Revisiting the combinatorics of the 2D Ising model

  • Dmitry Chelkak

    Ecole Normale Supérieure, Paris, France
  • David Cimasoni

    Université de Genève, Switzerland
  • Adrien Kassel

    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.

Cite this article

Dmitry Chelkak, David Cimasoni, Adrien Kassel, Revisiting the combinatorics of the 2D Ising model. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4 (2017), no. 3, pp. 309–385

DOI 10.4171/AIHPD/42