 Boundary counting proof of percolation uniqueness

Independent bond percolation on ${\mathbb{Z}}^2$ has at most one infinite percolation cluster at every parameter, including a possible critical parameter. The number of infinite percolation clusters is constant almost surely by translation ergodicity of Bernoulli percolation. A finite constant larger than one is impossible: a box meeting two clusters can be made entirely open, joining them and decreasing that number with positive probability by finite modification of Bernoulli percolation.

If infinitely many clusters existed, a box would meet three with positive probability. Preserve one infinite exterior arm from each and replace the finitely many interior and boundary edges by a three-armed tree joining them, closing the remaining edges. Its branch graph vertex becomes a trifurcation vertex in percolation. Translation invariance therefore gives a positive density $q$. But the trifurcation boundary-counting lemma gives $q|W|\le |{\partial }^{+}W|$ for every large box, contradicting vanishing boundary-to-volume ratio. This proves uniqueness without assuming absence of an infinite percolation cluster at criticality.