Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / a4 / Solution

On $A_N$, additionally require every site of $h_{N+1}$ to be open and every site of $h_{N+2}$ to be closed. The open ring joins every component of $C_N$ to the origin component, and the closed outer ring makes that component finite. It contains at least $\theta \#H_N/2$ sites. These ring states are independent of $A_N$, and their probability isSince both ring sizes are linear in $N$, this is at least $c(p)e^{-u(p)N}$. Multiplication by $\mathbb{P}(A_N)\ge {\theta }^2/2$ proves the claimed lower bound.