Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 1 / c2

Fix the finite set $S=\{x_1,\dots ,x_n\}\subset B$. If an infinite half-space cluster met $B$ exactly in $S$ with positive probability, take independent reflected occurrences in $H$ and $H^{\prime }$ and force the finitely many intervening sites adjacent to $S$ closed. The finite-energy property of Bernoulli percolation gives this combined event positive probability, while it creates two distinct whole-space infinite clusters, contradicting uniqueness. Thus the probability is zero for every finite $S$. Since $B$ has only countably many finite subsets, almost surely every infinite cluster that meets $B$ meets it infinitely often.