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For nearest-neighbour bond percolation with $0<p<1$ on the strip $T_k=\mathbb{Z}\times \{-k,\dots ,k\}$, put $q_k(n)={\mathbb{P}}_p((0,0)\leftrightarrow (n,0)\text{ in }{T}_k)$. The Harris-FKG inequality and translation invariance imply $q_k(n+m)\ge q_k(n)q_k(m)$. Thus $-\log q_k(n)$ is a subadditive sequence, and the Fekete lemma givesThe direct horizontal graph path gives $0\le f_k(p)\le -\log p$. 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.

The entropy rate of $\xi$ isThe numerator is a subadditive sequence, so the limit exists and equals the infimum of the displayed ratios.

For a real subadditive sequence $(x_n{)}_{n\ge 1}$, the normalized sequence has the extended-real limit of a sequenceNo lower-bound hypothesis is needed for this conclusion. The limit of a sequence is finite precisely when the ratios are bounded below. The example $x_n=-n^2$ is a subadditive sequence with normalized limit of a sequence $-\infty$.

To prove the result, fix $k\ge 1$ and, for $n\ge k$, write $n=qk+r$ with $q\ge 1$ and $0\le r<k$. Iterating subadditivity gives $x_n\le q{x}_k+C_k$, where $C_k=\max (0,x_1,\dots ,x_{k-1})$ and $C_1=0$. Hence ${\mathrm{limsup}}_n{x}_n/n\le x_k/k$ for every $k$. Every ratio is at least their infimum. If that infimum is finite these bounds identify the limit of a sequence; if it is $-\infty$, choosing $k$ with an arbitrarily negative ratio gives convergence to $-\infty$.

Set $L={\inf }_{k\ge 1}{x}_k/k$. It may be $-\infty$, but cannot be $+\infty$ because $x_1$ is finite. Fix $k$ and write $n=qk+r$, $0\le r<k$. Repeated use of the subadditive sequence inequality giveswhere $C_1=0$. Dividing by $n$ and letting $n\to \infty$ gives $\mathrm{limsup}{x}_n/n\le x_k/k$. Since this holds for every $k$, the upper limit of a sequence is at most $L$, while every ratio is at least $L$. If $L$ is finite these bounds agree. If $L=-\infty$, choose $k$ with $x_k/k$ below any prescribed real number; the same upper-limit bound proves convergence to $-\infty$. This establishes the extended-real Fekete lemma:A finite real limit of a sequence requires a lower linear bound. For example $x_n=-n^2$ satisfies the premise but has normalized limit of a sequence $-\infty$, so that qualification cannot be dropped.

Write $\varphi ={\varphi }_{p,q}^1$ and $C_n=\varphi (0\leftrightarrow e_n)$. We use these standard facts about the infinite-volume wired random-cluster measure for $q\ge 1$: it is translation invariant and invariant under coordinate reflections/permutations; it has positive association of the random-cluster model; and every conditional edge-open probability is at leastThe last bound follows because an edge either leaves its component count unchanged on opening, giving probability $p$, or merges two components, giving $p/(p+q(1-p))$. These are standard infinite-volume properties of the limit obtained from the finite wired laws by weak convergence of probability measures. Positive association of random variables applies to connection events by increasing approximation with finite-path events.

Opening the $n$ edges of the straight segment gives $C_n\ge r^n$, so its logarithm is finite. The events $A=\{0\leftrightarrow e_m\}$ and $B=\{e_m\leftrightarrow e_{m+n}\}$ are increasing, and their intersection implies connection to $e_{m+n}$. ThusThe sequence $a_n=-\log C_n$ is a subadditive sequence, with $0\le a_n\le -n\log r$. Part (a) proves the random-cluster inverse correlation length exists and is finite:Since $a_n/n\ge \alpha$ for every $n$, exponentiating yields the requested bound

This is the Fekete lemma, with its possible extended-real limit made explicit. Put $L={\inf }_{m\ge 1}{x}_m/m$. Since $x_n\le n{x}_1$, $L$ cannot be $+\infty$; it may be $-\infty$.

