Entropic Repulsion and the Maximum of the two-dimensional harmonic
Erwin Bolthausen, Jean-Dominique Deuschel, and Giambattista Giacomin
Source: Ann. Probab. Volume 29, Number 4
(2001), 1670-1692.
Abstract
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.
First Page:
Show
Hide
Keywords: Free field; effective interface models; entropic repulsion; large deviations; extrema of Gaussian fields; multiscale decomposition
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1015345767
Digital Object Identifier: doi:10.1214/aop/1015345767
Mathematical Reviews number (MathSciNet): MR1880237
Zentralblatt MATH identifier: 1034.82018
References
[1] Ben Arous, G. and Deuschel, J.-D. (1996). The construction of the d + 1 -dimensional Gaussian droplet. Comm. Math. Phys. 179 467-488.
Mathematical Reviews (MathSciNet): MR97h:60047
Zentralblatt MATH: 0858.60096
Digital Object Identifier: doi:10.1007/BF02102597
Project Euclid: euclid.cmp/1104287000
[2] Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33-45.
Zentralblatt MATH: 0104.11905
[3] Biggins, J. D. (1997). Chernoff's theorem in the branching random walk. J. Appl. Probab. 14 630-636.
Mathematical Reviews (MathSciNet): MR464415
Zentralblatt MATH: 0373.60090
Digital Object Identifier: doi:10.2307/3213469
JSTOR: links.jstor.org
[4] Bolthausen, E. and Deuschel, J.-D. (1993). Critical large deviations for Gaussian fields in the phase transition regime. Ann. Probab. 21 1876-1920.
Mathematical Reviews (MathSciNet): MR95c:60023
Zentralblatt MATH: 0801.60018
Digital Object Identifier: doi:10.1214/aop/1176989003
Project Euclid: euclid.aop/1176989003
[5] Bolthausen, E., Deuschel, J.-D. and Zeitouni, O. (1995). Entropic repulsion of the lattice free field. Comm. Math. Phys. 170 417-443.
Mathematical Reviews (MathSciNet): MR96g:82012
Digital Object Identifier: doi:10.1007/BF02108336
Project Euclid: euclid.cmp/1104273128
[6] Bolthausen, E. and Ioffe, D. (1997). Harmonic crystal on the wall: a microscopic approach. Comm. Math. Phys. 187 523-566.
Mathematical Reviews (MathSciNet): MR99b:82034
Zentralblatt MATH: 0893.60062
Digital Object Identifier: doi:10.1007/s002200050148
[7] Bramson, M. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531-581.
Mathematical Reviews (MathSciNet): MR58:13382
Zentralblatt MATH: 0361.60052
Digital Object Identifier: doi:10.1002/cpa.3160310502
[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.
Mathematical Reviews (MathSciNet): MR56:8774
Digital Object Identifier: doi:10.1016/0022-1236(76)90004-5
[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.
Zentralblatt MATH: 0868.60019
Mathematical Reviews (MathSciNet): MR1414304
Digital Object Identifier: doi:10.1007/BF02101291
Project Euclid: euclid.cmp/1104287906
[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.
Mathematical Reviews (MathSciNet): MR2001i:82019
Zentralblatt MATH: 0951.60091
Digital Object Identifier: doi:10.1007/s002200050713
[12] Deuschel, J.-D. and Giacomin, G. (2000). Entropic repulsion for massless fields. Stochastic Process. Appl. 89 333-354.
Mathematical Reviews (MathSciNet): MR2001j:82020
Zentralblatt MATH: 1045.60103
Digital Object Identifier: doi:10.1016/S0304-4149(00)00030-2
[13] Deuschel, J.-D. and Stroock, D. (1989). Large Deviations. Academic Press, Boston.
Mathematical Reviews (MathSciNet): MR90h:60026
Zentralblatt MATH: 0675.60086
[14] Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
Mathematical Reviews (MathSciNet): MR89k:82010
Zentralblatt MATH: 0657.60122
[15] Lawler, G. F. (1991). Intersections of Random Walks. Birkh¨auser, Boston.
[16] Spitzer, F. (1964). Principles of Random Walks. Van Nostrand, New York.
Mathematical Reviews (MathSciNet): MR30:1521
