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Wetting and Layering for Solid-on-Solid I: Identification of the Wetting Point and Critical Behavior

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Abstract

We provide a complete description of the low temperature wetting transition for the two dimensional solid-on-solid model. More precisely, we study the integer-valued field \({(\phi(x))_{x\in \mathbb{Z}^2}}\), associated associated with the energy functional

$$V(\phi)=\beta \sum_{x \sim y}|\phi(x)-\phi(y)|-\sum_{x} \left( h\mathbf{1}_{\{\phi(x)=0\}}-\infty\mathbf{1}_{\{\phi(x) < 0\}} \right).$$

Since the pioneering work Chalker [15], it is known that for every \({\beta}\), there exists \({h_{w}(\beta) > 0}\) delimiting a transition between a delocalized phase (\({h < h_{w}(\beta)}\)) where the proportion of points at level zero vanishes, and a localized phase (\({h > h_{w}(\beta)}\)) where this proportion is positive. We prove in the present paper that for \({\beta}\) sufficiently large we have

$$h_w(\beta)= \log \left(\frac{e^{4\beta}}{e^{4\beta}-1} \right).$$

Furthermore, we provide a sharp asymptotic for the free energy at the vicinity of the critical line: We show that close to \({h_w(\beta)}\), the free energy is approximately piecewise affine and that the points of discontinuity for the derivative of the affine approximation forms a geometric sequence accumulating on the right of \({h_w(\beta)}\). This asymptotic behavior provides strong evidence for the conjectured existence of countably many “layering transitions” at the vicinity of the wetting line, corresponding to jumps for the typical height of the field.

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References

  1. Abraham D.B.: Solvable model with a roughening transition for a planar Ising ferromagnet. Phys. Rev. Lett. 44, 1165–1168 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  2. Alexander S., Dunlop F., Miracle-Solé S.: Layering in the Ising model. J. Stat. Phys. 141, 217–241 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Alexander K.S., Dunlop F., Miracle-Solé S.: Layering and Wetting Transitions for an SOS Interface. J. Stat. Phys. 142, 524–576 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Armitstead K., Yeomans J.M.: A series approach to wetting and layering transitions. II. Solid-on-solid models. J. Phys. A Math. Gen. 21, 159–171 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  5. Binder K., Landau D.P.: Wetting versus layering near the roughening transition in the three-dimensional Ising model. Phys. Rev. B 46, 4844–4854 (1992)

    Article  ADS  Google Scholar 

  6. Bolthausen E., Deuschel J.-D., Zeitouni O.: Absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger. J. Math. Phys. 41, 1211–1223 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Brandenberger R., Wayne C.E.: Decay of correlations in surface models. J. Stat. Phys. 27, 425–440 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  8. Bricmont J., Fontaine J.-R., Lebowitz J.L.: Surface tension, percolation and roughening. J. Stat. Phys. 29, 193–203 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  9. Burton W.K., Cabrera N., Frank F.C.: The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. R. Soc. A 243, 299–358 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Caputo P., Lubetzky E., Martinelli F., Sly A., Toninelli F.L.: Dynamics of (2 + 1)-dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion. Ann. Probab. 42, 151–1589 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caputo P., Lubetzky E., Martinelli F., Sly A., Toninelli F.L.: Scaling limit and cube-root fluctuations in SOS surfaces above a wall. J. Eur. Math. Soc. 18, 931–995 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Caputo P., Martinelli F., Toninelli F.L.: On the probability of staying above a wall for the (2 + 1)-dimensional SOS model at low temperature. Prob. Theory Relat. Fields 163, 803–831 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Caputo P., Velenik Y.: A note on wetting transition for gradient field. Stoch. Proc. Appl. 87, 107–113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cesi F., Martinelli F.: On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results. J. Stat. Phys. 82, 823–916 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Chalker J.T.: The pinning of an interface by a planar defect. J. Phys. A 15, 481485 (1982)

    Article  MathSciNet  Google Scholar 

  16. Dinaburg E.I., Mazel A.E.: Layering transition in SOS model with external magnetic field. J. Stat. Phys. 74, 533–563 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Dobrushin R.L.: Gibbs states describing a coexistence of phases for the three-dimensional Ising model. Theory Probab. Appl. 17, 582600 (1972)

    Google Scholar 

  18. Fisher M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–729 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Friedli S., Velenik Y.: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2017)

    Book  MATH  Google Scholar 

  20. Fröhlich J., Pfister C.-E.: Semi-infinite Ising model. II. The wetting and layering transitions. Commun. Math. Phys. 112, 51–74 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Fröhlich J., Spencer T.: The Kosterlitz–Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas. Commun. Math. Phys. 81, 527–602 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  22. Giacomin G.: Random Polymer Models. Imperial College Press, London (2007)

    Book  MATH  Google Scholar 

  23. Giacomin G., Lacoin H.: Disorder and wetting transition: the pinned harmonic crystal in dimension three or larger. Ann. Appl. Probab. 28, 577–606 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  24. Harris T.E.: A lower bound for the critical probability in a certain percolation. Proc. Camb. Philos. Soc. 56, 13–20 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Holley R.: Remarks on the FKG Inequalities. Commun. Math. Phys. 36, 227–231 (1974) (math-ph)

    Article  ADS  MathSciNet  Google Scholar 

  26. Ioffe D., Velenik Y.: Low-temperature interfaces: prewetting, layering, faceting and Ferrari–Spohn diffusions. Markov Proc. Relat. Fields

  27. Kotecký R., Preiss D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103, 491–498 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Lacoin H.: Wetting and layering for solid-on-solid II: layering transitions, Gibbs states, and regularity of the free energy. arXiv:1712.03736 [math-ph] (preprint)

  29. Pandit R., Schick M., Wortis M.: Systematics of multilayer adsorption phenomena on attractive substrates. Phys. Rev. B 26, 5112–5140 (1982)

    Article  ADS  Google Scholar 

  30. Peierls, R.: On Isings model of ferromagnetism. Proc. Camb. Philos. Soc. 32, 477–181 (1936)

  31. Temperley H.N.V.: Statistical mechanics and the partition of numbers II. The form of crystal surfaces. Proc. Camb. Philos. Soc. 48, 683–697 (1952)

    Article  ADS  MathSciNet  Google Scholar 

  32. Weeks J.D., Gilmer G.H., Leamy H.J.: Structural transition in the Ising-model interface. Phys. Rev. Lett. 31, 549–551 (1973)

    Article  ADS  Google Scholar 

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  1. IMPA, Institudo de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, CEP-22460-320, Brazil

    Hubert Lacoin

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  1. Hubert Lacoin
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Correspondence to Hubert Lacoin.

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Communicated by H. Spohn

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Lacoin, H. Wetting and Layering for Solid-on-Solid I: Identification of the Wetting Point and Critical Behavior. Commun. Math. Phys. 362, 1007–1048 (2018). https://doi.org/10.1007/s00220-018-3162-4

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