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 K_n the mixing time of the chain is at most O(√n) for all non-critical temperatures. In this paper we show that the mixing time is θ(1) in high temperatures, θ(log n) in low temperatures and θ(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 of n vertices.