Hydrodynamic limit equation for a lozenge tiling Glauber dynamics

TL;DR

This paper derives an explicit nonlinear PDE as the hydrodynamic limit for a reversible lozenge tiling Glauber dynamics, revealing new analytic properties and a connection to surface free energy.

Contribution

It provides the first explicit hydrodynamic PDE for the lozenge tiling Glauber dynamics, including a detailed expression for the mobility coefficient and its relation to surface free energy.

Findings

The hydrodynamic limit is a nonlinear parabolic PDE.

The PDE contracts the L2 distance between solutions.

The mobility coefficient depends explicitly on the interface slope.

Abstract

We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. 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 enjoy especially nice features: a certain Hamming distance between configurations contracts with time on average and the relaxation time of the Markov chain is diffusive, 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 time scale, a fully explicit hydrodynamic limit equation for the height function (in the form of a non-linear 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.