1-2 model, dimers, and clusters

A 1-2 model configuration is a subgraph of the hexagonal lattice, in which the edge-degree of each vertex is either 1 or 2. We prove that for any translation invariant Gibbs measure of the 1-2 model configurations on the whole-plane hexagonal lattice, almost surely there are no infinite paths. Using a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and perfect matchings on a decorated graph, we construct an explicit translation invariant Gibbs measure for 1-2 model configurations on the bi-periodic hexagonal lattice. We prove that the behaviors of infinite clusters are different for small and large local weights, which shows the existence of a phase transition.