Random-cluster dynamics in .2 Z 2 : 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 .n of the integer lattice graph .2, 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 .n known as boundary conditions. For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when q>1 and p.pc(q), the Glauber dynamics on .n mixes in optimal O(n2logn) 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 .2..n. 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 (stretched-exponential) mixing at p.pc.