Rapid mixing of Swendsen–Wang dynamics in two dimensions

Abstract

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid (i.e. polynomial) mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature.As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.Acknowledgements. First and foremost, I would like to thank the advisor of my PhD thesis (the present paper is based on it), Erich Novak, for his constant support and the many discussions we had, not only about mathematics. This has certainly led to the development of my present interest in and fun with mathematics.I am also grateful to Aicke Hinrichs and Daniel Rudolf for helpful comments and suggestions during the work on my thesis, as well as for the valuable discussions we had about various topics.Finally, I want to express my thanks to my colleagues and friends Erich, Aicke, Daniel, Henning Kempka, Lev Markhasin, Philipp Rudolph, Winfried Sickel, Markus Weimar and Heidi Weyhausen who were (more or less regular) members of our daily coffee break that served with its friendly and relaxed atmosphere as a source of extra energy every day.

Mixing of the Glauber dynamics for the ferromagnetic Potts model

“…Although our results are not as sharp as those discussed in [31], using the structure of the grid we are able to prove an upper bound on μ + which is close to the lower bound given in Proposition 2. Note that the toroidal L-grid has n := L 2 vertices.…”

“…2 )) and that rapid mixing occurs for finite grids when β is below this threshold; see [26] and Theorem 2.10 of [31]. It is conjectured that the Glauber dynamics mixes slowly when β is above this threshold; see Remark 2.11 of [31]).…”

“…It is conjectured that the Glauber dynamics mixes slowly when β is above this threshold; see Remark 2.11 of [31]). Borgs, Chayes and Tetali [4] proved that for q sufficiently large and for β > log(q) 2…”

Random cluster dynamics for the Ising model is rapidly mixing

“…For complete graphs, the mixing time is very well understood for all q ≥ 1 [20,1,11]. For Z 2 , for all q ≥ 1, the dynamics is fast mixing at all temperatures other than the critical one [26,27,2].…”

Swendsen‐Wang algorithm on the mean‐field Potts model

“…Long et al showed more refined results for the complete graph establishing that the mixing time is for for , and for . For square boxes of , Ullrich proved that the mixing time of Swendsen‐Wang is polynomial for all temperatures (building upon results for the Glauber dynamics by Martinelli and Olivieri and Lubetzky and Sly ). Very recently, Guo and Jerrum showed that the mixing time of Swendsen‐Wang is polynomial for any graph G for all temperatures.…”