= Solution
The strips are nested. Every connection permitted inside $T_k$ is also permitted inside $T_{k+1}$, so $q_k(n)\leq q_{k+1}(n)$ for every fixed separation. Consequently
$$
f_{k+1}(p)=\inf_{n\geq1}\frac{-\log q_{k+1}(n)}n\leq\inf_{n\geq1}\frac{-\log q_k(n)}n=f_k(p).
$$
The preceding nonnegative bound makes this a decreasing <sequence> bounded below. By monotone convergence of real <sequences>,
$$
\boxed{\lim_{k\to\infty}f_k(p)=\inf_{k\geq1}f_k(p)\in[0,-\log p].}
$$
There is no assertion that this <limit of a sequence> is strictly positive: positivity at each fixed width need not survive an increasing-width <limit of a sequence>.
One can also identify the <limit of a sequence>. Let $q(n)$ be the <percolation two-point connection probability> in the whole <square lattice>. Every finite connecting <graph path> has a bounded vertical extent, so $q_k(n)\uparrow q(n)$. Commuting infima gives the <strip approximation to the planar connection decay rate>:
$$
\inf_k f_k(p)=\inf_k\inf_{n\geq1}\frac{-\log q_k(n)}n=\inf_{n\geq1}\inf_k\frac{-\log q_k(n)}n=\inf_{n\geq1}\frac{-\log q(n)}n.
$$
The same positive-association argument identifies the final <infimum> with the whole-plane normalized logarithmic <limit of a sequence>. This is an interchange of infima justified by monotonicity at fixed $n$. For example, when $p>1/2$, the threshold proved in question 2 and uniqueness give an <infinite percolation cluster> with root <probability> $\theta(p)>0$. The <Harris-FKG inequality> makes the <probability> that both endpoints belong to it at least $\theta(p)^2$, so $q(n)\geq\theta(p)^2$. The whole-plane rate, and hence $\lim_k f_k(p)$, is then zero, despite the strict positivity of every finite-strip rate.
Back to article page
