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

## TL;DR

This paper investigates the asymptotic behavior of the coupling time in the heat-bath process for the Fortuin-Kasteleyn random-cluster model on lattices, revealing universal distributional limits and relationships with relaxation time.

## Contribution

It provides new conjectures and empirical evidence on the distribution and scaling of coupling times, including universality and behavior at criticality.

## Key findings

- Coupling time distribution converges to Gumbel distribution.
- Standard deviation of coupling time scales with relaxation time.
- Results hold both off criticality and below the discontinuous transition point.

## Abstract

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.