Exponential decay of subcritical percolation

Below the percolation threshold, the probability that a fixed vertex has an open path to graph distance $n$ is at most $C{e}^{-cn}$ for positive constants $C,c$.

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.