### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

^{2}

*Probability Theory and Related Fields*,

*168*(3-4), 821-847. https://doi.org/10.1007/s00440-016-0725-1

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^{2}',

*Probability Theory and Related Fields*, vol. 168, no. 3-4, pp. 821-847. https://doi.org/10.1007/s00440-016-0725-1

**Random-cluster dynamics in Z ^{2} .** / Pimentel, Antonio Blanca; Sinclair, Alistair.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Random-cluster dynamics in Z2

AU - Pimentel, Antonio Blanca

AU - Sinclair, Alistair

PY - 2017/8/1

Y1 - 2017/8/1

N2 - 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 Z2. Our main result is a O(n2log 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 Z2. 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.

AB - 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 Z2. Our main result is a O(n2log 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 Z2. 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.

UR - http://www.scopus.com/inward/record.url?scp=84976351926&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976351926&partnerID=8YFLogxK

U2 - 10.1007/s00440-016-0725-1

DO - 10.1007/s00440-016-0725-1

M3 - Article

AN - SCOPUS:84976351926

VL - 168

SP - 821

EP - 847

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -

^{2}Probability Theory and Related Fields. 2017 Aug 1;168(3-4):821-847. https://doi.org/10.1007/s00440-016-0725-1