Hydrodynamic limit of a $(2+1)$-dimensional crystal growth model in the anisotropic KPZ class

Abstract

We study a model, introduced initially by Gates and Westcott [11] to describe crystal growth evolution, which belongs to the Anisotropic KPZ universality class [19]. 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: $\partial _{t}u = v(\nabla u)$ with $v$ an explicit non-convex speed function. The convergence holds in the strong almost sure sense.