(2 + 1)-Dimensional Interface Dynamics: Mixing Time, Hydrodynamic Limit and Anisotropic KPZ Growth

Abstract

Stochastic interface dynamics serve as mathematical models for diverse time-dependent physical phenomena: the evolution of boundaries between thermodynamic phases, crystal growth, random deposition... Interesting limits arise at large space-time scales: after suitable rescaling, the randomly evolving interface converges to the solution of a deterministic PDE (hydrodynamic limit) and the fluctuation process to a (in general non-Gaussian) limit process. In contrast with the case of (1 + 1)-dimensional models, there are very few mathematical results in dimension (d+1), d ≥ 2. As far as growth models are concerned, the (2 + 1)-dimensional case is particularly interesting: Wolf [45] conjectured the existence of two different universality classes (called KPZ and Anisotropic KPZ), with different scaling exponents. Here, we review recent mathematical results on (both reversible and irreversible) dynamics of some (2 + 1)-dimensional discrete interfaces, mostly defined through a mapping to two-dimensional dimer models. In particular, in the irreversible case, we discuss mathematical support and remaining open problems concerning Wolf's conjecture [45] on the relation between the Hessian of the growth velocity on one side, and the universality class of the model on the other.2010 Mathematics Subject Classification: 82C20, 60J10, 60K35, 82C24

Gaussian fluctuations from the 2D KPZ equation

“…It is a very interesting question to study the asymptotics of the anisotropic version of (1.1), where the Hopf-Cole transformation to the stochastic heat equation is unavailable and all existing methods break down. Some recent work on the interacting particle systems belonging to this class can be found in [2,3,9,26].…”

A (2 + 1)-Dimensional Anisotropic KPZ Growth Model with a Smooth Phase

“…In the (2 + 1)-dimensional case we consider here, the picture is different. On the basis of a renormalization-group analysis of (1.4) by D. Wolf [34], of numerical simulations [19,20,31] and on a few mathematically treatable models (see references in point (ii) below), the following conjectural picture has emerged (see also the introduction of [32] for a more detailed discussion):…”

Fluctuations of the solutions to the KPZ equation in dimensions three and higher

“…We refer to the reviews [15,36] for a more complete list of references. In d = 2, some relevant results can be found in [4,7,8,9,10,17,37].…”