= Weighted BK boundary-splitting estimate
{title2=$g_n\leq g_{n-m}\sum_{y\in\partial\Lambda_m}\tau_p(0,y)$}
For independent <bond percolation> on the <cubic lattice>, define $\tau_p(x,y)=\mathbb P_p(x\leftrightarrow y)$ and $g_n=\mathbb P_p(0\leftrightarrow\partial\Lambda_n)$, with $g_0=1$. Split an open <self-avoiding walk> at its first radius-$m$ boundary vertex $y$. The remaining segment reaches distance at least $n-m$ from $y$, using disjoint edges. The <BK inequality>, translation invariance and the <union bound> yield the displayed estimate. Keeping the individual connection weights can be stronger than replacing their sum by $|\partial\Lambda_m|g_m$.
Back to article page
