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# memo_has_moved_text();A power law of order 1/4 for critical mean field Swendsen-Wang dynamics

Yun Long, Asaf Nachmias, Weiyang Ning and Yuval Peres

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 232, Number 1092
ISBNs: 978-1-4704-0910-4 (print); 978-1-4704-1895-3 (online)
DOI: http://dx.doi.org/10.1090/memo/1092
Published electronically: March 11, 2014
Keywords:Markov chains, Mixing time, Ising model

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Chapters

• Chapter 1. Introduction
• Chapter 2. Statement of the results
• Chapter 3. Mixing time preliminaries
• Chapter 4. Outline of the proof of Theorem 2.1
• Chapter 5. Random graph estimates
• Chapter 6. Supercritical case
• Chapter 7. Subcritical case
• Chapter 8. Critical Case
• Chapter 9. Fast mixing of the Swendsen-Wang process on trees
• Acknowledgements

### Abstract

The Swendsen-Wang dynamics is a Markov chain widely used by physicists to sample from the Boltzmann-Gibbs distribution of the Ising model. Cooper, Dyer, Frieze and Rue proved that on the complete graph $K_n$ the mixing time of the chain is at most $O(\sqrt {n})$ for all non-critical temperatures. In this paper we show that the mixing time is $\Theta (1)$ in high temperatures, $\Theta (\log n)$ in low temperatures and $\Theta (n^{1/4})$ at criticality. We also provide an upper bound of $O(\log n)$ for Swendsen-Wang dynamics for the $q$-state ferromagnetic Potts model on any tree of $n$ vertices.