Scientific article
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A generalized Kac-Ward formula

ContributorsCimasoni, David
Publication date2010

The Kac-Ward formula allows to compute the Ising partition function on a planar graph G with straight edges from the determinant of a matrix of size 2N, where N denotes the number of edges of G. In this paper, we extend this formula to any finite graph: the partition function can be written as an alternating sum of the determinants of 2^{2g} matrices of size 2N, where g is the genus of an orientable surface in which G embeds. We give two proofs of this generalized formula. The first one is purely combinatorial, while the second relies on the Fisher-Kasteleyn reduction of the Ising model to the dimer model, and on geometric techniques. As a consequence of this second proof, we also obtain the following fact: the Kac-Ward and the Fisher-Kasteleyn methods to solve the Ising model are one and the same.

  • arxiv : math-ph
Note8 figures
Affiliation Not a UNIGE publication
Citation (ISO format)
CIMASONI, David. A generalized Kac-Ward formula. In: Journal of statistical mechanics, 2010, p. 23 p. doi: 10.1088/1742-5468/2010/07/P07023
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Article (Accepted version)
ISSN of the journal1742-5468

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