Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 1 / c1

Use independent percolation configurations in $H$ and in its reflected copy $H^{\prime }=(-\infty ,-2]\times {\mathbb{Z}}^{d-1}$. Require event $A$ in both copies and close the intervening site $(-1,0,\dots ,0)$. This has probability $(1-p)Q^2$. Because either distinguished infinite cluster meets its boundary hyperplane only at its distinguished origin, the closed intervening site prevents it from leaving its own half-space. The resulting whole-space configuration therefore has two distinct infinite clusters. Whole-space supercritical Bernoulli percolation has an almost surely unique infinite cluster, so $(1-p)Q^2=0$ and hence $Q=0$.