A generalisation of the honeycomb dimer model to higher dimensions

TL;DR

This paper extends the honeycomb dimer model to higher dimensions, establishing structural properties, a generalized Kasteleyn theory, and new insights into the covariance matrix and surface tension convexity.

Contribution

It introduces a higher-dimensional generalization of the honeycomb dimer model with a new hypergraph correspondence and a generalized Kasteleyn theory, plus novel covariance and convexity results.

Findings

Samples correspond to perfect matchings of hypergraphs

Partition function equals Cayley hyperdeterminant of the adjacency hypermatrix

New identity relates covariance matrix to geometric structure

Abstract

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.