Journal of Statistical Physics

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

Wetting Transitions for a Random Line in Long-Range Potential

  • P. Collet
  • F. DunlopEmail author
  • T. Huillet


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.


Random walks Solid-on-Solid model  Wetting transition 



The authors are grateful to the referees for useful remarks about the physical background.


  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)MathSciNetADSCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Coninck, J., Dunlop, F., Huillet, T.: Random walk weakly attracted to a wall. J. Stat. Phys. 133, 271–280 (2008)MathSciNetADSCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)ADSCrossRefzbMATHGoogle 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)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Karlin, S., McGregor, J.: Random walks. Ill. J. Math. 3, 66–81 (1959)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lamperti, J.: A new class of probability limit theorems. J. Math. Mech. 11, 749–772 (1962)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lamperti, J.: Criteria for stochastic processes II: passage-time moments. J. Math. Anal. Appl. 7, 127–145 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Levinson, N.: The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  25. 25.
    Nussbaum, R.: The radius of the essential spectrum. Duke Math. J. 37, 473–478 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Palais, R.: A simple proof of the Banach contraction principle. J. Fixed Point Theory Appl. 2, 221–223 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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

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© 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

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