= Forest proof of trifurcation boundary counting
Delete a finite set of degree-three <trifurcation vertices in percolation> in each <infinite percolation cluster> and contract the remaining components to nodes. The incidence graph is a <tree>: a cycle would bypass an original <trifurcation vertex in percolation>. All leaf components are infinite, otherwise a <graph vertex> would have a finite branch after deletion. The <tree> degree identity gives at least two more infinite leaves than degree-three <graph vertices>. Distinct infinite components escape a containing box through distinct outer boundary <graph vertices>. Hence the number of counted trifurcations is at most the outer boundary size.
Back to article page
