Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

The six-vertex model in statistical physics is a weighted generalization of the ice model on Z^2 (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model represents its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the first rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and significantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks

Hydrodynamic limit equation for a lozenge tiling Glauber dynamics

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.Comment: 31 pages, 8 figures. v2: typos corrected, some proofs clarified. To appear on Annales Henri Poincar

Quenched Invariance Principle for the Random Walk on the Penrose Tiling

We consider the simple random walk on the graph corresponding to a Penrose tiling. We prove that the path distribution of the walk converges weakly to that of a non-degenerate Brownian motion for almost every Penrose tiling with respect to the appropriate invariant measure on the set of tilings. Our tool for this is the corrector method.Comment: 15 pages, 1 figur