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(...

• Type: Journal
• Year: 2015
• Where: APPROX
• Authors: Andreas Galanis, Daniel Stefankovic, Eric Vigoda
• Keywords: Algorithms, APPROX 2015