### Abstract

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an n × n box in the Cartesian lattice ℤ^{2}. Our main result is a O(n^{2} log n) upper bound for the mixing time at all values of the model parameter p except the critical point p = pc(q), and for all values of the second model parameter q ≤ 1. We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in ℤ^{2}. It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense.

Original language | English (US) |
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Title of host publication | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Editors | Robert Krauthgamer |

Publisher | Association for Computing Machinery |

Pages | 498-513 |

Number of pages | 16 |

ISBN (Electronic) | 9781510819672 |

DOIs | |

State | Published - Jan 1 2016 |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: Jan 10 2016 → Jan 12 2016 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 1 |

### Conference

Conference | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
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Country | United States |

City | Arlington |

Period | 1/10/16 → 1/12/16 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

^{2}. In R. Krauthgamer (Ed.),

*27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016*(pp. 498-513). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 1). Association for Computing Machinery. https://doi.org/10.1137/1.9781611974331.ch37