= Sharpness of the percolation transition
{title2=$p<p_c\ \Longrightarrow\ \mathbb P_p(0\leftrightarrow\partial\Lambda_n)\leq Ce^{-cn}$}
= Percolation sharpness theorem
{synonym}
Independent <bond percolation> on the <cubic lattice> has exponential connection decay at every $p<p_c$ and positive <percolation probability> at every $p>p_c$. The <finite-set criterion for percolation sharpness> gives decay below its auxiliary threshold. The <Margulis–Russo formula> and exploration from a finite box boundary give $g_n'(p)\geq(1-g_n(p))\inf_{S\ni0}\varphi_p(S)/(p(1-p))$, with the infimum over interior finite sets. Above the auxiliary threshold the infimum is at least one; integrating gives positive <percolation probability> and identifies the two thresholds. This theorem does not identify the value of $p_c$ for a particular lattice.
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