= Reflection lower bound for two-point percolation
{title2=$(g_k/(8k))^2\leq h_{2k}\leq g_{2k}$}
For <bond percolation> on the <square lattice>, write $h_n=\mathbb P_p(0\leftrightarrow(n,0))$ and $g_k=\mathbb P_p(0\leftrightarrow\partial[-k,k]^2)$. A maximal-probability boundary <graph vertex> can be placed at $(k,j)$ by symmetry. Reflection about $x=k$ identifies its connection <probability> from $0$ with that from $(2k,0)$. The <Harris-FKG inequality> yields $(g_k/(8k))^2\leq h_{2k}\leq g_{2k}$ for $k\geq1$.
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