A generalisation of the honeycomb dimer model to higher dimensions

Piet Lammers

Published 2019-05-30Version 1

This paper studies a generalisation of the honeycomb dimer model to higher dimensions. The generalisation was introduced by Linde, Moore, and Nordahl. Each sample of the model is both a tiling and a height function. First, we derive a surprising identity for the covariance structure of the model. Second, we prove that the surface tension associated with the model is strictly convex, in any dimension. This greatly streamlines the original proof for strict convexity by Sheffield. It implies a large deviations result with a unique minimiser for the rate function, and consequently a variational principle with a unique limit shape. Third, we demonstrate that the model is a perfect matching model on a hypergraph with a generalised Kasteleyn theory: the partition function is given by the Cayley hyperdeterminant of the appropriate hypermatrix. The formula so obtained is very challenging: the author does not expect a closed-form solution for the surface tension. The first two results rely on the development of the boundary swap, which is a versatile technique for understanding the model; it is inspired by the double dimer model, works in any dimension, and may be of independent interest.