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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.

Declare an edge of the planar dual graph open exactly when its crossed primal edge is closed. This dual bond percolation has parameter $q=1-p<1/2=p_c$, using the Harris-Kesten theorem. We use the standard exponential tail of subcritical cluster size: for some $K,a>0$, uniformly in the dual vertex $v$,This is decay of the number of vertices in the percolation cluster, not merely its radius.

On $\{|C|=n\}$, the geometric fact allowed in the paper supplies a simple dual cycle in a graph surrounding $C$, with length $\ell \ge \alpha \sqrt{n}$. Every crossed edge is in the edge boundary of $C$ and is therefore closed, so the dual cycle is open. Its $\ell$ distinct vertices lie in one dual percolation cluster of size at least $\ell$.

A dual cycle in a graph of length $\ell$ surrounding the origin has all its vertices within sup-norm distance $\ell +1$ of the origin: its coordinate spans are at most $\ell$, and the origin lies between each pair of extreme coordinates. There are at most $K_0(\ell +1{)}^2$ possible dual vertices there. The union bound over lengths and possible vertices yieldsfor some $K_1,b>0$, since a polynomial factor can be absorbed into a slower exponential decay.

To remove the constant prefactor for all $n\ge 1$, first choose $n_0$ so that the bound is at most $e^{-(b/2)\sqrt{n}}$ for $n\ge n_0$. For the remaining finitely many $n$, use ${\mathbb{P}}_p(|C|=n)\le 1-\theta (p)<1$ and chooseFor $1\le n<n_0$, the chosen $c_2$ ensures $e^{-c_2\sqrt{n}}\ge 1-\theta (p)$.