  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.
