A Pfaffian Formula for Monomer–Dimer Partition Functions

Alessandro Giuliani, Ian Jauslin, Elliott H. Lieb

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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 languageEnglish (US)
Pages (from-to)211-238
Number of pages28
JournalJournal of Statistical Physics
Issue number2
StatePublished - Apr 1 2016

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Boundary monomers
  • Monomer–dimer problem
  • Pfaffian formula


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