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Swendsen‐Wang algorithm on the mean‐field Potts model
Author(s) -
Galanis Andreas,
Štefankovič Daniel,
Vigoda Eric
Publication year - 2019
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20768
Subject(s) - potts model , glauber , chiral potts curve , ising model , mathematics , mean field theory , ferromagnetism , combinatorics , statistical physics , physics , condensed matter physics , quantum mechanics , scattering
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 β c 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 cthat 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 ) ) forβ 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.