The monopole-dimer model on Cartesian products of plane graphs

TL;DR

This paper extends the monopole-dimer model to Cartesian products of planar graphs, providing explicit determinant formulas for the partition function, including for 3D and higher-dimensional grids, and discusses asymptotic behaviors.

Contribution

It introduces a generalized monopole-dimer model for Cartesian product graphs, deriving explicit determinant and product formulas for their partition functions.

Findings

Partition function expressed as a determinant of a generalized signed adjacency matrix.

Partition function is independent of orientations if they are Pfaffian.

Explicit product formulas for 3D and higher-dimensional grid graphs.

Abstract

The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. We extend the monopole-dimer model for planar graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof and show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs so long as the orientations are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs a la Kasteleyn and Temperley--Fischer, which turns out to be fourth power of a polynomial when all grid lengths are even. Finally, we generalise this product formula to $d$ dimensions, again obtaining an explicit product formula. We conclude with a discussion on asymptotic formulas for the free energy and monopole densities.