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Title 
A generalized KacWard formula 

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Published in  Journal of Statistical Mechanics: Theory and Experiment. 2010, p. 23 p.  
Abstract  The KacWard 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 FisherKasteleyn 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 KacWard and the FisherKasteleyn methods to solve the Ising model are one and the same.  
Note  8 figures  
Stable URL  http://archiveouverte.unige.ch/unige:12098  
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arXiv: 1004.3158 