 Exponential decay of subcritical percolation

For independent nearest-neighbour bond percolation on ${\mathbb{Z}}^d$, $d\ge 2$, and $p<p_c$, there are constants $C,c>0$ such thatHere $p_c$ is the percolation critical probability, and the event asks for an open path from the origin to maximum-norm distance $n$. The constants may depend on $p$ and $d$. The theorem strengthens absence of an infinite percolation cluster: it quantifies how unlikely a long finite connection becomes.^[ref]

The finite-set criterion for percolation sharpness explains the mechanism. Below the threshold, a finite set $S$ can be chosen with exit quantity ${\phi }_p(S)<1$. Repeatedly splitting a long open path as it exits translated copies of $S$, and using the BK inequality, yields a geometric contraction in the number of such steps. Each step spans a bounded distance, converting the geometric bound into exponential decay in $n$. Decay of the percolation cluster radius is distinct from the stronger exponential tail of subcritical cluster size.