FM 13-15; arXiv:1510.05027

Authors : Alessandro Giuliani, Ian Jauslin, Elliott H. Lieb

Title: A Pfaffian formula for monomer-dimer partition functions

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.

Alessandro Giuliani
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre,
L.go S.L. Murialdo, 1, 00146 Roma, Italy.

Ian Jauslin
University of Rome "Sapienza", Dipartimento di Fisica,
P.le Aldo Moro, 2, 00185 Roma, Italy

Elliott H. Lieb
Departments of Mathematics and Physics, Jadwin Hall, Princeton University,
Princeton 08544 NJ, USA