A 1-2 model configuration is a subset of edges of the hexagonal lattice such that each vertex is incident to one or two edges. We prove that for any translation-invariant Gibbs measure of 1-2 model, almost surely the infinite homogeneous cluster is unique.
"Uniqueness of the infinite homogeneous cluster in the 1-2 model." Electron. Commun. Probab. 19 1 - 8, 2014. https://doi.org/10.1214/ECP.v19-3105