= Finite-set criterion for percolation sharpness
{title2=$\varphi_p(S)=p\sum_{x\in S,\ y\notin S,\ x\sim y}\mathbb P_p(0\leftrightarrow x\text{ in }S)$}
For a finite vertex set $S$ containing the origin, this quantity is the expected number of potentially open exit edges reached from the origin within $S$. If $\varphi_p(S)<1$, the <BK inequality> applied to the first exit of a long open <self-avoiding walk> gives a contraction at the scale of $S$ and hence exponential connection decay. The threshold $\widetilde p_c=\sup\{p:\varphi_p(S)<1\text{ for some finite }S\ni0\}$ equals the <percolation critical probability> by the pivotal-edge differential inequality in <sharpness of the percolation transition>.
Back to article page
