= 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 $k\ge1$, 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 $(2k-3)!!\le2^{k-1}(k-1)!$, this proves $\mathbb E e^{t|C_0|}\le1-(2\chi)^{-1}\log(1-2t\chi^2)$ for $0<t<(2\chi^2)^{-1}$. Finite <percolation susceptibility> therefore gives an exponential cluster-volume tail.
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