Mixing properties of the Swendsen-Wang process on classes of graphs

Abstract

We consider the mixing properties of the widely used Swendsen–Wang process for the Markov chain Monte Carlo estimation of the partition function of the ferromagnetic Q-state Potts model, for certain classes of graphs. In the paper “The Swendsen–Wang Process Does Not Always Mix Rapidly,” V. Gore and M. Jerrum obtained results for the mixing properties of the Swendsen–Wang process on the complete graph Kn. Our main results for graphs with n vertices are the following: For graphs with small maximum degree, the mixing time is polynomial in n for small enough values of the coupling constant β. For trees, the mixing time is O(n) for any β. For cycles, the mixing time is O(n log n) for any β. For random graphs Gn, p, p=Ω(n−1/3), there are values of the coupling constant β for which whp the Swendsen–Wang process does not mix rapidly. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 242–261, 1999