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

For a finite graph $G=(V,E)$, the $q=2$ random-cluster model iswhere $o(\omega )$ is the number of open edges and $k(\omega )$ the number of open connected components. In the Edwards-Sokal coupling, first sample $\omega$, assign an independent uniform spin $\pm 1$ to each open cluster, and give every vertex its cluster's spin. The resulting spin law is the Ising model withConversely, from an Ising configuration, close every edge joining unequal spins and independently open each edge joining equal spins with probability $p$.