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The strips are nested. Every connection permitted inside $T_k$ is also permitted inside $T_{k+1}$, so $q_k(n)\le q_{k+1}(n)$ for every fixed separation. ConsequentlyThe preceding nonnegative bound makes this a decreasing sequence bounded below. By monotone convergence of real sequences,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: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 $q(n)\ge \theta (p{)}^2$. The whole-plane rate, and hence ${\lim }_k{f}_k(p)$, is then zero, despite the strict positivity of every finite-strip rate.

For bond percolation on the square lattice, the one-arm probability $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$ has a root limit of a sequence $\gamma (p)\in [p,1]$. The BK boundary-splitting estimate implies that $32n^2{g}_n$ is submultiplicative for positive integers. Applying the Fekete lemma to its logarithm proves existence when $p>0$; at $p=0$ the rate is zero. The same rate is the root limit of the percolation two-point connection probability along a coordinate axis, by the reflection lower bound for two-point percolation.

On a connected locally finite recurrent graph, root Wilson's algorithm at a fixed graph vertex $r$ and process a countable enumeration of the graph vertices. Every walk hits the existing tree almost surely, since it contains $r$. Each attached graph path is finite, and the union is a connected acyclic spanning subgraph. Its law is the infinite uniform spanning tree.

It is also the common free and wired finite-volume limit of a sequence along every graph exhaustion. For a finite edge set, run a finite initial segment of the chosen enumeration containing all its endpoints. Its membership is then permanently decided, because later attachments have new internal graph vertices. Only finitely many finite random-walk trajectories were used, and their graph vertices and graph neighbours fit inside all sufficiently large exhaustion sets. Coupling then gives convergence of those finite-dimensional edge laws for both boundary conventions. Finite Wilson's algorithm is root/order independent, so the limiting law is too. There is no uniform counting measure on all infinite spanning trees; the definition is this limiting probability law.