Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
The critical Z-invariant Ising model via dimers: the periodic case
Download PDF
Download PDF
  • Published: 13 March 2009

The critical Z-invariant Ising model via dimers: the periodic case

  • Cédric Boutillier1 &
  • Béatrice de Tilière2 

Probability Theory and Related Fields volume 147, pages 379–413 (2010)Cite this article

  • 144 Accesses

  • 39 Citations

  • Metrics details

Abstract

We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical \({\mathbb{Z}^2}\) , triangular and honeycomb lattice at the critical temperature. Fisher (J Math Phys 7:1776–1781, 1966) introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model. We prove that the dimer characteristic polynomial is equal (up to a constant) to the critical Laplacian characteristic polynomial, and defines a Harnack curve of genus 0. We prove an explicit expression for the free energy, and for the Gibbs measure obtained as weak limit of Boltzmann measures.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Baxter R.J.: Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics. Proc. R. Soc. Lond. Ser. A 404(1826), 1–33 (1986)

    Article  MathSciNet  Google Scholar 

  2. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press Inc. (Harcourt Brace Jovanovich Publishers), London (1989) (Reprint of the 1982 original)

  3. Boutillier, C., de Tiliè re, B.: The critical Z-invariant ising model via dimers: locality properties. arXiv:0902.1882, February (2009)

  4. Biggs N.: Algebraic graph theory. Cambridge Mathematical Library, 2nd edn. University Press, Cambridge (1993)

    Google Scholar 

  5. Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitres 1 à 3. Hermann, Paris(1970)

  6. Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Am. Math. Soc. 14(2), 297–346 (2001) (electronic)

    Google Scholar 

  7. Cimasoni D., Reshetikhin N.: Dimers on surface graphs and spin structures. I. Comm. Math. Phys. 275(1), 187–208 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Costa-Santos R.: Geometrical aspects of the Z-invariant Ising model. Eur. Phys. J. B 53(1), 85–90 (2006)

    Article  Google Scholar 

  9. de Tilière B.: Quadri-tilings of the plane. Probab. Theory Related Fields 137(3–4), 487–518 (2007)

    MATH  MathSciNet  Google Scholar 

  10. Dolbilin, N.P., Zinov′ev, Yu.M., Mishchenko, A.S., Shtan′ko, M.A., Shtogrin, M.I.: Homological properties of two-dimensional coverings of lattices on surfaces. Funktsional. Anal. i Prilozhen. 30(3):19–33, 95, 1996

    Google Scholar 

  11. Fisher M.E.: On the dimer solution of planar Ising models. J. Math. Phys. 7, 1776–1781 (1966)

    Article  Google Scholar 

  12. Ishikawa, M., Wakayama, M.: Minor summation formulas of Pfaffians, survey and a new identity. In: Combinatorial Methods in Representation Theory (Kyoto, 1998), vol 28 of Adv. Stud. Pure Math., Kinokuniya, Tokyo, pp 133–142 (2000)

  13. Kasteleyn P.W.: The statistics of dimers on a lattice : I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)

    Article  Google Scholar 

  14. Kasteleyn, P.W.: Graph theory and crystal physics. In: Graph Theory and Theoretical Physics, pp.43–110. Academic Press, London, 1967

  15. Kenyon R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Stat. 33(5), 591–618 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kenyon R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kirchhoff G.: Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Annalen der Physik 148, 497–508 (1847)

    Article  Google Scholar 

  18. Kenyon R., Okounkov A.: Planar dimers and Harnack curves. Duke Math. J. 131(3), 499–524 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math. (2) 163(3):1019–1056, 2006

    Google Scholar 

  20. Kenyon R., Schlenker J.-M.: Rhombic embeddings of planar quad-graphs. Trans. Amer. Math. Soc. 357(9), 3443–3458 (2005) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kramers H.A., Wannier G.H.: Statistics of the two-dimensional ferromagnet. I. Phys. Rev. (2) 60, 252–262, 1941

    Google Scholar 

  22. Mercat C.: Discrete Riemann surfaces and the Ising model. Comm. Math. Phys. 218(1), 177–216 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. McCoy B., Wu F.: The Two-dimensional Ising Model. Harvard University Press, Cambridge (1973)

    MATH  Google Scholar 

  24. Ryshik I.M., Gradšteĭn I.S.: Summen-, Produkt- und Integral-tafeln. VEB Deutscher Verlag der Wissenschaften, Berlin (1957)

    Google Scholar 

  25. Sheffield, S.: Random surfaces. Astérisque (304):vi+175, 2005

  26. Tesler G.: Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B 78(2), 198–231 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wannier G.H.: The statistical problem in cooperative phenomena. Rev. Mod. Phys. 17(1), 50–60 (1945)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI Pierre et Marie Curie, Case courrier 188, 4 place Jussieu, 75252, Paris Cedex 05, France

    Cédric Boutillier

  2. Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2007, Neuchâtel, Switzerland

    Béatrice de Tilière

Authors
  1. Cédric Boutillier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Béatrice de Tilière
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cédric Boutillier.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Boutillier, C., de Tilière, B. The critical Z-invariant Ising model via dimers: the periodic case. Probab. Theory Relat. Fields 147, 379–413 (2010). https://doi.org/10.1007/s00440-009-0210-1

Download citation

  • Received: 08 December 2008

  • Published: 13 March 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00440-009-0210-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (2000)

  • 82B20
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature