Tree-graph moment bound for a percolation cluster

For translation-invariant independent bond percolation with finite percolation susceptibility $\chi ={\sum }_y\mathbb{P}(0\leftrightarrow y)$, the BK inequality gives $\mathbb{E}|C_0{|}^k\le (2k-3)!!{\chi }^{2k-1}$ for $k\ge 1$, with $(-1)!!=1$. Prune a connecting open tree for the $k+1$ labelled vertices, suppress unmarked degree-two vertices, and split higher branching into binary branching with zero-length connections. There are $(2k-3)!!$ binary tree topologies and $2k-1$ connections. The connections have disjoint bond witnesses, so the BK inequality bounds their joint probability by a product of two-point connection probabilities. Summing successive leaves contributes $\chi$ for each connection. Since $(2k-3)!!\le 2^{k-1}(k-1)!$, this proves $\mathbb{E}{e}^{t|C_0|}\le 1-(2\chi {)}^{-1}\log (1-2t{\chi }^2)$ for $0<t<(2{\chi }^2{)}^{-1}$. Finite percolation susceptibility therefore gives an exponential cluster-volume tail.