Random-cluster dynamics in ℤ2: Rapid mixing with general boundary conditions

Antonio Blanca, Reza Gheissari, Eric Vigoda

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 ℤ2, 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(n2 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 ℤ2 \ 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(n2 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 languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
EditorsDimitris Achlioptas, Laszlo A. Vegh
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771252
DOIs
StatePublished - Sep 2019
Event22nd 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 2019Sep 22 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume145
ISSN (Print)1868-8969

Conference

Conference22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
CountryUnited States
CityCambridge
Period9/20/199/22/19

All Science Journal Classification (ASJC) codes

  • Software

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