The Annals of Probability

Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension

Noemi Kurt

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Abstract

We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green’s function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. d=4 is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.

Article information

Source
Ann. Probab., Volume 37, Number 2 (2009), 687-725.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1241099926

Digital Object Identifier
doi:10.1214/08-AOP417

Mathematical Reviews number (MathSciNet)
MR2510021

Zentralblatt MATH identifier
1166.60060

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 31B30: Biharmonic and polyharmonic equations and functions

Keywords
Random interfaces membrane model entropic repulsion discrete biharmonic Green’s function

Citation

Kurt, Noemi. Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension. Ann. Probab. 37 (2009), no. 2, 687--725. doi:10.1214/08-AOP417. https://projecteuclid.org/euclid.aop/1241099926


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References

  • [1] Bolthausen, E., Deuschel, J.-D. and Giacomin, G. (2001). Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 1670–1692.
  • [2] Bolthausen, E., Deuschel, J.-D. and Zeitouni, O. (1995). Entropic repulsion of the lattice free field. Comm. Math. Phys. 170 417–443.
  • [3] Caravenna, F. and Deuschel, J. D. Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction. Ann. Probab. To appear. Available at arXiv.org: math/0703434[math.PR]
  • [4] Caravenna, F. and Deuschel, J. D. Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction. Available at arXiv.org: math/08023154[math.PR]
  • [5] Daviaud, O. (2006). Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 962–986.
  • [6] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
  • [7] Giacomin, G. Aspects of statistical mechanics of random surfaces. Notes of the lectures given at the IHP in the fall 2001. Available at www.proba.jussieu.fr/pageperso/giacomin/pub/publicat.html.
  • [8] Herbst, I. and Pitt, L. (1991). Diffusion equation techniques in stochastic monotonicity and positive correlations. Probab. Theory Related Fields 87 275–312.
  • [9] Hiergeist, C. and Lipowsky, R. (1997). Local contacts of membranes and strings. Physica A 244 164–175.
  • [10] Kurt, N. (2007). Entropic repulsion for a class of Gaussian interface models in high dimensions. Stochastic Process. Appl. 117 23–34.
  • [11] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
  • [12] Lebowitz, J. L. and Maes, C. (1987). The effect of an external field on an interface, entropic repulsion. J. Statist. Phys. 46 39–49.
  • [13] Sakagawa, H. (2003). Entropic repulsion for a Gaussian lattice field with certain finite range interaction. J. Math. Phys. 44 2939–2951.
  • [14] Volmer, A., Seifert, U. and Lipowsky, R. (1998). Critical behavior of interacting surfaces with tension. Eur. Phys. J. B 5 193–203.
  • [15] Wloka, J. (1987). Partial Differential Equations. Cambridge Univ. Press, Cambridge.