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The Swendsen-Wang dynamics is a Markov chain widely used by physicists tosample 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 thechain is at most O(n^{1/2}) for all non-critical temperatures. In this paper weshow that the mixing time is Theta(1) in high temperatures, Theta(log n) in lowtemperatures and Theta(n^{1/4}) at criticality. We also provide an upper boundof O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Pottsmodel on any tree with n vertices.

A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics