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

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 $p$.

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

The conditional open probability in part (b) is increasing in the states of all other edges, because adding edges can only connect the two endpoints. A common-uniform update therefore preserves the coordinatewise order. Start two copies from all closed and use the same updates, with the second copy additionally conditioned through an increasing event $B$ by the corresponding monotone censored heat-bath chain. The monotone grand coupling and convergence to stationarity show that conditioning on $B$ stochastically increases the configuration. Hence for increasing $A$,which is the Harris-FKG inequality $\varphi (A\cap B)\ge \varphi (A)\varphi (B)$. Applying this to the complement of a decreasing event $B^{\prime }$ gives

Couple the free-boundary random-cluster heat-bath chains in ${\Lambda }_N$ and ${\Lambda }_{N+1}$. Restricted to common edges, the larger box has extra possible open paths through its outer annulus. These can only turn a disconnected-endpoint update into a connected-endpoint update and raise its open probability from $p/(2-p)$ to $p$. The monotone coupling therefore giveson common increasing events. Consequently $P_N(A)$ is nondecreasing and, being bounded by one, has a limit.

The stochastic bounds from part (b) hold uniformly in every box. Passing to the increasing limit for the local increasing event $A$ givesThus the infinite-volume free random-cluster measure lies stochastically between Bernoulli bond percolation at parameters $p/(2-p)$ and $p$.