Combinatorial Theory

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.

Mathematics Subject Classifications: 82B20, 05A15, 05C70

Keywords: Monopole-dimer model, cartesian products, determinantal formula, Kasteleyn orientation, bipartite, cycle decomposition, partition function, grid graphs, free energy