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Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function. (English) Zbl 1405.82006
Summary: A streamlined derivation of the Kac-Ward formula for the planar Ising model’s partition function is presented and applied in relating the kernel of the Kac-Ward matrices’ inverse with the correlation functions of the Ising model’s order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on $$\mathbb{Z}^2$$ for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.

MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
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