= 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 $q|W|\leq|\partial^+W|$ for every large box, contradicting vanishing boundary-to-volume ratio. This proves uniqueness without assuming absence of an <infinite percolation cluster> at criticality.
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