Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions

Authors Antonio Blanca, Reza Gheissari, Eric Vigoda

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Antonio Blanca
  • Department of Computer Science and Engineering, Pennsylvania State University, USA
Reza Gheissari
  • Courant Institute of Mathematical Sciences, New York University, USA
Eric Vigoda
  • School of Computer Science, Georgia Institute of Technology, USA


The authors thank the anonymous referees for their helpful suggestions.

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Antonio Blanca, Reza Gheissari, and Eric Vigoda. Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 beta<beta_c(q) of the q-state 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 single-site 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 FK-dynamics, the analogous Glauber dynamics for the random-cluster model. On Lambda_n the random-cluster model with parameters (p,q) has a sharp phase transition at p = p_c(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 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 long-range 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 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov-chain Monte Carlo convergence measures
  • Theory of computation → Random walks and Markov chains
  • Markov chain
  • mixing time
  • random-cluster model
  • Glauber dynamics
  • spatial mixing


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