Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

Abstract

We study a reversible continuous-time Markov dynamics of a discrete $(2+1)$-dimensional interface. This can be alternatively viewed as a dynamics of lozenge tilings of the $L\times L$ torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in [Luby-Randall-Sinclair]: in terms of interlaced particles, a particle jump of length $n$ that preserves the interlacement constraints has rate $1/(2n)$. This dynamics presents special features: the average mutual volume between two interface configurations decreases with time and a certain one-dimensional projection of the dynamics is described by the heat equation. In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as $L\to\infty$ to the solution of a non-linear parabolic PDE. The initial profile is assumed to be $C^2$ differentiable and to contain no "frozen region". The explicit form of the PDE was recently conjectured on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg-Landau model [Funaki-Spohn,Nishikawa], here the mobility coefficient turns out to be a non-trivial function of the interface slope.