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

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.