Dual bond percolation

In planar bond percolation, declare a dual edge open exactly when the primal edge it crosses is closed. Independent primal parameter $p$ then gives independent dual parameter $1-p$.

A planar finite network with three boundary terminals induces one of five connection partitions: all three connected, all three separate, or one connected pair and an isolated third terminal. Put three dual terminals on the intervening boundary arcs. Passing to the planar dual graph interchanges the all-connected and all-separate partitions and permutes the three pair partitions. If the network is symmetric under all terminal permutations, equality $P_3=P_0$ of the first two probabilities makes its whole partition law dual-invariant. Independent cells on a self-dual triangular terminal arrangement then give identical primal and dual coarse connection laws. Infinite connectivity is preserved because cells are uniformly finite. For a martini lattice cell with six independent bonds of common parameter $p$, $P_3=3p^5-2p^6$ and $P_3-P_0=(2p^2-1)(p^4-3p^3+2p^2+1)$. The unique zero in $[0,1]$ is $1/\sqrt{2}$.

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 $C{e}^{-c\sqrt{n}}$ for a finite primal cluster of size at least $n$.