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

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.