work we use primarily the language of the height representation.

To map the square-lattice classical dimer model onto a height model, we follow the prescription given in Ref. . One first assigns a height variable to each plaquette. In going around a vertex on the even sub-lattice clockwise, the height changes by $+3$ if a dimer is present on the link between the plaquettes, and by $-1$ if no dimer is present on that link. On the odd sub-lattice, the heights change by $-3$ and $+1$ respectively. The dimer constraint, that one lattice site belongs to one and only one dimer, implies (for the square lattice) that, for the fully packed case, there are four possible configurations of dimers for each lattice site. In the dual height model this is reproduced by the period four property $ h \equiv h+4$ of the allowed height configurations. It is easy to see that for the allowed configuration the average values that the height variables can take at a given site of the direct lattice (a vertex) are $\pm 3/2, \pm 1/2$. On the other hand, a uniform shift [*of all*]{} the heights by one unit, $h
\to h+1$, leads to an equivalent 