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

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