Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 3 / b / Solution

Let $A_n$ be the increasing cylinder event that an open path joins $0$ to the boundary of the box $[-n,n{]}^d$. If two edge-disjoint open paths run from $0$ to infinity, then $A_n\square A_n$ occurs for every $n$. Therefore the van den Berg-Kesten inequality giveswhere continuity from above identifies ${\lim }_n{\mathbb{P}}_p(A_n)=\theta (p)$.