Hydrodynamic limit and viscosity solutions for a 2D growth process in the anisotropic KPZ class

Abstract

We study a $(2+1)$-dimensional stochastic interface growth model, that is believed to belong to the so-called Anisotropic KPZ (AKPZ) universality class [Borodin and Ferrari, 2014]. It can be seen either as a two-dimensional interacting particle process with drift, that generalizes the one-dimensional Hammersley process [Aldous and Diaconis 1995, Seppalainen 1996], or as an irreversible dynamics of lozenge tilings of the plane [Borodin and Ferrari 2014, Toninelli 2015]. Our main result is a hydrodynamic limit: the interface height profile converges, after a hyperbolic scaling of space and time, to the solution of a non-linear first order PDE of Hamilton-Jacobi type with non-convex Hamiltonian (non-convexity of the Hamiltonian is a distinguishing feature of the AKPZ class). We prove the result in two situations: (i) for smooth initial profiles and times smaller than the time $T_{shock}$ when singularities (shocks) appear or (ii) for all times, including $t>T_{shock}$, if the initial profile is convex. In the latter case, the height profile converges to the viscosity solution of the PDE. As an important ingredient, we introduce a Harris-type graphical construction for the process.