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

Andrea Collevecchio, Eren Metin Elçi, Timothy M. Garoni, Martin Weigel

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    2 Citations (Scopus)
    34 Downloads (Pure)


    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.

    Original languageEnglish
    Pages (from-to)22-61
    Number of pages40
    JournalJournal of Statistical Physics
    Issue number1
    Early online date10 Nov 2017
    Publication statusPublished - 1 Jan 2018

    Bibliographical note

    The final publication is available at Springer via http://dx.doi.org/10.1007/s10955-017-1912-x


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

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematical Physics


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