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Critical Temperature of Periodic Ising Models

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Abstract

A periodic Ising model has interactions which are invariant under translations of a full-rank sublattice \({\mathfrak{L}}\) of \({\mathbb{Z}^2}\). We prove an exact, quantitative characterization of the critical temperature, defined as the supremum of temperatures for which the spontaneous magnetization is strictly positive. For the ferromagnetic model, the critical temperature is the solution of a certain algebraic equation, resulting from the condition that the spectral curve of the corresponding dimer model on the Fisher graph has a real zero on the unit torus. With our technique we provide a simple proof for the exponential decay of spin-spin correlations above the critical temperature, as well as the exponential decay of the edge-edge correlations for all non-critical edge weights of the corresponding dimer model on periodic Fisher graphs.

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References

  1. Aizenman, M.: Rigorous Results of Critical Behavior II. In: Statistical Physics and Dynamical Systems (Köszeg, 1984), Progr. Phys., 10, Boston, MA: Birkhäuser Boston, 1985, pp. 453–481

  2. Aizenman M., Barsky D., Fernández R.: The Phase Transition in a General Class of Ising-type Models is Sharp. J. Stat. Phys. 47(3-4), 343–374 (1987)

    Article  ADS  Google Scholar 

  3. Baxter R.: Exactly Solved Models in Statistical Mechanics. Academic Press, London-Newyork (1982)

    MATH  Google Scholar 

  4. Boutillier C., de Tilière B.: The Critical Z-invariant Ising Model via Dimers: the Periodic Case. Prob. Th. Rel. Fields 147, 379–413 (2010)

    Article  MATH  Google Scholar 

  5. Cimasoni D.: A Generalized Kac-Ward Formula. J. Stat. Mech. 2010, P07023 (2010)

    Article  Google Scholar 

  6. Cohn H., Kenyon R., Propp J.: A Variational Principle for Domino Tilings. J. Amer. Math. Soc. 14(2), 297–346 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fisher M.: On the Dimer Solution of Planar Ising Models. J. Math. Phys. 7, 1776–1781 (1966)

    Article  ADS  Google Scholar 

  8. Fortuin C., Kasteleyn P., Ginibre J.: Correlation Inequalities on some Partially Ordered Sets. Commun. Math. Phys. 22(2), 89–103 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Gohberg, I.C., Feldman, I.A.: Convolution Equations and Projection Methods for their Solutions. Trans. Math. Mon. 41, Providence, RI: Amer. Math. Soc., 1974

  10. Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Trans. Math. Mon. 18, Providence, RI: Amer. Math. Soc., 1969

  11. Griffiths, R.: Rigorous Results and Theorems. In: Phase Transition and Critical Phenomena, edited by C. Domb, M. Green, New York: Academic Press, 1972

  12. Holley R.: Remarks on the FKG Inequalities. Commun. Math. Phys. 36(3), 227–231 (1974)

    Article  MathSciNet  ADS  Google Scholar 

  13. Kasteleyn P.: The Statistics of Dimers on a Lattice. Physica 27, 1209–1225 (1961)

    Article  ADS  MATH  Google Scholar 

  14. Kasteleyn, P.: Graph Theory and Crystal Physics. In: Graph Theory and Theoretical Physics, London: Academic Press, 1967

  15. Kenyon, R.: An Introduction to the Dimer Model. http://arxiv.org/abs/math/0310326v1 [math.co], 2003

  16. Kenyon R.: Local Statistics on Lattice Dimers. Ann. Inst. H. Poicaré. Probabilités 33, 591–618 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Kenyon R., Okounkov A., Sheffield S.: Dimers and Amoebae. Ann. Math. 163(3), 1019–1056 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Lanford III, O.: Entropy and Equilibrium States in Classical Statistical Mechanics. In: Statistical Mechanics and Mathematical Problems, A. Lenard, ed., Berlin-Heidelberg-New York: Springer, 1973

  20. Lebowitz J.: Bounds on the Correlations and Analyticity Properties of Ferromagnetic Ising Spin Systems. Commun. Math. Phys. 28(4), 313–321 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  21. Lebowitz J.: Coexistence of Phases in Ising Ferromagnets. J. Stat. Phys. 16(6), 453–461 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  22. Lebowitz J., Martin-Löf A.: On the Uniqueness of the Equilibrium State for Ising Spin Systems. Commun. Math. Phys. 25(4), 276–282 (1972)

    Article  ADS  Google Scholar 

  23. Lee T.D., Yang C.N.: Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model. Phys. Rev. 87, 410–419 (1952)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Li, Z.: Spectral Curve of Periodic Fisher Graphs. http://arxiv.org/abs/1008.3936v5 [math.cv], 2012

  25. McCoy B., Wu T.: The Two Dimensional Ising Model. Harvard University Press, Cambridge, MA (1973)

    MATH  Google Scholar 

  26. Sheffield, S.: Random Surfaces. Arterisque, No. 304 (2005)

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Widom H.: Asymptotic Behavior of Block Toeplitz Matrices and Determinants. II. Adv. Math. 21, 1–29 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, GB30WA, UK

    Zhongyang Li

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  1. Zhongyang Li
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Correspondence to Zhongyang Li.

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Communicated by H. Spohn

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Li, Z. Critical Temperature of Periodic Ising Models. Commun. Math. Phys. 315, 337–381 (2012). https://doi.org/10.1007/s00220-012-1571-3

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  • DOI: https://doi.org/10.1007/s00220-012-1571-3

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