Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 214 / 1 / b / i

Put $q_k(n)={\mathbb{P}}_p((0,0)\leftrightarrow (n,0)\text{ inside }{T}_k)$, with $q_k(0)=1$. The connection event is increasing in the open edges. Apply the Harris-FKG inequality to connection from $(0,0)$ to $(n,0)$ and connection from $(n,0)$ to $(n+m,0)$, both constrained to the strip. Their intersection implies connection between the outer endpoints, while horizontal translation invariance makes the second event's probability $q_k(m)$. HenceThese events may depend on infinitely many strip edges. To justify the inequality directly, first restrict each connecting graph path to a finite box, apply Harris-FKG inequality in the finite independent edge family, and then let the box grow. Every connection has a finite witness graph path, so continuity from below gives the stated inequality.

The direct horizontal graph path supplies $q_k(n)\ge p^n>0$, and probabilities are at most one. Thus $a_k(n)=-\log q_k(n)$ is a finite nonnegative subadditive sequence with $a_k(n)\le -n\log p$. The Fekete lemma now gives a finite connection decay rate in a percolation strip:Each individual ratio is at least its infimum, so $-\log q_k(n)\ge n{f}_k(p)$ and thereforeAt $n=0$ this reads $1\le 1$. Only nonnegative separations are intended by this notation; horizontal reflection supplies the analogous bound with $|n|$ for negative separations.