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A Power Law of Order 1/4 for Critical Mean Field Swendsen-Wang Dynamics

Yun Long University of California, Berkeley
Asaf Nachmias University of British Columbia, Vancouver, British Columbia, Canada
Weiyang Ning University of Washington, Seattle, Washington
Yuval Peres Microsoft Research, Redmond, Washington
Available Formats:
Electronic ISBN: 978-1-4704-1895-3
Product Code: MEMO/232/1092.E
List Price: $65.00 MAA Member Price:$58.50
AMS Member Price: $39.00 Click above image for expanded view A Power Law of Order 1/4 for Critical Mean Field Swendsen-Wang Dynamics Yun Long University of California, Berkeley Asaf Nachmias University of British Columbia, Vancouver, British Columbia, Canada Weiyang Ning University of Washington, Seattle, Washington Yuval Peres Microsoft Research, Redmond, Washington Available Formats:  Electronic ISBN: 978-1-4704-1895-3 Product Code: MEMO/232/1092.E  List Price:$65.00 MAA Member Price: $58.50 AMS Member Price:$39.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2322014; 84 pp
MSC: Primary 60; Secondary 82;

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 the authors show that the mixing time is $\Theta(1)$ in high temperatures, $\Theta(\log n)$ in low temperatures and $\Theta(n^{1/4})$ at criticality. They 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.

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

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Volume: 2322014; 84 pp
MSC: Primary 60; Secondary 82;

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 the authors show that the mixing time is $\Theta(1)$ in high temperatures, $\Theta(\log n)$ in low temperatures and $\Theta(n^{1/4})$ at criticality. They 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.

• Chapters
• 1. Introduction
• 2. Statement of the results
• 3. Mixing time preliminaries
• 4. Outline of the proof of Theorem
• 5. Random graph estimates
• 6. Supercritical case
• 7. Subcritical case
• 8. Critical Case
• 9. Fast mixing of the Swendsen-Wang process on trees
• Acknowledgements
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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