Swendsen-Wang Algorithm on the Mean-Field Potts Model

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. The case q = 2 (the Swendsen-Wang algorithm for the ferromagnetic Ising model) undergoes a slow-down at the uniqueness/non-uniqueness critical temperature for the infinite ∆-regular tree ([16]) but yet still has polynomial mixing time at all (inverse) temperatures β > 0 ([7]). In contrast for q ≥ 3 there are two critical temperatures 0 < βu < βrc that are relevant, these two critical points relate to phase transitions in the infinite tree. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts model on the nvertex complete graph satisfies: (i) O(logn) for β < βu, (ii) O(n1/3) for β = βu, (iii) exp(nΩ(1)) for βu < β < βrc, and (iv) O(logn) for β ≥ βrc. These results complement refined results of Cuff et al. [10] on the mixing time of the Glauber dynamics for the ferromagnetic Potts model. The most interesting aspect of our analysis is at the critical temperature β = βu, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over. 1998 ACM Subject Classification G.3 Probability and Statistics