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Examples of DLR States Which are Not Weak Limits of Finite Volume Gibbs Measures with Deterministic Boundary Conditions

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Abstract

We review what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and in particular we prove that the mixture \(\frac{1}{2}(\mu ^\pm +\mu ^\mp )\) of two reflection-symmetric Dobrushin states of the 3-dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finite-volume measures with deterministic boundary conditions. Finally we point out what the issues are in order to extend the analysis to the Potts model, and give a few conjectures.

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Notes

  1. We emphasize that the natural monotonicity of the fluctuations that we could expect with respect to the temperature is not true in general. Indeed, positive temperature may result in reduction of the fluctuations [6].

  2. The boundary of a fixed region \(\Omega \subset \mathbb {R}^d\) must be partitioned in such a way that for each \(n\), on the boundary of \(\Omega _n=\Omega \cap \frac{1}{n}\mathbb {Z}^d\), the number of nearest-neighbor pairs of vertices having different spins is \(o(n^{d-1})\).

  3. More precisely in a region of size \(f(n)\) such that \(\log n\ll f(n)\ll n^{1/(d-1)}\).

References

  1. Abraham, D.B., Reed, P.: Interface profile of the Ising ferromagnet in two dimensions. Commun. Math. Phys. 49, 35–46 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  2. Aizenman, M.: Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Commun. Math. Phys. 73, 83–94 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  3. Albeverio, S., Zegarliński, B.: Markov Global property in quantum field theory and statistical mechanics: a review on results and problems, in Ideas and methods in quantum and statistical physics (Oslo : Cambridge Univ. Press, Cambridge 1992, 331–369 (1988)

  4. Alfaro, M., Conger, M., Hodges, K.: The structure of singularities in \(\Phi \)-minimizing networks in \(\mathbb{R}^{2}\). Pacific J. Math. 149, 201–210 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bodineau, T.: Translation invariant Gibbs states for the Ising model. Probab. Theory Relat. Fields 135, 153–168 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bodineau, T., Giacomin, G., Velenik, Y.: On entropic reduction of fluctuations. J. Stat. Phys. 102, 1439–1445 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bodineau, T., Ioffe, D., Velenik, Y.: Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41, 1033–1098 (2000). (Probabilistic techniques in equilibrium and nonequilibrium statistical physics)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Campanino, M., Ioffe, D., Velenik, Y.: Fluctuation theory of connectivities for subcritical random cluster models. Ann. Probab. 36, 1287–1321 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cerf, R., Pisztora, Á.: Phase coexistence in Ising, Potts and percolation models. Ann. Inst. H. Poincaré Probab. Stat. 37, 643–724 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Černý, J., Kotecký, R.: Interfaces for random cluster models. J. Stat. Phys. 111, 73–106 (2003)

    Article  MATH  Google Scholar 

  11. Coquille, L., Duminil-Copin, H., Ioffe, D., Velenik, Y.: On the Gibbs states of the noncritical Potts model on \(\mathbb{Z}^{2}\). Probab. Theory Relat. Fields 158, 477–512 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. Coquille, L., Velenik, Y.: A finite-volume version of Aizenman-Higuchi theorem for the 2d Ising model. Probab. Theory Relat. Fields 153, 25–44 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dobrushin, R.L.: Description of a random field by means of conditional probabilities and the conditions governing its regularity. Theory Probab. Appl. 13, 197–224 (1968)

    Article  Google Scholar 

  14. Dobrushin, R.L.: The Gibbs state that describes the coexistence of phases for a three-dimensional Ising model. Teor. Verojatnost. i Primenen. 17, 619–639 (1972)

    MathSciNet  Google Scholar 

  15. Edwards, R.G., Sokal, A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  16. Gallavotti, G.: The phase separation line in the two-dimensional Ising model. Commun. Math. Phys. 27, 103–136 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  17. Georgii, H. -O.: Gibbs measures and phase transitions, vol. 9 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, (1988)

