### 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 (J. Stat. Phys. 97 (1999) 67.86) 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 point β_{c}(q), Swendsen. Wang dynamics has tMIX ≥ exp(c √ n). Galanis et al. (In Proc. of the 19th International Workshop on Randomization and Computation (RANDOM 2015) (2015) 815.828) showed that t_{MIX} ≥ exp(cn^{1/3}) throughout the critical window (β_{s},β_{S}) around β_{c}, and Blanca and Sinclair (In Proc. of the 19th International Workshop on Randomization and Computation (RANDOM 2015) (2015) 528.543) established that t_{MIX} ≥ exp(c √ n) in the critical window for the 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} ≤ exp(c′ n) was known. Here we show that the mixing time is truly exponential in n: namely, t_{MIX} ≥ exp(cn) for Swendsen-Wang dynamics when q ≥ 3 and β ∈ (β_{s},β_{S}), 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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Pages (from-to) | 68-86 |

Number of pages | 19 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 56 |

Issue number | 1 |

DOIs | |

State | Published - 2020 |

### Keywords

- FK model
- Mixing time
- Potts model
- Random graphs
- Swendsen-Wang

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*56*(1), 68-86. https://doi.org/10.1214/18-AIHP955