Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 1 / b / ii

By the union bound,Hence some $z\in \partial B(n)$ has connection probability at least the left side divided by $|\partial B(n)|$. Some coordinate of $z$ equals $n$ or $-n$. A coordinate permutation and reflection of ${\mathbb{Z}}^d$ sends that face to the face $x_1=n$ and preserves the bond percolation law. Its image $x$ therefore satisfies