Memoirs of the American Mathematical Society 2014; 84 pp; softcover Volume: 232 ISBN10: 1470409100 ISBN13: 9781470409104 List Price: US$65 Individual Members: US$39 Institutional Members: US$52 Order Code: MEMO/232/1092
 The SwendsenWang dynamics is a Markov chain widely used by physicists to sample from the BoltzmannGibbs 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 noncritical 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 SwendsenWang dynamics for the \(q\)state ferromagnetic Potts model on any tree of \(n\) vertices. Table of Contents  Introduction
 Statement of the results
 Mixing time preliminaries
 Outline of the proof of Theorem 2.1
 Random graph estimates
 Supercritical case
 Subcritical case
 Critical Case
 Fast mixing of the SwendsenWang process on trees
 Acknowledgements
 Bibliography
