= Encounter box in percolation
An encounter box is a finite connected set of open vertices whose removal from its <infinite percolation cluster> leaves at least three infinite connected components. Unlike a <trifurcation vertex in percolation>, it allows branching to occur across several vertices, which is useful for <site percolation> on graphs with local cycles. Contract disjoint encounter boxes to junctions. In a finite region, each junction must connect at least three different exits. Pruning to a forest gives at most as many junctions as boundary exits, by the identity $\sum_v(\deg(v)-2)=-2$ for a finite <tree>. Positive density of encounter boxes is therefore incompatible with an <amenable graph>.
Back to article page
