A Power Law of Order 1/4 for Critical Mean Field Swendsen-wang Dynamics

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 Kn the mixing time of the chain is at most O( O n) for all non-critical temperatures. In this paper the authors show that the mixing time is Q (1) in high temperatures, Q (log n) in low temperatures and Q (n 1/4) at criticality. They also provide an upper bound of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on any tree of n vertices