Finite-set criterion for percolation sharpness (${\phi }_p(S)=p{\sum }_{x\in S,y\notin S,x\sim y}{\mathbb{P}}_p(0\leftrightarrow x\text{ in }S)$)

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 ${\phi }_p(S)<1$, 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 ${\tilde{p}}_c=\sup \{p:{\phi }_p(S)<1\text{ for some finite }S\ni 0\}$ equals the percolation critical probability by the pivotal-edge differential inequality in sharpness of the percolation transition.