  The random-cluster model has been widely studied as a unifying framework for
random graphs, spin systems and electrical networks, but its dynamics have so
far largely resisted analysis. In this paper we analyze the Glauber dynamics of
the 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 model
parameter $p$ except the critical point $p=p_c(q)$, and for all values of the
second model parameter $q\ge 1$. We also provide a matching lower bound proving
that our result is tight. Our analysis takes as its starting point the recent
breakthrough by Beffara and Duminil-Copin on the location of the random-cluster
phase transition in $\mathbb{Z}^2$. It is reminiscent of similar results for
spin systems such as the Ising and Potts models, but requires the reworking of
several standard tools in the context of the random-cluster model, which is not
a spin system in the usual sense.
