  For $\Delta \ge 5$ and $q$ large as a function of $\Delta$, we give a
detailed picture of the phase transition of the random cluster model on random
$\Delta$-regular graphs. In particular, we determine the limiting distribution
of the weights of the ordered and disordered phases at criticality and prove
exponential decay of correlations and central limit theorems away from
criticality.
  Our techniques are based on using polymer models and the cluster expansion to
control deviations from the ordered and disordered ground states. These
techniques also yield efficient approximate counting and sampling algorithms
for the Potts and random cluster models on random $\Delta$-regular graphs at
all temperatures when $q$ is large. This includes the critical temperature at
which it is known the Glauber and Swendsen-Wang dynamics for the Potts model
mix slowly. We further prove new slow-mixing results for Markov chains, most
notably that the Swendsen-Wang dynamics mix exponentially slowly throughout an
open interval containing the critical temperature. This was previously only
known at the critical temperature.
  Many of our results apply more generally to $\Delta$-regular graphs
satisfying a small-set expansion condition.
