Séminaire Lotharingien de Combinatoire, 86B.65 (2022), 12 pp.
Anita Arora and Arvind Ayyer
The Monopole-Dimer Model for Cartesian Products of Graphs: Extended Abstract
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 introduced by the second author (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 give an explicit product formula for three-dimensional grid graphs à la Kasteleyn and Temperley-Fischer, in which case the partition function turns out to be fourth power of a polynomial when all grid lengths are even. Finally, we generalise this product formula to k dimensions, again obtaining an explicit product formula.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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