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.

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.

For independent bond percolation on ${\mathbb{Z}}^d$ below the percolation critical probability, there are $K,c>0$ such that ${\mathbb{P}}_p(|C(x)|\ge n)\le K{e}^{-cn}$, uniformly in $x$. This concerns the number of vertices in a percolation cluster, a stronger conclusion than decay of its radius alone.