### 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 |

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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### 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.

}

^{2}. in R Krauthgamer (ed.),

*27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016.*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 1, Association for Computing Machinery, pp. 498-513, 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, United States, 1/10/16.

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

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

TY - GEN

T1 - Random-cluster dynamics in ℤ2

AU - Pimentel, Antonio Blanca

AU - Sinclair, Alistair

PY - 2016/1/1

Y1 - 2016/1/1

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(n2 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 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(n2 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 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=84962856959&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84962856959&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84962856959

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 498

EP - 513

BT - 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016

A2 - Krauthgamer, Robert

PB - Association for Computing Machinery

ER -

^{2}. In Krauthgamer R, editor, 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016. Association for Computing Machinery. 2016. p. 498-513. (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).