Fix $m\ge 1$ and write $n=qm+r$, with $0\le r<m$. Repeated subadditivity giveswhere for this calculation set $x_0=0$. The finitely many possible remainder terms are bounded. Since $q/n\to 1/m$,This holds for every $m$, whereas $x_n/n\ge L$ by the definition of an infimum. If $L$ is finite, the upper and lower bounds agree. If $L=-\infty$, choosing $m$ with arbitrarily negative $x_m/m$ shows that $x_n/n\to -\infty$. ThusA finite real limit is not guaranteed: $x_n=-n^2$ is a subadditive sequence and has $x_n/n=-n\to -\infty$. If the limit is intended to be real rather than extended real, one must add a linear lower bound.

Split a self-avoiding walk of length $m+n$ after its first $m$ steps. Its prefix is one of the $c_m$ walks, and its translated suffix is a self-avoiding walk of length $n$ from the origin. The pair determines the original walk, although not every pair can be concatenated without intersection. ThereforeAll $c_n$ are positive, so $x_n=\log c_n$ is a subadditive sequence. The Fekete lemma gives existence of ${\lim }_n{n}^{-1}\log c_n$ and consequently of the connective constant.

For a lower bound, all words of length $n$ using only north and east steps produce distinct self-avoiding walks: the sum of the two coordinates increases at each step. Thus $c_n\ge 2^n$. For an upper bound, there are four choices for the first step and at most three thereafter, since a self-avoiding walk cannot immediately reverse its previous edge. Hence $c_n\le 43^{n-1}$. Taking $n$th roots provesThe elementary upper bound counts all non-backtracking walks, including some that self-intersect, so it is an upper estimate rather than an equality.

A self-avoiding walk is a finite sequence of graph vertices $v_0,\dots ,v_n$ in which successive vertices are adjacent and no graph vertex is repeated; its length is $n$, the number of edges. Include the zero-length walk.

Split a length-$m+n$ walk from $v$ into its first $m$ edges and remaining $n$ edges. After fixing the first segment, the number of possible suffixes is at most ${\sigma }_n$, since forgetting the prohibition on revisiting its earlier vertices can only increase that number. Thus ${\sigma }_{m+n}(v)\le {\sigma }_m(v){\sigma }_n$, and taking the supremum givesEvery finite-radius ball is finite because degrees are bounded. An infinite connected graph has vertices arbitrarily far from each root, so shortest paths give ${\sigma }_n(v)\ge 1$. Also ${\sigma }_n\le \Delta (\Delta -1{)}^{n-1}$ for $n\ge 1$: after the first edge, an immediate reversal is forbidden. In particular $\Delta \ge 2$ and these suprema are finite.

Apply the Fekete lemma: a real subadditive sequence $a_{m+n}\le a_m+a_n$ satisfies $\lim a_n/n={\inf }_{n\ge 1}{a}_n/n$, allowing negative infinity. Here $a_n=\log {\sigma }_n\ge 0$, so the limit is finite. The uniform connective constant of a bounded-degree graph is therefore

Use integer graph vertices in the boxes: ${\Lambda }_n=[-n,n{]}^2\cap {\mathbb{Z}}^2$, and set ${\Lambda }_{-1}=\varnothing$. Thus $\partial {\Lambda }_0=\{0\}$, $g_0=1$, and $|\partial {\Lambda }_m|=8m$ for $m\ge 1$.

If $0\leftrightarrow \partial {\Lambda }_{m+n}$, erase loops from an open graph path to obtain a self-avoiding walk. Let $x$ be its first graph vertex on $\partial {\Lambda }_m$. Split at $x$. The prefix certifies $A_x=\{0\leftrightarrow x\}$, while a segment of the suffix certifies $B_x=\{x\leftrightarrow x+\partial {\Lambda }_n\}$: the final graph vertex is at supremum norm distance at least $n$ from $x$, so the suffix first hits that translated boundary. The two certificates use disjoint edges. Therefore the BK boundary-splitting estimate, the union bound, the Van den Berg-Kesten inequality, and translation invariance giveThe cases $m=0$ or $n=0$ follow directly from $g_0=1$, so this proves the bound including the endpoints.

