## Abstract

We study the Glauber dynamics for the random cluster (FK) model on the torus (Formula presented.) with parameters (p,q), for q ∈ (1,4] and p the critical point p_{c}. The dynamics is believed to undergo a critical slowdown, with its continuous-time mixing time transitioning from (Formula presented.) for p ≠ p_{c} to a power-law in n at p = p_{c}. This was verified at p ≠ p_{c} by Blanca and Sinclair, whereas at the critical p = p_{c}, with the exception of the special integer points q = 2,3,4 (where the model corresponds to the Ising/Potts models) the best-known upper bound on mixing was exponential in n. Here we prove an upper bound of (Formula presented.) at p = p_{c} for all q ∈ (1,4], where a key ingredient is bounding the number of nested long-range crossings at criticality.

Original language | English (US) |
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Pages (from-to) | 517-556 |

Number of pages | 40 |

Journal | Random Structures and Algorithms |

Volume | 56 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2020 |

## Keywords

- Glauber dynamics
- critical phenomena
- mixing time
- random cluster model

## ASJC Scopus subject areas

- Software
- General Mathematics
- Computer Graphics and Computer-Aided Design
- Applied Mathematics