# A Pfaffian Formula for Monomer–Dimer Partition Functions

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## Abstract

We consider the monomer–dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on the boundary. We prove a Pfaffian formula for the corresponding partition function. As a consequence of this result, multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics. Our proof is based on the celebrated Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the authors, and a careful labeling and directing procedure of the vertices and edges of the graph.

## Keywords

Monomer–dimer problem Pfaffian formula Boundary monomers## Notes

### Acknowledgments

Thanks go to Jan Philip Solovej and Lukas Schimmer for devoting their time to some preliminary calculations that helped put us on the right track. We also would like to thank Tom Spencer and Joel Lebowitz for their hospitality at the IAS in Princeton and their continued interest in this problem. In addition, we thank Michael Aizenman and Hugo Duminil-Copin for discussing their work in progress on the random current representation for planar lattice models with us. We thank Jacques Perk and Fa Yueh Wu for very useful historical comments. We gratefully acknowledge financial support from the A*MIDEX project ANR-11-IDEX-0001-02 (A.G.), from the PRIN National Grant *Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions* (A.G. and I.J.), and NSF grant PHY-1265118 (E.H.L.).

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