Abstract
The Kac–Ward formula allows the Ising partition function to be computed on a planar graphG with straight edges from the determinant of a matrix of size2N, whereN denotes the numberof edges of G. In this paper, we extend this formula to any finite graph: the partitionfunction can be written as an alternating sum of the determinants of22g matricesof size 2N,where g is the genus of an orientable surface in whichG embeds. We give two proofs of this generalized formula. The first one is purelycombinatorial, while the second relies on the Fisher–Kasteleyn reduction of the Ising modelto 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 forsolving the Ising model are one and the same.
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