Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 3 / b

Use the single-edge heat-bath Markov chain: choose an edge uniformly and resample it from its conditional random-cluster law. If its endpoints are already connected without that edge, its conditional open probability is $p$; otherwise opening it merges two components and the probability isDetailed balance makes ${\varphi }_{p,2}$ stationary. Driving this chain and Bernoulli heat-bath chains by the same update edges and uniforms gives the stochastic domination