= Exponential decay of subcritical percolation
For <independent> nearest-neighbour <bond percolation> on $\mathbb Z^d$, $d\geq2$, and $p<p_c$, there are constants $C,c>0$ such that
$$
\mathbb P_p(0\leftrightarrow\partial[-n,n]^d)\leq Ce^{-cn}\qquad(n\geq1).
$$
Here $p_c$ 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.<A new proof of the sharpness of the phase transition for Bernoulli percolation on Zd by Hugo Duminil-Copin and Vincent Tassion>[Theorem 1.1, subcritical exponential decay; notation and section 2.1]{ref}
The <finite-set criterion for percolation sharpness> explains the mechanism. Below the threshold, a finite set $S$ can be chosen with exit quantity $\varphi_p(S)<1$. 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>.
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