Journal of Statistical Physics

, Volume 160, Issue 6, pp 1545–1622 | Cite as

Wetting Transitions for a Random Line in Long-Range Potential

Article

Abstract

We consider a restricted Solid-on-Solid interface in \(\mathbb {Z}_{+}\), subject to a potential \(V\left( n\right) \) behaving at infinity like \(-\text {w} /n^{2}\). Whenever there is a wetting transition as \(b_{0}\equiv \exp V\left( 0\right) \) is varied, we prove the following results for the density of returns \(m\left( b_{0}\right) \) to the origin: if \(\text {w}<-3/8\), then \( m\left( b_{0}\right) \) has a jump at \(b_{0}^{c}\); if \(-3/8<\text {w}<1/8\), then \(m\left( b_{0}\right) \sim \left( b_{0}^{c}-b_{0}\right) ^{\theta /\left( 1-\theta \right) }\) where \(\theta =1-\frac{\sqrt{1-8\text {w}}}{2}\); if \(\text {w}>1/8\), there is no wetting transition.

Keywords

Random walks Solid-on-Solid model  Wetting transition 

References

  1. 1.
    Abraham, D.B.: Solvable model with a roughening transition for a planar ising ferromagnet. Phys. Rev. Lett. 44, 1165 (1980)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Abraham, D.B., Smith, E.R.: An exactly solved model with a wetting transition. J. Stat. Phys. 43(3–4), 621–643 (1986)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series, vol. 55. US Government Printing Office, Washington, DC (1964)Google Scholar
  4. 4.
    Bender, C.M., Boettcher, S., Moshe, M.: Spherically symmetric random walks in noninteger dimension. J. Math. Phys. 35(9), 4941–4963 (1994)MathSciNetADSCrossRefMATHGoogle Scholar
  5. 5.
    Bender, C.M., Cooper, F., Meisinger, P.N.: Spherically symmetric random walks. I. Representation in terms of orthogonal polynomials. Phys. Rev. E 54(1), 100–111 (1996)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Burchnall, J.L., Chaundy, T.W.: The hypergeometric identities of Cayley, Orr, and Bailey. Proc. Lond. Math. Soc. 50, 56–74 (1948)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    De Coninck, J., Dunlop, F., Huillet, T.: Random walk weakly attracted to a wall. J. Stat. Phys. 133, 271–280 (2008)MathSciNetADSCrossRefMATHGoogle Scholar
  8. 8.
    De Coninck, J., Dunlop, F., Huillet, T.: Random walk versus random line. Phys. A 388(19), 4034–4040 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dette, H., Fill, J.A., Pitman, J., Studden, W.J.: Wall and Siegmund duality relations for birth and death chains with reflecting barrier. Dedicated to Murray Rosenblatt. J. Theor. Probab. 10(2), 349–374 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New-York (1953)Google Scholar
  11. 11.
    Fisher, M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34(5–6), 667–729 (1984)ADSCrossRefMATHGoogle Scholar
  12. 12.
    Gradshteyn, I., Ryzhik, I.: Table of Integrals, Series and Products. Academic Press, New York (1965)Google Scholar
  13. 13.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 2. Wiley, New York (1974)MATHGoogle Scholar
  14. 14.
    Jacobsen, L., Masson, D.: On the convergence of limit periodic continued fractions \(\mathbf{K}(a_{n}/1)\) when \(a_{n}\rightarrow -1/4\). Part III. Constr. Approx. 6, 363–374 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Karlin, S., McGregor, J.: Random walks. Ill. J. Math. 3, 66–81 (1959)MathSciNetMATHGoogle Scholar
  16. 16.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, Berlin (1966)CrossRefGoogle Scholar
  17. 17.
    Kroll, D.M., Lipowsky, R.: Universality classes for the critical wetting transition in two dimensions. Phys. Rev. B 28(9), 5273–5280 (1983)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Lamperti, J.: Criteria for the recurrence or transience of stochastic process. I. J. Math. Anal. Appl. 1, 314–330 (1960)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lamperti, J.: A new class of probability limit theorems. J. Math. Mech. 11, 749–772 (1962)MathSciNetMATHGoogle Scholar
  20. 20.
    Lamperti, J.: Criteria for stochastic processes II: passage-time moments. J. Math. Anal. Appl. 7, 127–145 (1963)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Levinson, N.: The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lipowsky, R., Fisher, M.E.: Scaling regimes and functional renormalization for wetting transitions. Phys. Rev. B 36, 2126–2141 (1987)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Lipowsky, R., Nieuwenhuizen, ThM: Intermediate fluctuation regime for wetting transitions in two dimensions. J. Phys. A 21, L89–L94 (1988)ADSCrossRefGoogle Scholar
  24. 24.
    Littin, J., Martínez, S.: R-positivity of nearest neighbor matrices and applications to Gibbs states. Stoch. Process. Appl. 120(12), 2432–2446 (2010)CrossRefMATHGoogle Scholar
  25. 25.
    Nussbaum, R.: The radius of the essential spectrum. Duke Math. J. 37, 473–478 (1970)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Palais, R.: A simple proof of the Banach contraction principle. J. Fixed Point Theory Appl. 2, 221–223 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Privman, V., Svrakic, N.M.: Wetting phenomena with long-range forces: exact results for the solid-on-solid model with the 1/r substrate potential. Phys. Rev. B 37, 5974–5977 (1988)ADSCrossRefGoogle Scholar
  28. 28.
    Vere-Jones, D.: Ergodic properties of nonnegative matrices I. Pac. J. Math. 22, 361–386 (1967)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Watson, G.N.: Asymptotic expansions of hypergeometric functions. Trans. Camb. Philos. Soc. 22, 277–308 (1918)Google Scholar
  30. 30.
    Yosida, K.: Functional Analysis, 2nd edn. Springer, New York (1968)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CPHT, CNRS UMR-7644Ecole PolytechniquePalaiseau CedexFrance
  2. 2.LPTM, CNRS UMR-8089 and University of Cergy-PontoiseCergy-PontoiseFrance

Personalised recommendations