Abstract
AbstractWe 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) for , (ii) for , (iii) for , where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for there are two critical temperatures 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) for , (ii) for , (iii) for , and (iv) for . These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.
Highlights
The mixing time of Markov chains is of critical importance for simulations of statistical physics models
We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Potts model on the complete graph
Our main result is a complete classification of the mixing time of the Swendsen-Wang dynamics on the complete graph when the parameter B is a constant independent of n
Summary
The mixing time of Markov chains is of critical importance for simulations of statistical physics models. The focus of this paper is the Curie-Weiss model which in computer science terminology is the n-vertex complete graph G (V , E) The interest in this model is that it allows more detailed results and these results are believed to extend to other graphs of particular interest such as random regular graphs. (Note that ln(1 B / n) ~ B / n for large n.) The following critical points Bu Bo Brc for the parameter B are well-studied and relevant to our study of the Potts model on the complete graph:. These thresholds correspond to the critical points for the infinite Δ-regular tree and random Δ-regular graphs by taking appropriate limits as. Our main result is a complete classification of the mixing time of the Swendsen-Wang dynamics on the complete graph when the parameter B is a constant independent of n
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