A Pfaffian formula for the monomer–dimer model on surface graphs
We consider the monomer–dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani et al. (J Stat Phys 163(2):211–238, 2016) for graphs embedded in the disc. Our proof is based on an extension of a bijective method mentioned in Giuliani et al. (2016), together with the Pfaffian formula for the dimer partition function of Cimasoni–Reshetikhin (Commun Math Phys 275(1):187–208, 2007).
KeywordsMonomer–dimer model Partition function Surface graph Pfaffian
Mathematics Subject ClassificationPrimary 82B20 Secondary 05C70 05C10 57M15
This work was supported by a Grant of the Swiss National Science Foundation (SNSF). The author would like to thank his advisor David Cimasoni for helpful discussions. The author would also like to thank Hanoi National University of Education for supporting his work.
- 8.Garey, M.R., Johnson, D.S.: Computers and Intractability, vol. 29. WH Freeman, New York (2002)Google Scholar
- 10.Heilmann, O.J., Lieb, E.H.: Monomers and dimers. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds.) Statistical Mechanics, pp. 41–43. Springer, Berlin, Heidelberg (1970)Google Scholar
- 11.Heilmann, O.J., Lieb, E.H.: Theory of monomer-dimer systems. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds.) Statistical Mechanics, pp. 45–87. Springer, Berlin, Heidelberg (1972)Google Scholar
- 17.Temperley, H.N.V.: Enumeration of graphs on a large periodic lattice. In: Combinatorics: Proceedings of the British Combinatorial Conference, pp. 155–159 (1973)Google Scholar