Probability Theory and Related Fields

, Volume 174, Issue 1–2, pp 235–305 | Cite as

The Z-invariant Ising model via dimers

  • Cédric BoutillierEmail author
  • Béatrice de Tilière
  • Kilian Raschel


The Z-invariant Ising model (Baxter in Philos Trans R Soc Lond A Math Phys Eng Sci 289(1359):315–346, 1978) is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k. When \(k=0\) the model is critical, and as k varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers (Boutillier and de Tilière in Probab Theory Relat Fields 147:379–413, 2010; Commun Math Phys 301(2):473–516, 2011) to the fullZ-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of Boutillier and de Tilière (2011): it involves a local function and the massive discrete exponential function introduced in Boutillier et al. (Invent Math 208(1):109–189, 2017). This shows in particular that Z-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the Z-invariant spanning forests of Boutillier et al.  (2017), and deduce that the two models have the same second order phase transition in k. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this Z-invariant Ising model.

Mathematics Subject Classification




We acknowledge support from the Agence Nationale de la Recherche (Projet MAC2: ANR-10-BLAN-0123) and from the Région Centre-Val de Loire (Projet MADACA). We are grateful to the referee for his/her many insightful comments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Cédric Boutillier
    • 1
    Email author
  • Béatrice de Tilière
    • 2
  • Kilian Raschel
    • 3
  1. 1.Laboratoire de Probabilités, Statistique et ModélisationSorbonne UniversitéParisFrance
  2. 2.Laboratoire d’Analyse et de Mathématiques AppliquéesUniversité Paris-Est CréteilCréteilFrance
  3. 3.CNRS, Institut Denis PoissonUniversité de ToursToursFrance

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