### 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.

Original language | English (US) |
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Pages (from-to) | 211-238 |

Number of pages | 28 |

Journal | Journal of Statistical Physics |

Volume | 163 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2016 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Boundary monomers
- Monomer–dimer problem
- Pfaffian formula

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## Cite this

Giuliani, A., Jauslin, I., & Lieb, E. H. (2016). A Pfaffian Formula for Monomer–Dimer Partition Functions.

*Journal of Statistical Physics*,*163*(2), 211-238. https://doi.org/10.1007/s10955-016-1484-1