Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

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 et al. (SIAM J Comput 31:167–192, 2001): 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 (Luby et al. 2001) and a certain one-dimensional projection of the dynamics is described by the heat equation (Wilson in Ann Appl Probab 14:274–325, 2004). 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 C2 differentiable and to contain no “frozen region”. The explicit form of the PDE was recently conjectured (Laslier and Toninelli in Ann Henri Poincare Theor Math Phys 18:2007–2043, 2017) on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg–Landau model (Funaki and Spohn in Commun Math Phys 85:1–36, 1997; Nishikawa in Commun Math Phys 127:205–227, 2003), here the mobility coefficient turns out to be a non-trivial function of the interface slope.