Hydrodynamic Limit Equation for a Lozenge Tiling Glauber Dynamics

Abstract

We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. (SIAM J Comput 31:167–192, 2001). Single updates consist in concatenations of n elementary lozenge rotations at adjacent vertices. The dynamics can also be seen as a reversible stochastic interface evolution. When the update rate is chosen proportional to 1 / n, the dynamics is known to have special features: a certain Hamming distance between configurations contracts with time on average (Luby et al. in SIAM J Comput 31:167–192, 2001), and the relaxation time of the Markov chain is diffusive (Wilson in Ann Appl Probab 14:274–325, 2004), growing like the square of the diameter of the system. Here, we present another remarkable feature of this dynamics, namely we derive, in the diffusive timescale, a fully explicit hydrodynamic limit equation for the height function (in the form of a nonlinear parabolic PDE). While this equation cannot be written as a gradient flow w.r.t. a surface energy functional, it has nice analytic properties, for instance it contracts the ${\mathbb {L}}^2$ distance between solutions. The mobility coefficient $\mu$ in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system’s surface free energy. The derivation of the hydrodynamic limit is not fully rigorous, in that it relies on an unproven assumption of local equilibrium.