Fermions and Loops on Graphs. II. Monomer-Dimer Model as Series of Determinants

TL;DR

This paper introduces a graphical gauge model for fermions on graphs, showing its equivalence to the monomer-dimer model and expressing its partition function as a series over directed cycles involving determinants, with implications for analysis and algorithms.

Contribution

It presents a novel graphical gauge model that links fermion models to the monomer-dimer model and expresses the partition function as a cycle series involving determinants.

Findings

Partition function expressed as a series over disjoint directed cycles

Model equivalence between the GGM and the MD model

Potential implications for analytic and algorithmic approaches

Abstract

We continue the discussion of the fermion models on graphs that started in the first paper of the series. Here we introduce a Graphical Gauge Model (GGM) and show that : (a) it can be stated as an average/sum of a determinant defined on the graph over $\mathbb{Z}_{2}$ (binary) gauge field; (b) it is equivalent to the Monomer-Dimer (MD) model on the graph; (c) the partition function of the model allows an explicit expression in terms of a series over disjoint directed cycles, where each term is a product of local contributions along the cycle and the determinant of a matrix defined on the remainder of the graph (excluding the cycle). We also establish a relation between the MD model on the graph and the determinant series, discussed in the first paper, however, considered using simple non-Belief-Propagation choice of the gauge. We conclude with a discussion of possible analytic and algorithmic consequences of these results, as well as related questions and challenges.