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We consider the coupling from the past implementation of the random-clusterheat-bath process, and study its random running time, or coupling time. Wefocus on hypercubic lattices embedded on tori, in dimensions one to three, withcluster fugacity at least one. We make a number of conjectures regarding theasymptotic behaviour of the coupling time, motivated by rigorous results in onedimension and Monte Carlo simulations in dimensions two and three. Amongst ourfindings, we observe that, for generic parameter values, the distribution ofthe appropriately standardized coupling time converges to a Gumbeldistribution, and that the standard deviation of the coupling time isasymptotic 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 alsoat the critical point, provided the cluster fugacity is below the value atwhich the transition becomes discontinuous. Finally, we consider analogousquestions for the single-spin Ising heat-bath process.

On the coupling time of the heat-bath process for the Fortuin-Kasteleyn random-cluster model