### Abstract

Swendsen-Wang dynamics for the Potts model was proposed in the late 1980's as an alternative to single-site heat-bath dynamics, in which global updates allow this MCMC sampler to switch between metastable states and ideally mix faster. Gore and Jerrum (1997) found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with q ≥ 3 colors on the complete graph on n vertices at the critical pointc(q), Swendsen-Wang dynamics has t_{mix} ≥ exp(√c/n). Galanis et al. (2015) showed that tmix exp(cn1=3) throughout the critical window (βs; βS) around βc, and Blanca and Sinclair (2015) established that t_{mix} ≥ exp(c p n) in the critical window for corresponding mean-field FK model, which implied the same bound for Swendsen-Wang via known comparison estimates. In both cases, an upper bound of t_{mix} β exp(c0n) was known. Here we show that the mixing time is truly exponential in n: namely, t_{mix} β exp(cn) for Swendsen-Wang dynamics when q ≥ 3 and β ϵ (βs; βS), and the same bound holds for the related MCMC samplers for the mean-field FK model when q > 2.

Original language | English (US) |
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Title of host publication | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |

Editors | Artur Czumaj |

Publisher | Association for Computing Machinery |

Pages | 1981-1988 |

Number of pages | 8 |

ISBN (Electronic) | 9781611975031 |

DOIs | |

State | Published - 2018 |

Event | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States Duration: Jan 7 2018 → Jan 10 2018 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |
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Country | United States |

City | New Orleans |

Period | 1/7/18 → 1/10/18 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

*29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018*(pp. 1981-1988). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery. https://doi.org/10.1137/1.9781611975031.129