Hydrodynamic limit for a 2D interlaced particle process

Abstract

The Markov dynamics of interlaced particle arrays, introduced by A. Borodin and P. L. Ferrari in (Comm. Math. Phys. 325 (2014) 603–684), is a classical example of $(2+1)$-dimensional random growth model belonging to the so-called Anisotropic KPZ universality class. In (Comm. Pure Appl. Math. 72 (2018) 620–666), a hydrodynamic limit—the convergence of the height profile, after space/time rescaling, to the solution of a deterministic Hamilton–Jacobi PDE with nonconvex Hamiltonian—was proven when either the initial profile is convex, or for small times, before the solution develops shocks. In the present work, we give a simpler proof, that works for all times and for all initial profiles for which the limit equation makes sense. In particular, the convexity assumption is dropped. The main new idea is a new viewpoint about “finite speed of propagation” that allows to bypass the need of a priori control of the interface gradients, or equivalently of inter-particle distances.