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For bond percolation on the square lattice, put $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$. Splitting an open self-avoiding walk at its first visit to the radius-$m$ boundary gives disjoint certificates for reaching a boundary graph vertex and then reaching radius $n$ about that graph vertex. The Van den Berg-Kesten inequality and the union bound give $g_{m+n}\le 8m{g}_m{g}_n$ for positive $m,n$. With $g_0=1$ and $|\partial {\Lambda }_0|=1$, the boundary-cardinality form holds for zero indices as well.

A finite open coordinate set witnesses an increasing event when prescribing those coordinates open forces the event, regardless of the remaining configuration. The disjoint occurrence of increasing events $A\square B$ means that $A$ and $B$ have disjoint finite witnesses. For finite-coordinate events this is the usual disjoint-occurrence definition. The Van den Berg-Kesten inequality bounds its probability by $\mathbb{P}(A)\mathbb{P}(B)$ under an independent Bernoulli distribution product measure.

In bond percolation on the square lattice, every nearest-neighbour edge independently has state one, meaning open, with probability $p$, and state zero, meaning closed, with probability $1-p$. Thus the configuration law is the product measure ${\mathbb{P}}_p={⨂}_{e\in E}\mathrm{Bernoulli}(p)$. An open path in bond percolation uses only open edges. An increasing event $A$ satisfies $\omega \in A,{\omega }^{\prime }\ge \omega \Rightarrow {\omega }^{\prime }\in A$: opening further edges cannot destroy the event.

The Harris-FKG inequality for two increasing events isFor disjoint occurrence of increasing events, write $A\square B$ when disjoint sets of open edges separately witness $A$ and $B$, so each event is forced by its own witness regardless of the other edges. The Van den Berg-Kesten inequality statesFor events given by cylinder sets these are finite-coordinate assertions. Connection events have finite open-path witnesses; truncating to finite boxes and taking increasing limits gives the same disjoint-occurrence bound for them. Sharing a vertex between the two witness paths is permitted; sharing an edge is not.

For $1\le m<n$, take a simple open path from the origin to the radius-$n$ boundary and let $x$ be its first vertex on the radius-$m$ boundary. Its initial segment witnesses $0\leftrightarrow x$. If $z$ is its final vertex, then $\Vert z-x{\Vert }_{\infty }\ge n-m$. Consequently the remaining segment reaches $x+\partial {\Lambda }_{n-m}$, and its initial portion until that visit witnesses $x\leftrightarrow x+\partial {\Lambda }_{n-m}$. The two segments use disjoint edges. HenceThe union bound, Van den Berg-Kesten inequality and translation invariance of the product measure now give the weighted BK boundary-splitting estimateFor $m=n$ the same bound follows directly from the union bound, with $g_0=1$. With the conventional $\partial {\Lambda }_0=\{0\}$, $m=0$ gives equality.

Write $C(0)$ for the percolation cluster of the origin. By summing its membership indicators, the percolation susceptibility isIf it is finite, then $a_m\to 0$. For $0<p<1$, choose $m$ with $0<a_m=a<1$. Iterating the boundary-splitting estimate with this fixed $m$ givesFor $k\ge 2m$, $\lfloor k/m\rfloor \ge k/(2m)$, so $g_k\le \exp [-(-\log a)k/(2m)]$. To handle all smaller positive radii without an unspecified prefactor, observe that $g_k\le g_1=1-(1-p{)}^4<1$. ThereforeFor $1\le k<2m$, the second chosen rate gives $e^{-\gamma k}\ge g_1$, and for larger $k$ the first rate applies. At $k=0$ both sides equal one. The endpoint $p=0$ has $g_k=0$ for every $k\ge 1$, so any positive rate works; $p=1$ is excluded by finite susceptibility. This proves exponential one-arm decay from finite susceptibility with the required unit prefactor.

We use the Van den Berg-Kesten inequality: for increasing events $A,B$ under independent bond percolation, their disjoint occurrence $A\square B$, meaning that they have disjoint sets of open edges witnessing the two events, satisfies ${\mathbb{P}}_p(A\square B)\le {\mathbb{P}}_p(A){\mathbb{P}}_p(B)$. The finite-edge statement applies here; connection to a finite-box boundary can always be witnessed before the first exit from that box.

On the event $0\leftrightarrow \partial {\Lambda }_{m+n}$ choose a simple open path to that boundary, and let $y$ be its first point on $\partial {\Lambda }_m$. Its initial segment witnesses $0\leftrightarrow \partial {\Lambda }_m$ through $y$, inside ${\Lambda }_m$. The remaining segment eventually reaches maximum-norm distance at least $n$ from $y$, because its endpoint has norm $m+n$ while $\Vert y{\Vert }_{\infty }=m$. Stop that segment when it first reaches $y+\partial {\Lambda }_n$. These two segments use disjoint edges.