Here is an explicit way to remove the polynomial boundary factor. For $p>0$ put $b_n=32n^2{g}_n$, $n\ge 1$. By symmetry in $m,n$, assume $m\le n$. ThenConsequently $a_n=\log b_n$ is a subadditive sequence. The Fekete lemma states that any real subadditive sequence satisfies $\lim a_n/n={\inf }_{n\ge 1}{a}_n/n$, possibly $-\infty$. In this case the direct horizontal open graph path gives $g_n\ge p^n$, so this limit of a sequence is bounded below by $\log p$. Since $g_n\le 1$, its upper bound is zero. Finally,has the same limit of a sequence. Thus the percolation one-arm decay rate exists andFor $p=0$, every $g_n$ with $n\ge 1$ is zero and $\gamma =0$. No exponential-decay theorem or assumption that $p$ is below the percolation critical probability is needed.

On a probability measure-preserving system, for a finite measurable partition $\xi =\{A_1,\dots ,A_r\}$, the entropy of a finite measurable partition isUse natural logarithms, so information entropy is measured in nats; another fixed logarithm base rescales all answers. The join of measurable partitions is their common refinement, and write ${\xi }_a^b={\bigvee }_{j=a}^b{T}^{-j}\xi$, with an empty join the trivial partition. The entropy rate of a measurable partition and Kolmogorov-Sinai entropy are respectivelyThe block entropies form a subadditive sequence, so the first limit exists by the Fekete lemma. For finite partitions the conditional entropy of finite measurable partitions is $H(\eta \mid \zeta )=H(\eta \vee \zeta )-H(\zeta )$.

Put $c_0=H(\xi )$ and $c_k=H(\xi \mid {\xi }_1^k)$ for $k\ge 1$. The chain rule for information entropy, applied from the last coordinate backwards, and measure preservation giveThe second equality uses invariance of the joint partition atom probabilities under the common pullback $T^{-j}$; invertibility is unnecessary. Since conditioning reduces entropy, $c_k$ decreases to a nonnegative limit $c$. The Cesaro convergence of a sequence of this convergent sequence has the same limit. ThereforeEquivalently $h_{\mu }(T,\xi )=H_{\mu }(\xi \mid {\mathcal{F}}_1)$, where ${\mathcal{F}}_1=\sigma ({\bigvee }_{j\ge 1}{T}^{-j}\xi )$. Here conditional entropy of a countable measurable partition conditioned on a sigma-algebra is computed using conditional partition atom probabilities; the martingale convergence theorem gives continuity under increasing conditioning sigma-algebras. This is the infinite-future formula for partition entropy rate.

The Kolmogorov-Sinai generator theorem states that if a finite or countable measurable partition $\eta$ has finite entropy of a countable measurable partition and its iterates generate the whole completed sigma-algebra modulo null sets, then $h_{\mu }(T)=h_{\mu }(T,\eta )$. For an invertible system, generating means $\sigma ({\bigvee }_{j\in \mathbb{Z}}{T}^{-j}\eta )=\mathcal{B}$ modulo null sets. For a noninvertible system a one-sided generator, using $j\ge 0$, suffices. The two-sided and one-sided versions must not be confused.

For a Bernoulli shift with discrete symbol probabilities $(p_i)$, the coordinate-zero measurable partition has independent coordinate iterates. ThusFor a finite alphabet the coordinate partition is a generator of finite entropy of a finite measurable partition; on the two-sided sequence space use all integer coordinate iterates, and on the one-sided space use the nonnegative ones. The Kolmogorov-Sinai generator theorem proves the displayed answer in both cases. The same calculation applies to countably many symbols when their Shannon entropy is finite, using the countable finite-entropy version of the theorem. If the Shannon entropy is infinite, merge all but the first $r$ symbols into one cell. These finite coordinate partitions have entropy rate $-{\sum }_{i\le r}{p}_i\log p_i-p_{>r}\log p_{>r}\to \infty$, so the system entropy is infinite. Zero-probability symbols contribute zero. In particular a fair $r$-symbol shift has entropy $\log r$.

