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Annales de l’Institut Henri Poincaré D

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Volume 4, Issue 3, 2017, pp. 309–385
DOI: 10.4171/AIHPD/42

Published online: 2017-09-26

Revisiting the combinatorics of the 2D Ising model

Dmitry Chelkak[1], David Cimasoni[2] and Adrien Kassel[3]

(1) Ecole Normale Supérieure, Paris, France
(2) Université de Genève, Switzerland
(3) ETH Zürich, Switzerland

We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants, using the language of Kac–Ward matrices. We also give a brief account of the relations between various alternative formalisms which have been used in the combinatorial study of the planar Ising model: dimers and Grassmann variables, spin and disorder operators, and, more recently, s-holomorphic observables. In addition, we point out that these formulas can be extended to the double-Ising model, defined as a pointwise product of two Ising spin congurations on the same discrete domain, coupled along the boundary.

Keywords: Ising model, Kac–Ward matrix, spin correlations, fermionic observables, discrete holomorphic functions, spin structures, double-Ising model

Chelkak Dmitry, Cimasoni David, Kassel Adrien: Revisiting the combinatorics of the 2D Ising model. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4 (2017), 309-385. doi: 10.4171/AIHPD/42