A generalized Kac-Ward formula

TL;DR

This paper generalizes the Kac-Ward formula for the Ising model's partition function from planar graphs to graphs embedded on surfaces of any genus, linking it to the Fisher-Kasteleyn method.

Contribution

It extends the Kac-Ward formula to all finite graphs embedded on surfaces of arbitrary genus, unifying it with the Fisher-Kasteleyn approach through two different proofs.

Findings

Generalized Kac-Ward formula for any finite graph

Equivalence of Kac-Ward and Fisher-Kasteleyn methods

Partition function expressed as sum of determinants of 2^{2g} matrices

Abstract

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.