For $j=0,\ldots,k-1$, let $E_j$ be the increasing event that $ju\longleftrightarrow(j+1)u$. Their intersection implies $0\longleftrightarrow ku$. By translation invariance and the <Harris-FKG inequality>,
$$
\mathbb P_p(0\longleftrightarrow ku)
\geq\mathbb P_p\left(\bigcap_{j=0}^{k-1}E_j\right)
\geq\prod_{j=0}^{k-1}\mathbb P_p(E_j)
=\mathbb P_p(0\longleftrightarrow u)^k.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 212 / 4 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-212/4/c/solution