Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A remark on the low-temperature behavior of the SOS interface in half-space


We investigate the low-temperature phase diagram of thed-dimensional (d≥2) solid-on-solid (SOS) interface constrained to lie above a rigid wall to which it is attracted by a constant force. We prove uniqueness of the Gibbs state and exponentially fast convergence (in the diameter of the domain) of the finite-volume expectation of local observables, for all values of parameters where uniqueness in the class of translation-periodic limit Gibbs states was established previously. These domains of uniqueness are bordered by lines on which the system undergoes a layering transition.

This is a preview of subscription content, log in to check access.


  1. 1.

    J. Bricmont, A. El. Mellouki, and J. Fröhlich, Random surfaces in statistical mechanics: Roughening, rounding, wetting, ...,J. Stat. Phys. 42:743–798 (1986).

  2. 2.

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

  3. 3.

    F. Cesi and F. Martinelli, On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results,J. Stat. Phys. (1995).

  4. 4.

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

  5. 5.

    G. Forgacs, R. Lipowsky, and T. M. Nieuwenhuizen, The behavior of interfaces in ordered and disordered systems, inPhase Transitions and Critical Phenomena, Vol. 14, C. Comb and J. L. Lebowitz, eds. (Academic Press, New York, 1991), pp. 136–363.

  6. 6.

    C. Fortuin, P. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets,Commun. Math. Phys. 22:89–103 (1971).

  7. 7.

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

  8. 8.

    R. Holley, Remarks on the FKG inequalities,Commun. Math. Phys. 23:87–99 (1971).

  9. 9.

    J. L. Lebowitz and A. Martin-Löf, On the uniqueness of the equilibrium state for Ising spin systems,Commun. Math. Phys. 25:276–282 (1971).

  10. 10.

    J. L. Lebowitz and E. Presutti, Statistical mechanics of systems of unbounded spins,Commun. Math. Phys. 50:195–218 (1976).

  11. 11.

    M. Zahradnik, An alternate version of Pirogov-Sinai theory,Commun. Math. Phys. 93:559–581 (1984).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lebowitz, J.L., Mazel, A.E. A remark on the low-temperature behavior of the SOS interface in half-space. J Stat Phys 84, 379–397 (1996).

Download citation

Key Words

  • Gibbs state
  • SOS model
  • layering transition
  • uniqueness
  • FKG
  • cluster expansion