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Squareness for the Monopole-Dimer Model

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Abstract

The monopole-dimer model introduced recently is an exactly solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. We give a combinatorial interpretation of the square root of the partition function for such graphs in terms of a monopole-dimer model on a new kind of graph with two types of edges which we call a dicot. The partition function of the latter can be written as a determinant, this time of a complex adjacency matrix. This formulation generalises Wu’s assignment of imaginary orientation for the grid graph to planar dicots. As an application, we compute the partition function for a family of non-planar dicots with positive weights.

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Notes

  1. 1.

    Short for dicotyledons, a plant whose embryo contains two leaves.

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Acknowledgements

We thank Jeremie Bouttier for discussions and anonymous referees for many useful comments. This work is supported in part by the UGC Centre for Advanced Studies and by Department of Science and Technology Grants DST/INT/SWD/VR/P-01/2014 and EMR/2016/006624.

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Correspondence to Arvind Ayyer.

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Ayyer, A. Squareness for the Monopole-Dimer Model. Ann. Comb. (2020). https://doi.org/10.1007/s00026-019-00480-5

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Keywords

  • Monopole-dimer model
  • Dicots
  • Dimer model
  • Determinantal formula
  • Kasteleyn orientation
  • Partition function
  • Free energy

Mathematics Subject Classification

  • 82B20
  • 05C70