On loops in the complement of dimers

We consider ergodic translation-invariant Gibbs measures for the dimer model (i.e., perfect matchings) on the hexagonal lattice. The complement of a dimer configuration is a fully-packed loop configuration: each vertex has degree two. This is also known as the loop O(1) model at x=∞. We show that, if the measure is non-frozen, then it exhibits either infinitely many loops around every face or a unique bi-infinite path. Our main tool is the flip (or XOR) operation: if a hexagon contains exactly three dimers, one can replace them by the other three edges. Classical results in dimer theory imply that such hexagons appear with a positive density. To some extent, this replaces the finite-energy property and allows to make use of tools from the percolation theory, in particular the Burton–Keane argument, to exclude existence of more than one bi-infinite path.