Solution
= Solution
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.