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A power law of order 1/4 for critical mean field Swendsen-Wang dynamics

About this Title

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)
Published electronically: March 11, 2014
Keywords:Markov chains, Mixing time, Ising model

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Table of Contents


  • 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


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 the mixing time of the chain is at most for all non-critical temperatures. In this paper we show that the mixing time is in high temperatures, in low temperatures and at criticality. We also provide an upper bound of for Swendsen-Wang dynamics for the -state ferromagnetic Potts model on any tree of vertices.

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