## Abstract

The random-cluster (FK) model is a key tool for the study of phase transitions and for the design of efficient Markov chain Monte Carlo (MCMC) sampling algorithms for the Ising/Potts model. It is well-known that in the high-temperature region β < βc(q) of the q-state Ising/Potts model on an n × n box Λn of the integer lattice ℤ^{2}, spin correlations decay exponentially fast; this property holds even arbitrarily close to the boundary of Λn and uniformly over all boundary conditions. A direct consequence of this property is that the corresponding single-site update Markov chain, known as the Glauber dynamics, mixes in optimal O(n^{2} log n) steps on Λn for all choices of boundary conditions. We study the effect of boundary conditions on the FK-dynamics, the analogous Glauber dynamics for the random-cluster model. On Λn the random-cluster model with parameters (p, q) has a sharp phase transition at p = pc(q). Unlike the Ising/Potts model, the random-cluster model has non-local interactions which can be forced by boundary conditions: external wirings of boundary vertices of Λn. We consider the broad and natural class of boundary conditions that are realizable as a configuration on ℤ^{2} \ Λn. Such boundary conditions can have many macroscopic wirings and impose long-range correlations even at very high temperatures (p ≪ pc(q)). In this paper, we prove that when q > 1 and p 6= pc(q) the mixing time of the FK-dynamics is polynomial in n for every realizable boundary condition. Previously, for boundary conditions that do not carry long-range information (namely wired and free), Blanca and Sinclair (2017) had proved that the FK-dynamics in the same setting mixes in optimal O(n^{2} log n) time. To illustrate the difficulties introduced by general boundary conditions, we also construct a class of non-realizable boundary conditions that induce slow (stretched-exponential) convergence at high temperatures.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019 |

Editors | Dimitris Achlioptas, Laszlo A. Vegh |

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

ISBN (Electronic) | 9783959771252 |

DOIs | |

State | Published - Sep 2019 |

Event | 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 - Cambridge, United States Duration: Sep 20 2019 → Sep 22 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 145 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 |
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Country | United States |

City | Cambridge |

Period | 9/20/19 → 9/22/19 |

## All Science Journal Classification (ASJC) codes

- Software