Revisiting the combinatorics of the 2D Ising model. (English) Zbl 1380.82017

Chelkak, Dmitry; Cimasoni, David; Kassel, Adrien

Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD) 4, No. 3, 309-385 (2017).

The combinatorial method constitutes an important approach for the study of classical Ising models. The present paper provides short proofs of several combinatorial formulas (such as the Kac-Ward formula), which are useful in evaluating the partition function, multi-point fermionic observables, and spin and energy density correlations for the planar Ising model. A self-contained overview of the connections between several formalisms (dimer representation, Grassmann variables, disorder insertions, and combinatorial s-holomorphic observables) in the study of the planar Ising model is given. In the last section, the authors also make an attempt to generalize their results to the double-Ising model, which describes two identical planar Ising models with couplings at the boundary.

Reviewer: Hong-Hao Tu (Dresden)

MSC:

82B20

Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Keywords:

Ising model; Kac-Ward matrix; spin correlations; fermionic observables; discrete holomorphic functions; spin structures; double-Ising model