### 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 |

Publisher | Association for Computing Machinery |

Pages | 1981-1988 |

Number of pages | 8 |

ISBN (Electronic) | 9781611975031 |

DOIs | |

State | Published - Jan 1 2018 |

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

### Other

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

Country | United States |

City | New Orleans |

Period | 1/7/18 → 1/10/18 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

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

**Exponentially slow mixing in the mean-field Swendsen-Wang dynamics.** / Gheissari, Reza; Lubetzky, Eyal; Peres, Yuval.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018.*Association for Computing Machinery, pp. 1981-1988, 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, United States, 1/7/18. https://doi.org/10.1137/1.9781611975031.129

}

TY - GEN

T1 - Exponentially slow mixing in the mean-field Swendsen-Wang dynamics

AU - Gheissari, Reza

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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 tmix ≥ 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 tmix ≥ 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 tmix β exp(c0n) was known. Here we show that the mixing time is truly exponential in n: namely, tmix 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.

AB - 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 tmix ≥ 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 tmix ≥ 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 tmix β exp(c0n) was known. Here we show that the mixing time is truly exponential in n: namely, tmix 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.

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U2 - 10.1137/1.9781611975031.129

DO - 10.1137/1.9781611975031.129

M3 - Conference contribution

SP - 1981

EP - 1988

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

PB - Association for Computing Machinery

ER -