A (2\xa0+\xa01)-Dimensional Anisotropic KPZ Growth Model with a Smooth Phase

Sunil Chhita,Fabio Lucio Toninelli

doi:10.1007/s00220-019-03402-x

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 {{rm det} H_rho > 0} (“isotropic KPZ class”) or {{rm 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 differentiable, 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 {rhone0} , 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 {{rm 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 {rhosim 0} .

Highlights

A statistical physicist’s view of stochastic interface growth models is that they describe the diverse phenomena of interface growth and crystal deposition [1]
We prove that fluctuations grow at most logarithmically in time and that det Hρ < 0: the model belongs to the AKPZ class
(ii) If instead det Hρ ≤ 0 (“Anisotropic KPZ” or AKPZ class) one expects that α = β = 0 and that the growth of (1.2), (1.3) is logarithmic, exactly like for the two-dimensional stochastic heat equation

Summary

Introduction

A statistical physicist’s view of stochastic interface growth models is that they describe the diverse phenomena of interface growth and crystal deposition [1]. (ii) If instead det Hρ ≤ 0 (“Anisotropic KPZ” or AKPZ class) one expects that α = β = 0 and that the growth of (1.2), (1.3) is logarithmic, exactly like for the two-dimensional stochastic heat equation This belief is supported by the mathematical analysis of various (2 + 1)-dimensional growth models [2,5,7,33] that share the following features: stationary states can be found explicitly and their height fluctuations behave on large space scales as a massless Gaussian Field (GFF), with α = 0 and logarithmic correlations; the speed of growth v(·) can be computed and det Hρ turns out to be negative; the growth exponent β is zero and the height variance grows at most logarithmically with time.

Notations

Dynamics

Stationarity of Gibbs Measures