= Connection decay rate in a percolation strip
{title2=$f_k(p)$}
For nearest-neighbour <bond percolation> with $0<p<1$ on the strip $T_k=\mathbb Z\times\{-k,\ldots,k\}$, put $q_k(n)=\mathbb P_p((0,0)\leftrightarrow(n,0)\text{ in }T_k)$. The <Harris-FKG inequality> and translation invariance imply $q_k(n+m)\geq q_k(n)q_k(m)$. Thus $-\log q_k(n)$ is a <subadditive sequence>, and the <Fekete lemma> gives
$$
f_k(p)=\inf_{n\geq1}-\frac1n\log q_k(n),\qquad q_k(n)\leq e^{-nf_k(p)}.
$$
The direct horizontal <graph path> gives $0\leq f_k(p)\leq-\log p$. Enlarging the strip increases every <percolation two-point connection probability>, so $f_k(p)$ is nonincreasing in $k$ and has a nonnegative <limit of a sequence>.
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