 Solution

Use the coordinatewise partial order: $\omega \le \eta$ means $\omega (e)\le \eta (e)$ for every $e\in E$. An increasing event $A$ is upward closed: $\omega \in A$ and $\omega \le \eta$ imply $\eta \in A$.

Under the independent Bernoulli distribution product measure ${\mathbb{P}}_p$ on these coordinates, the Harris-FKG inequality statesfor increasing events $A,B$. For $K\subseteq E$, let $[\omega {]}_K$ be the cylinder of configurations agreeing with $\omega$ on $K$. The disjoint occurrence of increasing events $A\square B$ consists of configurations for which there exist disjoint $K,L$ with $[\omega {]}_K\subseteq A$ and $[\omega {]}_L\subseteq B$. For increasing events, witnesses can be taken to prescribe only open coordinates. The BK inequality isPositive association rewards simultaneous occurrence; the disjoint-witness requirement in the BK inequality gives a bound in the opposite direction. The inequalities also apply to different independent Bernoulli parameters in different coordinates.

Regard the boxes as sets of lattice graph vertices. Write ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$, and put $g_0=1$. On the event defining $g_n$, choose an open self-avoiding walk from the origin to its first visit to $\partial {\Lambda }_n$. Let $y$ be its first visit to $\partial {\Lambda }_m$. Its initial segment witnesses $0\leftrightarrow y$. Its remaining segment ends at a point $z$ withFor $m<n$, truncate this remaining segment on its first visit to $y+\partial {\Lambda }_{n-m}$. The two segments use disjoint edges, so they are disjoint witnesses. For $m=n$, the second event is the sure event with empty witness. Thus the union bound, the BK inequality and translation invariance giveAn unrestricted connection event can depend on infinitely many edges. Apply the finite-coordinate BK inequality first to connections confined to growing boxes and take increasing limits; each occurrence has a finite path witness. This justifies its use here. The estimate is the weighted BK boundary-splitting estimate.

By summing cluster indicators and using Tonelli theorem, the percolation susceptibility isIf $\chi (p)<\infty$, then $S_m\to 0$. At $p=0$ all $g_k$ vanish, so any positive exponential rate works. Otherwise choose $m\ge 1$ with $a=S_m\in (0,1)$. Iterating the weighted BK boundary-splitting estimate gives, for $k=jm+r$ with $0\le r<m$,For $k\ge m$, $\lfloor k/m\rfloor \ge k/(2m)$, so $g_k\le \exp [-(-\log a)k/(2m)]$. To include the finitely many smaller indices without a prefactor, note that $p<1$ under finite susceptibility andTherefore one may takeFor $k<m$ this follows from $\mu (p)k\le -\log g_1$, and for $k\ge m$ it follows from the iterated estimate. This proves exponential one-arm decay from finite susceptibility with exactly the requested unit prefactor.