Swendsen‐Wang algorithm on the mean‐field Potts model. Issue 1 (6th March 2018) - Galanis, Andreas; Štefankovič, Daniel; Vigoda, Eric
Abstract: 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 < β c, (ii) Θ ( n 1 / 4 ) for β = β c, (iii) Θ ( log ⁡ n ) for β > β c, where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for q ≥ 3 there are two critical temperatures 0 < β u < β r c 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 < β u, (ii) Θ ( n 1 / 3 ) for β = β u, (iii) exp ⁡ ( n Ω ( 1 ) ) for β u < β < β r c, and (iv) Θ ( log ⁡ n ) for β ≥ β r c . These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.