Rzandom-cluster dynamics in ℤ²: Rapid mixing with general boundary conditions

The random-cluster model with parameters (p, q) is a random graph model that generalizes bond percolation (q = 1) and the Ising and Potts models (q ≥ 2). We study its Glauber dynamics on n × n boxes Λn of the integer lattice graph ℤ2, where the model exhibits a sharp phase transition at p = pc(q). Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of Λn, known as boundary conditions. For select boundary conditions that do not carry longrange information (namely, wired and free), Blanca and Sinclair proved that when q > 1 and p ≠ pc(q), the Glauber dynamics on Λn mixes in optimal O(n2 log n) time. In this paper, we prove that this mixing time is polynomial in n for every boundary condition that is realizable as a configuration on ℤ2\Λn. We then use this to prove near-optimal Õ(n2) mixing time for "typical"boundary conditions. As a complementary result, we construct classes of nonrealizable (nonplanar) boundary conditions inducing slow (stretchedexponential) mixing at p≪pc.