The Annals of Probability

Entropic Repulsion and the Maximum of the two-dimensional harmonic

Erwin Bolthausen, Jean-Dominique Deuschel, and Giambattista Giacomin

Full-text: Open access


We consider the lattice version of the free field in two dimensions (also called harmonic crystal). The main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall. The basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure.

Article information

Ann. Probab., Volume 29, Number 4 (2001), 1670-1692.

First available in Project Euclid: 5 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G15: Gaussian processes 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Free field effective interface models entropic repulsion large deviations extrema of Gaussian fields multiscale decomposition


Bolthausen, Erwin; Deuschel, Jean-Dominique; Giacomin, Giambattista. Entropic Repulsion and the Maximum of the two-dimensional harmonic. Ann. Probab. 29 (2001), no. 4, 1670--1692. doi:10.1214/aop/1015345767.

Export citation


  • [1] Ben Arous, G. and Deuschel, J.-D. (1996). The construction of the d + 1 -dimensional Gaussian droplet. Comm. Math. Phys. 179 467-488.
  • [2] Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33-45.
  • [3] Biggins, J. D. (1997). Chernoff's theorem in the branching random walk. J. Appl. Probab. 14 630-636.
  • [4] Bolthausen, E. and Deuschel, J.-D. (1993). Critical large deviations for Gaussian fields in the phase transition regime. Ann. Probab. 21 1876-1920.
  • [5] Bolthausen, E., Deuschel, J.-D. and Zeitouni, O. (1995). Entropic repulsion of the lattice free field. Comm. Math. Phys. 170 417-443.
  • [6] Bolthausen, E. and Ioffe, D. (1997). Harmonic crystal on the wall: a microscopic approach. Comm. Math. Phys. 187 523-566.
  • [7] Bramson, M. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531-581.
  • [8] Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn-Minkowski and Pr´ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366-389.
  • [9] Derrida, B. and Gardner, E. (1986). Solution of the generalized random energy model. J. Phys. C 19 2253-2274.
  • [10] Deuschel, J.-D. (1996). Entropic repulsion of the lattice free field. II. The 0-boundary case. Comm. Math. Phys. 181 647-665.
  • [11] Deuschel, J.-D. and Giacomin, G. (1999). Entropic repulsion for the free field: pathwise characterization in d 3. Comm. Math. Phys. 206 447-462.
  • [12] Deuschel, J.-D. and Giacomin, G. (2000). Entropic repulsion for massless fields. Stochastic Process. Appl. 89 333-354.
  • [13] Deuschel, J.-D. and Stroock, D. (1989). Large Deviations. Academic Press, Boston.
  • [14] Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
  • [15] Lawler, G. F. (1991). Intersections of Random Walks. Birkh¨auser, Boston.
  • [16] Spitzer, F. (1964). Principles of Random Walks. Van Nostrand, New York.