Two dimensional dimers beyond planarity

Abstract: We consider a generalisation of the double dimer model which includes several models of interest, such as the monomer double dimer model, the dimer model, the Spin O(N) model, and it is related to the loop O(N) model. We prove that on two-dimension like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices converge to zero as the distance between such vertices gets large. Our analysis is by introducing a new (complex) spin representation for all models belonging to this class, and by deriving a new proof of the Mermin-Wagner theorem which does not require the positivity of the Gibbs measure. Even for the well studied dimer and double dimer model our results are new since - not relying on solvability and Kasteleyn's theorem - they hold beyond the framework of planar graphs.