Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 1 / d1 / Solution

By part (c2), $C\cap B_{n_0+1}$ is infinite almost surely. For each $x$ in this intersection, its neighbor $x-e_1\in B_{n_0}$ is open independently with probability $p$. The probability that all infinitely many such neighbors are closed is zero. Hence some open site of $B_{n_0}$ is adjacent to $C$, and the open cluster $C^{\prime }$ containing their union in $H_{n_0}$ is infinite, contains $C$, and intersects $B_{n_0}$.