  The Swendsen-Wang dynamics is a Markov chain widely used by physicists to
sample from the Boltzmann-Gibbs distribution of the Ising model. Cooper, Dyer,
Frieze and Rue proved that on the complete graph K_n the mixing time of the
chain is at most O(n^{1/2}) for all non-critical temperatures. In this paper we
show that the mixing time is Theta(1) in high temperatures, Theta(log n) in low
temperatures and Theta(n^{1/4}) at criticality. We also provide an upper bound
of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts
model on any tree with n vertices.
