## Abstract

The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrized by an edge probability p and a cluster weight q. Our focus is on the critical regime: p = pc(q) and q ∈ (1, 2), where pc(q) is the threshold corresponding to the order-disorder phase transition of the model. We show that the mixing time of the Chayes-Machta dynamics is O(log n · log log n) in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the log log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021 |

Editors | Mary Wootters, Laura Sanita |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772075 |

DOIs | |

State | Published - Sep 1 2021 |

Event | 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021 - Virtual, Seattle, United States Duration: Aug 16 2021 → Aug 18 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 207 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021 |
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Country/Territory | United States |

City | Virtual, Seattle |

Period | 8/16/21 → 8/18/21 |

## All Science Journal Classification (ASJC) codes

- Software