Mermin-Wagner theorem for dimers, monomer double-dimers, and spatial random permutations

We study a generalisation of the double-dimer model that encompasses several models of interest, including the monomer double-dimer model, spatial random permutations, the dimer model, and the spin $O(N)$ model, and which is also related to the loop $O(N)$ model. We show that on two-dimensional-like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices decay to zero as the distance between the vertices diverges. Our approach is based on the introduction of a new complex spin representation for all models in this class, together with a new proof of the Mermin-Wagner theorem that does not require positivity of the Gibbs measure. Even for the well-studied dimer and double-dimer models our results are new: since they do not rely on exact solvability or Kasteleyn’s theorem, they apply beyond the planar-graph setting.