License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.67
URN: urn:nbn:de:0030-drops-112827
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Blanca, Antonio ; Gheissari, Reza ; Vigoda, Eric

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

LIPIcs-APPROX-RANDOM-2019-67.pdf (0.6 MB)


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.

BibTeX - Entry

  author =	{Antonio Blanca and Reza Gheissari and Eric Vigoda},
  title =	{{Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{67:1--67:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-112827},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.67},
  annote =	{Keywords: Markov chain, mixing time, random-cluster model, Glauber dynamics, spatial mixing}

Keywords: Markov chain, mixing time, random-cluster 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

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