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.
The author is grateful to Nathanaël Berestycki and James Norris for their enthusiastic supervision of the writing of this paper. The author thanks Amir Dembo, Richard Kenyon, Scott Sheffield and Martin Tassy for helpful discussions. Finally, the author would like to thank the anonymous referees for their insightful feedback.
The author was supported by the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge and the UK Engineering and Physical Sciences Research Council grant EP/L016516/1.
"A generalisation of the honeycomb dimer model to higher dimensions." Ann. Probab. 49 (2) 1033 - 1066, March 2021. https://doi.org/10.1214/20-AOP1469