  18. Georgii, H.-O., Häggström, O., Maes, C.: The random geometry of equilibrium phases, in Phase transitions and critical phenomena. Phase Transit. Crit. Phenom. 18, 1–142 (2001). (Academic Press, San Diego)

    Article  Google Scholar 

  19. Gielis, G., Grimmett, G.: Rigidity of the interface in percolation and random-cluster models. J. Stat. Phys. 109, 1–37 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Goldstein, S.: Remarks on the global Markov property. Commun. Math. Phys. 74, 223–234 (1980)

    Article  ADS  MATH  Google Scholar 

  21. Greenberg, L., Ioffe, D.: On an invariance principle for phase separation lines. Ann. Inst. H. Poincaré Probab. Stat. 41, 871–885 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. G. Grimmett, The random-cluster model, vol. 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2006

  23. Higuchi, Y.: On limiting Gibbs states of the two-dimensional Ising models. Publ Res. Inst. Math. Sci. 14, 53–69 (1978)

    Article  MathSciNet  Google Scholar 

  24. Higuchi, Y.: On some limit theorems related to the phase separation line in the two-dimensional Ising model. Z. Wahrsch. Verw. Gebiete 50, 287–315 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  25. Higuchi, Y.: On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model, in Random fields, Vol. I, II (Esztergom : vol. 27 of Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam 1981, pp 517–534 (1979)

  26. Israel, R.B.: Some examples concerning the global markov property. Commun. Math. Phys. 105, 669–673 (1986)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Kahn, J., Weininger, N.: Positive association in the fractional fuzzy Potts model. Ann. Probab. 35, 2038–2043 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kenyon, R.: Local statistics of lattice dimers. Annales de l’institut Henri Poincaré (B) 33, 591–618 (1997)

  29. Lanford III, O.E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  30. Martirosian, D.H.: Translation invariant Gibbs states in the \(q\)-state Potts model. Commun. Math. Phys. 105, 281–290 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  31. Messager, A., Miracle-Sole, S.: Correlation functions and boundary conditions in the Ising ferromagnet. J. Stat. Phys. 17, 245–262 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  32. Miracle-Sole, S.: Surface tension, step free energy, and facets in the equilibrium crystal. J. Stat. Phys. 79, 183–214 (1995)

    Article  ADS  MATH  Google Scholar 

  33. Miyamoto, M.: Statistical mechanics – a mathematical approach (in Japanese). Nippon-Hyoron-Sha Co., Ltd (2004)

    Google Scholar 

  34. Schonmann, R.H.: On two correlation inequalities for Potts models. J. Stat. Phys. 52, 61–67 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. van Beijeren, H.: Interface sharpness in the ising system. Commun. Math. Phys. 40, 1–6 (1975)

    Article  ADS  Google Scholar 

  36. van den Berg, J., Häggström, O., Kahn, J.: Some conditional correlation inequalities for percolation and related processes. Random Struct. Algorithms 29, 417–435 (2006)

    Article  MATH  Google Scholar 

  37. van Enter, A.C.D., Medved, I., Netočný, K.: Chaotic size dependence in the Ising model with random boundary conditions. Markov Process. Relat. Fields 8, 479–508 (2002)

    MATH  Google Scholar 

  38. van Enter, A.C.D., Netočný, K., Schaap, H.G.: On the Ising model with random boundary condition. J. Stat. Phys. 118, 997–1056 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I am grateful to A. van Enter for encouraging me to work on this question, and for valuable comments and suggestions. I also thank Y. Velenik for his advice, a stimulating discussion and a few references, A. Bovier for mentioning reference [3] to me, and V. Beffara for the simulations of Figs. 2 and 3. I am indebted to Y. Higuchi for pointing out reference [33]. This research was supported by the German Research Foundation (DFG) and the Hausdorff Center for Mathematics (HCM).

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  1. Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, 53115, Bonn, Germany

    Loren Coquille

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Coquille, L. Examples of DLR States Which are Not Weak Limits of Finite Volume Gibbs Measures with Deterministic Boundary Conditions. J Stat Phys 159, 958–971 (2015). https://doi.org/10.1007/s10955-015-1211-3

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