We study the q-state ferromagnetic Potts model on the n-vertex complete
graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang
algorithm which is a Markov chain that utilizes the random cluster
representation for the ferromagnetic Potts model to recolor large sets of
vertices in one step and potentially overcomes obstacles that inhibit
single-site Glauber dynamics. Long et al. studied the case q=2, the
Swendsen-Wang algorithm for the mean-field ferromagnetic Ising model, and
showed that the mixing time satisfies: (i) Θ(1) for β<βc,
(ii) Θ(n1/4) for β=βc, (iii) Θ(logn) for
β>βc, where βc is the critical temperature for the
ordered/disordered phase transition. In contrast, for q≥3 there are two
critical temperatures 0<βu<βrc that are relevant. We prove that
the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts
model on the n-vertex complete graph satisfies: (i) Θ(1) for
β<βu, (ii) Θ(n1/3) for β=βu, (iii)
exp(nΩ(1)) for βu<β<βrc, and (iv)
Θ(logn) for β≥βrc. These results complement refined
results of Cuff et al. on the mixing time of the Glauber dynamics for the
ferromagnetic Potts model.Comment: To appear in Random Structures & Algorithm