= Exponential one-arm decay from finite susceptibility
{title2=$\chi(p)<\infty\ \Longrightarrow\ g_n\leq e^{-cn}$}
The <percolation susceptibility> is $\chi(p)=1+\sum_{m\geq1}\sum_{y\in\partial\Lambda_m}\tau_p(0,y)$. Finiteness gives a boundary sum $a<1$. The <weighted BK boundary-splitting estimate> then gives $g_k\leq a^{\lfloor k/m\rfloor}$. The strict bound $g_1=1-(1-p)^{2d}<1$ absorbs the finitely many smaller radii into a positive exponential rate with unit prefactor. At $p=0$, all positive-radius connection <probabilities> are zero.
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