= Monochromatic line-graph percolation counterexample
Color the vertices of a periodic <graph> independently red or blue with equal <probabilities>, and declare a vertex of its <line graph> open exactly when the corresponding edge has equal-colored endpoints. The resulting <site percolation> has marginal $1/2$ and is a <finite-range dependent random field>: sets at line-graph distance greater than one use disjoint color inputs. Red and blue open clusters cannot join. If the original graph has an infinite red component and an infinite blue component, its <line graph> consequently has at least two <infinite percolation clusters>. One explicit <amenable graph> with that property is obtained by replacing each square-lattice vertex by a <complete graph> on ten vertices, joining neighboring blocks completely. Blocks containing both colors have independent probability $511/512$; a <Peierls argument> proves that they percolate, and their infinite cluster carries both a red and a blue component. Thus finite-range dependence, even with all site marginals nondegenerate, does not substitute for finite energy in percolation uniqueness.
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