Hydrodynamic Limit of a (2+1)-Dimensional Crystal Growth Model in the Anisotropic KPZ Class

TL;DR

This paper establishes the hydrodynamic limit of a (2+1)-dimensional crystal growth model in the anisotropic KPZ class, showing convergence of the scaled interface profile to a viscosity solution of a Hamilton-Jacobi PDE.

Contribution

It proves the full hydrodynamic limit for a (2+1)-dimensional crystal growth model, extending the understanding of anisotropic KPZ universality class dynamics.

Findings

Convergence of the interface height profile to a viscosity solution.

Explicit non-convex speed function in the PDE.

Almost sure convergence in the hydrodynamic limit.

Abstract

We study a model, introduced initially by Gates and Westcott to describe crystal growth evolution, which belongs to the Anisotropic KPZ universality class. It can be thought of as a $(2+1)$-dimensional generalisation of the well known (1+1)-dimensional Polynuclear Growth Model (PNG). We show the full hydrodynamic limit of this process i.e the convergence of the random interface height profile after ballistic space-time scaling to the viscosity solution of a Hamilton-Jacobi PDE: $\partial_tu = v(\nabla u)$ with $v$ an explicit non-convex speed function. The convergence holds in the strong almost sure sense.