By the <union bound>,
$$
\mathbb P_p(0\leftrightarrow\partial B(n))
\leq\sum_{z\in\partial B(n)}
\mathbb P_p(0\leftrightarrow z).
$$
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
$$
\boxed{\mathbb P_p(0\leftrightarrow x)
\geq\frac{\mathbb P_p(0\leftrightarrow\partial B(n))}{|\partial B(n)|}.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-214/1/b/ii/solution