Random-cluster dynamics in $\mathbb{Z}^{2}$: Rapid mixing with general boundary conditions

Abstract

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 ($q\geq 2$). We study its Glauber dynamics on $n\times n$ boxes $\Lambda_{n}$ of the integer lattice graph $\mathbb{Z}^{2}$, where the model exhibits a sharp phase transition at $p=p_{c}(q)$. 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 $\Lambda_{n}$, 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 $p\neq p_{c}(q)$, the Glauber dynamics on $\Lambda_{n}$ mixes in optimal $O(n^{2}\log n)$ 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 $\mathbb{Z}^{2}\setminus\Lambda_{n}$. We then use this to prove near-optimal $\tilde{O}(n^{2})$ mixing time for “typical” boundary conditions. As a complementary result, we construct classes of nonrealizable (nonplanar) boundary conditions inducing slow (stretched-exponential) mixing at $p\ll p_{c}$.