Uniqueness and Mixing in the Low-Temperature Random-Cluster Model on Trees and Random Graphs

We study the random-cluster model on trees and treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p\in (0,1)$ and a clustering weight $q\in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, approximately sampling from this model on graphs of degree at most $\Delta$ is computationally hard. At parameter $p$ below the tree uniqueness threshold $p_{\mathsf{u}}(q,\Delta)$, it is expected that sampling is easy and local Markov chains mix rapidly on all bounded degree graphs. On typical graphs (e.g., random regular graphs), the same is predicted at $p > p_{\mathsf{s}}(q,\Delta)$, where $p_{\mathsf{s}}(q,\Delta)$ is a second uniqueness transition point on the $\Delta$-regular wired tree. Our first result establishes this non-uniqueness/uniqueness phase transition at $p_{\mathsf{s}}(q,\Delta)$ for all $q$ on the infinite $\Delta$-regular wired tree, resolving a conjecture of H{\"a}ggstr{\"o}m (1996). For this, we establish weak spatial mixing at $p>p_{\mathsf{s}}(q,\Delta)$ under sufficiently wired boundary conditions. We use this understanding of decay of correlations to show that on the wired tree on $n$ vertices, whenever $q>1$ and $p>p_{\mathsf{s}}(q,\Delta)$, the mixing time of random-cluster Glauber dynamics is a near-optimal $n^{1+o(1)}$. We then extend these results on spatial and temporal mixing from the tree to treelike geometries with mostly wired boundaries and use them to show that the random-cluster Glauber dynamics mix rapidly on the random $\Delta$-regular graph for all $p>p_{\mathsf{s}}(q,\Delta)$ as long as $q \ge C \log \Delta$, providing an efficient sampling algorithm for both the random-cluster and Potts models in this context.