Revisiting the combinatorics of the 2D Ising model

2015/07/31 by Dmitry Chelkak, David Cimasoni, Adrien Kassel · 44 citations
Mathematics · Physics and Astronomy · #Algebraic combinatorics #Combinatorics #Complex Network Analysis Techniques #Computer science #Ising model #Mathematics #Physics #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.CO #math.MP #math.PR

paper · pdf · doi:10.4171/aihpd/42

published in Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions 4(3), 309-385 · Minor change in the notation (definition of eta). 55 pages, 4 figures

arxiv created 2017/08/23 · openalex publication_date 2017/09/26 · arxiv updated 2019/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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.

Citations

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