A Pfaffian formula for monomer-dimer partition functions

TL;DR

This paper derives a Pfaffian formula for the monomer-dimer partition function on planar graphs with boundary monomer weights, revealing fermionic statistics in boundary correlations.

Contribution

It introduces a Pfaffian formula for the monomer-dimer partition function on arbitrary planar graphs with boundary monomer weights, extending previous results.

Findings

Pfaffian formula for partition function

Boundary monomer correlations satisfy fermionic statistics

Applicable to arbitrary finite planar graphs

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.