= BK boundary-splitting estimate
{c}
{title2=$g_{m+n}\leq|\partial\Lambda_m|g_mg_n$}
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 $g_{m+n}\leq8m g_mg_n$ for positive $m,n$. With $g_0=1$ and $|\partial\Lambda_0|=1$, the boundary-cardinality form holds for zero indices as well.
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