Journal of Statistical Physics

, Volume 170, Issue 1, pp 22–61 | Cite as

On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model

  • Andrea Collevecchio
  • Eren Metin Elçi
  • Timothy M. GaroniEmail author
  • Martin Weigel


We consider the coupling from the past implementation of the random–cluster heat-bath process, and study its random running time, or coupling time. We focus on hypercubic lattices embedded on tori, in dimensions one to three, with cluster fugacity at least one. We make a number of conjectures regarding the asymptotic behaviour of the coupling time, motivated by rigorous results in one dimension and Monte Carlo simulations in dimensions two and three. Amongst our findings, we observe that, for generic parameter values, the distribution of the appropriately standardized coupling time converges to a Gumbel distribution, and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time. Perhaps surprisingly, we observe these results to hold both off criticality, where the coupling time closely mimics the coupon collector’s problem, and also at the critical point, provided the cluster fugacity is below the value at which the transition becomes discontinuous. Finally, we consider analogous questions for the single-spin Ising heat-bath process.


Coupling from the past Relaxation time Random–cluster model Markov-chain Monte Carlo 



The authors thank Youjin Deng, Alan Sokal, and Ulli Wolff for useful discussions, and an anonymous referee for helpful comments. This work was supported under the Australian Research Council’s Discovery Projects funding scheme (project numbers DP140100559 & DP110101141), and T.G. is the recipient of an Australian Research Council Future Fellowship (project number FT100100494). A.C. would like to thank STREP project MATHEMACS. The work of EE and MW was partially supported by the European Commission through the IRSES network DIONICOS (PIRSES-GA-2013-612707).


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Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia
  2. 2.ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical SciencesMonash UniversityClaytonAustralia
  3. 3.Applied Mathematics Research CentreCoventry UniversityCoventryUnited Kingdom

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