The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Annales de l’Institut Henri Poincaré D


Full-Text PDF (841 KB) | Metadata | Table of Contents | AIHPD summary
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