  The random-cluster model is a unifying framework for studying random graphs,
spin systems in physics and random spanning trees. The model is closely related
to, though much more general than the classical Ising and Potts models, but its
dynamics are much less well understood. In this paper we study a natural
non-local Markov chain known as the Chayes-Machta dynamics for the mean-field
case of the random-cluster model, and identify a critical regime
$(\lambda_s,\lambda_S)$ of the model parameter $\lambda$ in which the dynamics
undergoes an exponential slowdown. Namely, we prove that the mixing time is
$\Theta(\log n)$ if $\lambda \not\in [\lambda_s,\lambda_S]$, and
$\exp(\Omega(\sqrt{n}))$ when $\lambda \in (\lambda_s,\lambda_S)$. These
results hold for all values of the second model parameter $q > 1$. Thus, we
obtain the first analysis of a dynamics for the random-cluster model for values
of $q$ other than the already well understood special case $q=2$ (which
corresponds to the Ising model) over almost the full range of values of
$\lambda$. In addition, we prove that the local heat-bath dynamics undergoes a
similar exponential slowdown in $(\lambda_s,\lambda_S)$.
