropic KPZ class") or $\det H_\rho\le 0$ ("anisotropic KPZ (AKPZ)" class). The former is characterized by strictly positive growth and roughness exponents, while in the AKPZ class fluctuations are logarithmic in time and space. It is natural to ask (a) if one can exhibit interesting growth models with "smooth" stationary states, i.e., with $O(1)$ fluctuations (instead of logarithmically or power-like growing, as in Wolf's picture) and (b) what new phenomena arise when $v(\cdot)$ is not smooth, so that $H_\rho$ is not defined. The two questions are actually related and here we provide an answer to both, in a specific framework. We define a $(2+1)$-dimensional interface growth process, based on the so-called shuffling algorithm for domino tilings. The stationary, non-reversible measures are translation-invariant Gibbs measures on perfect matchings of $\mathbb Z^2$, with $2$-periodic weights. If $\rho\ne0$, fluctuations are known to grow logarithmically in space and to behave like a two-dimensional GFF. We prove that fluctuations grow at most logarithmically in time and that $\det H_\rho<0$: the model belongs to the AKPZ class. When $\rho=0$, instead, the stationary state is "smooth", with correlations uniformly bounded in space and time; correspondingly, $v(\cdot)$ is not differentiable at $\rho=0$ and we extract the singularity of the eigenvalues of $H_\rho$ for $\rho\sim 0$.