= Dual contour bound for a finite planar cluster
In <bond percolation> on the square lattice, the outer boundary of a finite <percolation cluster>, after filling its holes, consists of closed bonds. Their crossing edges in the <planar dual graph> form a connected enclosing contour, and hence belong to a single <dual bond percolation> cluster $K$. If $|K|=m$, both its coordinate widths are at most $m-1$, so the enclosed primal cluster contains at most $m^2$ vertices. A contour enclosing the origin also places every vertex of $K$ within distance $m+1$ of the origin. An exponential volume tail for the subcritical <dual bond percolation> clusters thus gives a bound $Ce^{-c\sqrt n}$ for a finite primal cluster of size at least $n$.
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