Fermions and loops on graphs: II. A monomer–dimer model as a series of determinants

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 a (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.

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