For fixed $y$, let $A_y$ be connection from zero to $y$ inside ${\Lambda }_m$, and $B_y$ be connection from $y$ to $y+\partial {\Lambda }_n$ inside $y+{\Lambda }_n$. Then ${\mathbb{P}}_p(A_y)\le {\beta }_m$ and, by translation invariance, ${\mathbb{P}}_p(B_y)={\beta }_n$. The union bound and the stated disjoint-occurrence inequality give

Set $x_n=\log {\beta }_n$ and ${\alpha }_n=\log |\partial {\Lambda }_n|$. The previous part applies, sinceMoreover a straight path of $n$ open bonds has probability $p^n$, so $p^n\le {\beta }_n\le 1$. Thus the limit is finite andEquivalently, for the nonnegative decay rate $\kappa =-\gamma$, the bound is ${\beta }_n\ge |\partial {\Lambda }_n{|}^{-1}{e}^{-n\kappa }$.

The printed final exponent has the opposite sign while retaining the definition of $\gamma$ as the limit of $n^{-1}\log {\beta }_n$. With that definition it cannot be correct in general. Indeed, if $(2d-1)p<1$, count self-avoiding paths of length $\ell \ge n$ to getHence $\gamma \le \log ((2d-1)p)<0$. A lower bound with $e^{-n\gamma }$ would eventually exceed one, because the boundary factor grows only polynomially. The valid bound is the boxed expression with $e^{n\gamma }$, or the equivalent expression using $\kappa =-\gamma$.

Order configurations coordinatewise: $\omega \le {\omega }^{\prime }$ means ${\omega }_e\le {\omega }_e^{\prime }$ for every edge. An increasing event $A$ is a measurable set such that $\omega \in A$ and $\omega \le {\omega }^{\prime }$ imply ${\omega }^{\prime }\in A$. Opening additional edges cannot destroy an increasing event. The Harris-FKG inequality, the product measure version of the FKG inequality, statesfor two increasing events. Equivalently, bounded coordinatewise increasing random variables $f,g$ satisfy ${\mathbb{E}}_p(fg)\ge {\mathbb{E}}_{p}f{\mathbb{E}}_{p}g$. The event inequality also holds for two decreasing events, by taking complements.

For disjoint occurrence of increasing events, a finite open edge set $K$ witnesses $A$ in $\omega$ if every configuration agreeing with $\omega$ on $K$ belongs to $A$. Since $A$ is an increasing event and all edges of $K$ are open, this says that the configuration with exactly $K$ open already forces $A$. Define $A\square B$ to consist of configurations admitting disjoint finite witnesses $K,L$ for $A,B$. For events depending on finitely many edges, this is the usual disjoint-occurrence definition; it also applies to finite-connection events on the infinite graph. The Van den Berg-Kesten inequality statesOn a countable graph, its finite-witness version follows by taking increasing unions over finite edge sets. The Harris-FKG inequality concerns simultaneous occurrence, whereas the Van den Berg-Kesten inequality requires separate certificates using disjoint coordinates.

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.

Because $p<p_c={\tilde{p}}_c$, choose a finite set $S\ni 0$ with ${\varphi }_p(S)=\rho <1$. Let $L$ exceed the ${\ell }^{\infty }$-distance from $0$ to every endpoint of an edge in $\partial S$, and putOn the one-arm event to distance $n>L$, take the first oriented boundary edge $(x,y)\in \partial S$ used by an open self-avoiding path. The connection $0\stackrel{S}{\leftarrow }x$, the open edge $(x,y)$, and the remaining connection from $y$ to $\partial B(n)$ occur disjointly. The Van den Berg-Kesten inequality and translation invariance therefore giveIteration yields $q_n\le {\rho }^{\lfloor n/L\rfloor }$ up to an inessential finite-scale adjustment. Since $p<1$, each of the finitely many remaining $q_n$ is strictly below one, so reducing the exponent if necessary produces a constant $c>0$ valid for every $n\ge 1$:This is the finite-size proof of exponential decay of subcritical percolation.

The Van den Berg-Kesten inequality says that for increasing cylinder events $A$ and $B$ under product bond percolation,where $A\square B$ is the event that $A$ and $B$ have disjoint finite witness edge sets.

Let $A_n$ be the increasing cylinder event that an open path joins $0$ to the boundary of the box $[-n,n{]}^d$. If two edge-disjoint open paths run from $0$ to infinity, then $A_n\square A_n$ occurs for every $n$. Therefore the Van den Berg-Kesten inequality giveswhere continuity from above identifies ${\lim }_n{\mathbb{P}}_p(A_n)=\theta (p)$.

Set $a={\mathbb{E}}_p{M}_m={\sum }_{|x|=m}{\tau }_p(x)$. If $|u|\ge m$, an open self-avoiding path from $0$ to $u$ first meets the ${\ell }^1$ sphere of radius $m$ at some $x$. Its portions from $0$ to $x$ and from $x$ to $u$ use disjoint edge sets. The Van den Berg-Kesten inequality and translation invariance therefore giveLet $s_k={\sup }_{|u|\ge km}{\tau }_p(u)$. Since $|u-x|\ge |u|-m$, the last inequality gives $s_k\le a{s}_{k-1}$ and $s_0\le 1$. Induction yields