**Memoirs of the American Mathematical Society**

2014;
84 pp;
Softcover

MSC: Primary 60;
Secondary 82

Print ISBN: 978-1-4704-0910-4

Product Code: MEMO/232/1092

List Price: $65.00

AMS Member Price: $39.00

MAA Member Price: $58.50

**Electronic ISBN: 978-1-4704-1895-3
Product Code: MEMO/232/1092.E**

List Price: $65.00

AMS Member Price: $39.00

MAA Member Price: $58.50

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

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*Yun Long; Asaf Nachmias; Weiyang Ning; Yuval Peres*

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.

#### Table of Contents

# Table of Contents

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

- Cover 11 free
- Title page 22 free
- Chapter 1. Introduction 88
- Chapter 2. Statement of the results 1010
- Chapter 3. Mixing time preliminaries 1414
- Chapter 4. Outline of the proof of Theorem 2.1 1616
- Chapter 5. Random graph estimates 2020
- Chapter 6. Supercritical case 4444
- Chapter 7. Subcritical case 5252
- Chapter 8. Critical Case 5656
- Chapter 9. Fast mixing of the Swendsen-Wang process on trees 8686
- Acknowledgements 8888
- Bibliography 9090
- Back Cover 9696