Séminaire Lotharingien de Combinatoire, 93B.58 (2025), 12 pp.
Anita Arora
The Monopole-Dimer Model on High-Dimensional Cylindrical, Toroidal, Möbius and Klein Grids
Abstract.
The dimer (monomer-dimer) model deals with weighted enumeration of perfect matchings (matchings).
The monopole-dimer model is a signed variant of the monomer-dimer model whose partition function is a determinant.
In 1999, Lu and Wu evaluated the partition function of the dimer model on two-dimensional grids embedded on a Möbius strip and a Klein bottle.
While the partition function of the dimer model has been known for
the two-dimensional grids with different boundary conditions,
we present a similar product formula for the partition function of the monopole-dimer model on higher dimensional cylindrical and toroidal grid graphs. We also evaluate the same for the three-dimensional Möbius and Klein grid graphs and show that the formula does not generalise for the higher dimensions. Further, we present a relation between the product formula for the three-dimensional cylindrical and Möbius grid.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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