Abstract
The randomcluster (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 wellknown that in the hightemperature region beta<beta_c(q) of the qstate Ising/Potts model on an n x n box Lambda_n of the integer lattice Z^2, spin correlations decay exponentially fast; this property holds even arbitrarily close to the boundary of Lambda_n and uniformly over all boundary conditions. A direct consequence of this property is that the corresponding singlesite update Markov chain, known as the Glauber dynamics, mixes in optimal O(n^2 log{n}) steps on Lambda_{n} for all choices of boundary conditions. We study the effect of boundary conditions on the FKdynamics, the analogous Glauber dynamics for the randomcluster model.
On Lambda_n the randomcluster model with parameters (p,q) has a sharp phase transition at p = p_c(q). Unlike the Ising/Potts model, the randomcluster model has nonlocal interactions which can be forced by boundary conditions: external wirings of boundary vertices of Lambda_n. We consider the broad and natural class of boundary conditions that are realizable as a configuration on Z^2 \ Lambda_n. Such boundary conditions can have many macroscopic wirings and impose longrange correlations even at very high temperatures (p << p_c(q)). In this paper, we prove that when q>1 and p != p_c(q) the mixing time of the FKdynamics is polynomial in n for every realizable boundary condition. Previously, for boundary conditions that do not carry longrange information (namely wired and free), Blanca and Sinclair (2017) had proved that the FKdynamics 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 nonrealizable boundary conditions that induce slow (stretchedexponential) convergence at high temperatures.
BibTeX  Entry
@InProceedings{blanca_et_al:LIPIcs:2019:11282,
author = {Antonio Blanca and Reza Gheissari and Eric Vigoda},
title = {{RandomCluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages = {67:167:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771252},
ISSN = {18688969},
year = {2019},
volume = {145},
editor = {Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11282},
URN = {urn:nbn:de:0030drops112827},
doi = {10.4230/LIPIcs.APPROXRANDOM.2019.67},
annote = {Keywords: Markov chain, mixing time, randomcluster model, Glauber dynamics, spatial mixing}
}
Keywords: 

Markov chain, mixing time, randomcluster model, Glauber dynamics, spatial mixing 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019) 
Issue Date: 

2019 
Date of publication: 

17.09.2019 