**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