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The random-cluster model has been widely studied as a unifying framework forrandom graphs, spin systems and electrical networks, but its dynamics have sofar largely resisted analysis. In this paper we analyze the Glauber dynamics ofthe random-cluster model in the canonical case where the underlying graph is an$n \times n$ box in the Cartesian lattice $\mathbb{Z}^2$. Our main result is a$O(n^2\log n)$ upper bound for the mixing time at all values of the modelparameter $p$ except the critical point $p=p_c(q)$, and for all values of thesecond model parameter $q\ge 1$. We also provide a matching lower bound provingthat our result is tight. Our analysis takes as its starting point the recentbreakthrough by Beffara and Duminil-Copin on the location of the random-clusterphase transition in $\mathbb{Z}^2$. It is reminiscent of similar results forspin systems such as the Ising and Potts models, but requires the reworking ofseveral standard tools in the context of the random-cluster model, which is nota spin system in the usual sense.

Random-Cluster Dynamics in $\mathbb{Z}^2$