For the final assertion, let ${\mathcal{F}}_0=\sigma ({\bigvee }_{j\ge 0}{T}^{-j}\xi )$ and complete it modulo null sets. The approximation property forces ${\mathcal{F}}_0=\mathcal{B}$ modulo null sets. Indeed for each $A\in \mathcal{B}$, choose approximating sets from finite blocks with error tending to zero. Their indicators approach $1_A$ in $L^2$, and $L^2({\mathcal{F}}_0)$ is a closed vector subspace, so $A$ is ${\mathcal{F}}_0$-measurable modulo a null set.

Invertibility now gives ${\mathcal{F}}_1=T^{-1}{\mathcal{F}}_0=T^{-1}\mathcal{B}=\mathcal{B}$ modulo null sets. In particular the present partition $\xi$ is measurable with respect to its entire future, so $H(\xi \mid {\mathcal{F}}_1)=0$. The infinite-future formula gives $h_{\mu }(T,\xi )=0$. Since the given one-sided generator is also a two-sided generator, the Kolmogorov-Sinai generator theorem finishes the proof:This is finite one-sided generator of an invertible system forces zero entropy. Invertibility is essential: a fair binary one-sided Bernoulli shift has a finite one-sided generator and Kolmogorov-Sinai entropy $\log 2$.

Let $\alpha ={\inf }_{k\ge 1}{x}_k/k$. The relevant version of the Fekete lemma is the extended-real Fekete lemma: $\alpha$ may be $-\infty$, but cannot be $+\infty$ because $x_1$ is real. We show that this is exactly the limit of a sequence.

Fix $k\ge 1$ and, for $n\ge k$, write $n=qk+r$, where $q\ge 1$ and $0\le r<k$. Iterating the defining inequality gives $x_{qk}\le q{x}_k$. If $r>0$, it also gives $x_n\le q{x}_k+x_r$; if $r=0$, there is no remainder term. Thus, with $C_k=\max (0,x_1,\dots ,x_{k-1})$ and $C_1=0$,As $n\to \infty$, $q/n\to 1/k$, so ${\mathrm{limsup}}_n{x}_n/n\le x_k/k$. This holds for every positive $k$. On the other hand every $x_n/n\ge \alpha$. If $\alpha$ is finite, these two bounds give the desired equality. If $\alpha =-\infty$, for every real $M$ choose $k$ with $x_k/k<M-1$; the same upper bound makes $x_n/n<M$ for all sufficiently large $n$. ThereforeThe index $k=0$ is excluded because its ratio is undefined, and $x_0$ is irrelevant since the defining inequality was required only at positive indices. If the printed word “limit of a sequence” is interpreted as a finite real limit of a sequence, an additional lower bound is necessary: $x_n=-n^2$ defines a subadditive sequence but $x_n/n=-n\to -\infty$. The extended-real statement is the correct unrestricted conclusion.

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.

For a finite measurable partition $\xi$,Its conditional entropy of finite measurable partitions relative to $\eta$ isThe concavity of $-t\log t$, equivalently conditioning reduces entropy, gives

The atoms of $T^{-1}\xi$ are $T^{-1}A$ with the same measures as the atoms $A$ of $\xi$. Hence $\boxed{H_{\mu }(T^{-1}\xi )=H_{\mu }(\xi )}$.

SetThe chain rule and invariance give $a_{N+M}\le a_N+a_M$, so $(a_N)$ is a subadditive sequence. ThereforeThe Kolmogorov-Sinai entropy is $h_{\mu }(T)={\sup }_{\xi }{h}_{\mu }(T,\xi )$ over finite partitions.

Taking $F=\{0,\dots ,N-1\}$ immediately shows that the infimum over arbitrary finite $F$ is at most $h_{\mu }(T,\xi )$. For the reverse inequality, apply Shearer's inequality to translates of a fixed finite $F$ inside a long interval. Every interior coordinate is covered $|F|$ times, while only $O(\max F)$ boundary coordinates are lost. Subadditivity bounds the boundary contribution; division by the interval length and passage to the limit giveTaking the infimum proves

LetThe events $\{0\leftrightarrow e_n\}$ and $\{e_n\leftrightarrow e_{n+m}\}$ are increasing events of bond percolation. The FKG inequality and translation invariance giveThus $(a_n)$ is a subadditive sequence. Since every probability is positive for $p>0$, Fekete lemma applies and gives