Vladimir Y Chernyak and Michael Chertkov J. Stat. Mech. (2008) P12012 doi:10.1088/1742-5468/2008/12/P12012
Fermions and loops on graphs: II. A monomer–dimer model as a series of determinants
Vladimir Y Chernyak1,2 and Michael Chertkov2
Show affiliationsWe 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.
- Keywords
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E-print Number: 0809.3481
Cited: by |
Refers: to
- PACS
- MSC
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81T13 Yang-Mills and other gauge theories (See also 53C07, 58E15)
15A15 Determinants, permanents, other special matrix functions (See also 19B10, 19B14)
- Subjects
- Dates
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Issue 12 (December 2008)
Received 23 September 2008, accepted for publication 25 November 2008
Published 17 December 2008
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Fermions and loops on graphs: II. A monomer–dimer model as a series of determinants
Vladimir Y Chernyak and Michael Chertkov J. Stat. Mech. (2008) P12012
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Fermions and loops on graphs: I. Loop calculus for determinants
Vladimir Y Chernyak and Michael Chertkov J. Stat. Mech. (2008) P12011
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