  Linde, Moore, and Nordahl introduced a generalisation of the honeycomb dimer
model to higher dimensions. The purpose of this article is to describe a number
of structural properties of this generalised model. First, it is shown that the
samples of the model are in one-to-one correspondence with the perfect
matchings of a hypergraph. This leads to a generalised Kasteleyn theory: the
partition function of the model equals the Cayley hyperdeterminant of the
adjacency hypermatrix of the hypergraph. Second, we prove an identity which
relates the covariance matrix of the random height function directly to the
random geometrical structure of the model. This identity is known in the planar
case but is new for higher dimensions. It relies on a more explicit formulation
of Sheffield's cluster swap which is made possible by the structure of the
honeycomb dimer model. Finally, we use the special properties of this explicit
cluster swap to give a new and simplified proof of strict convexity of the
surface tension in this case.
