Journal of Statistical Physics

, Volume 147, Issue 1, pp 18–34 | Cite as

On the Free Energy of a Gaussian Membrane Model with External Potentials

  • Hironobu Sakagawa


We consider a class of d-dimensional Gaussian lattice field which is known as a model of semi-flexible membrane. We study the free energy of the model with external potentials and show the following:

(1) We consider the model with δ-pinning and prove that the field is always localized when d≥4.

(2) Consider the model confined between two hard walls. We give asymptotics of the free energy as the height of the wall goes to infinity.


Gaussian field Membrane Pinning Confinement 



The author would like to thank the referee for his very useful comments. This work was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B), No. 23740086.


  1. 1.
    Bolthausen, E., Deuschel, J.-D.: Critical large deviation for Gaussian field in the phase transition regime. Ann. Probab. 21, 1876–1920 (1993) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bolthausen, E., Deuschel, J.-D., Zeitouni, O.: Absence of a wetting transition for a pinned harmonic crystals in dimensions three and larger. J. Math. Phys. 41, 1211–1223 (2000) MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Bolthausen, E., Ioffe, D.: Harmonic crystal on the wall: a microscopic approach. Commun. Math. Phys. 187, 523–566 (1997) MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Bolthausen, E., Velenik, Y.: Critical behavior of the massless free field at the depinning transition. Commun. Math. Phys. 223, 161–203 (2001) MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Bricmont, J., El Mellouki, A., Fröhlich, J.: Random surfaces in Statistical Mechanics: roughening, rounding, wetting. J. Stat. Phys. 42(5/6), 743–798 (1986) ADSCrossRefGoogle Scholar
  6. 6.
    Caravenna, F., Deuschel, J.-D.: Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction. Ann. Probab. 36, 2388–2433 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Caravenna, F., Deuschel, J.-D.: Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction. Ann. Probab. 37, 903–945 (2009) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dunlop, F., Magnen, J., Rivasseau, V., Roche, P.: Pinning of an interface by a weak potential. J. Stat. Phys. 66, 71–98 (1992) MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Funaki, T.: Stochastic interface models. In: Picard, J. (ed.) Lectures on Probability Theory and Statistics, Ecole d’Eté de Probabilités de Saint-Flour XXXIII, 2003. Lect. Notes Math., vol. 1869, pp. 103–274. Springer, Berlin (2005) CrossRefGoogle Scholar
  10. 10.
    Ginibre, J.: General formulation of Griffith’s inequalities. Commun. Math. Phys. 16, 310–328 (1970) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988) MATHCrossRefGoogle Scholar
  12. 12.
    Hargé, G.: A particular case of correlation inequality for the Gaussian measure. Ann. Probab. 27, 1939–1951 (1999) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hryniv, O., Velenik, Y.: Some rigorous results on semiflexible polymers. I. Free and confined polymers. Stoch. Process. Appl. 119, 3081–3100 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kurt, N.: Entropic repulsion for a class of Gaussian interface models in high dimensions. Stoch. Process. Appl. 117, 23–34 (2007) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kurt, N.: Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension. Ann. Probab. 37, 687–725 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge (2010) MATHGoogle Scholar
  17. 17.
    Leiber, S.: Equilibrium statistical mechanics of fluctuating films and membranes. In: Statistical Mechanics of Membranes and Surfaces, 2nd edn., pp. 49–101. World Scientific, Singapore (2004) Google Scholar
  18. 18.
    Li, W.V., Shao, Q.M.: Gaussian processes: inequalities, small ball probabilities and applications. In: Stochastic Processes: Theory and Methods. Handbook of Statist, vol. 19, pp. 533–597. North-Holland, Amsterdam (2001) CrossRefGoogle Scholar
  19. 19.
    Lipowsky, R.: Generic interaction of flexible membranes. In: Structure and Dynamics of Membranes. Handbook of Biological Physics, vol. 1, pp. 521–602. North-Holland, Amsterdam (1995) Google Scholar
  20. 20.
    Ruiz-Lorenzo, J.J., Cuerno, R., Moro, E., Sánchez, A.: Phase transition in tensionless surfaces. Biophys. Chem. 115, 187–193 (2005) CrossRefGoogle Scholar
  21. 21.
    Sakagawa, H.: Entropic repulsion for a Gaussian lattice field with certain finite range interaction. J. Math. Phys. 44, 2939–2951 (2003) MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Sakagawa, H.: Bounds on the mass for the high dimensional Gaussian lattice field between two hard walls. J. Stat. Phys. 129, 537–553 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Sakagawa, H.: Confinement of the two dimensional discrete Gaussian free field between two hard walls. Electron. J. Probab. 14, 2310–2327 (2009) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Schechtman, G., Schlumprecht, Th., Zinn, J.: On the Gaussian measure of the intersection. Ann. Probab. 26, 346–357 (1998) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Velenik, Y.: Localization and delocalization of random interfaces. Probab. Surv. 3, 112–169 (2006) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and TechnologyKeio UniversityYokohamaJapan

Personalised recommendations