Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2008-09-20
J.Stat.Mech.0812:P12012,2008
Physics
Condensed Matter
Statistical Mechanics
11 pages, 2 figures; misprints corrected
Scientific paper
10.1088/1742-5468/2008/12/P12012
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.
Chernyak Vladimir Y.
Chertkov Michael
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