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.