## 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= p_{c}(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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Pages (from-to) | 821-847 |

Number of pages | 27 |

Journal | Probability Theory and Related Fields |

Volume | 168 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 1 2017 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty