Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 14 (2009) > Paper 79 open journal systems 


Confinement of the two dimensional discrete Gaussian free field between two hard walls

Hironobu Sakagawa, Department of Mathematics, Faculty of Science and Technology, Keio University


Abstract
We consider the two dimensional discrete Gaussian free field confined between two hard walls. We show that the field becomes massive and identify the precise asymptotic behavior of the mass and the variance of the field as the height of the wall goes to infinity. By large fluctuation of the field, asymptotic behaviors of these quantities in the two dimensional case differ greatly from those of the higher dimensional case studied by cite{S07}.


Full text: PDF

Pages: 2310-2327

Published on: October 30, 2009
Bibliography
  1. E. Bolthausen, J.-D. Deuschel and G. Giacomin. Entropic repulsion and the maximum of two dimensional harmonic crystal. Ann. Prob. 29 (2001), 1670-1692. Math. Review 2003a:82028
  2. E. Bolthausen and D. Ioffe. Harmonic crystal on the wall: a microscopic approach. Comm. Math. Phys. 187 (1997), 523-566. Math. Review 99b:82034
  3. E. Bolthausen and Y. Velenik. Critical behavior of the massless free field at the depinning transition. Comm. Math. Phys. 223 (2001), 161-203. Math. Review 2002k:82031
  4. J. Bricmont, A. El Mellouki and J. Fr"ohlich. Random surfaces in Statistical Mechanics: roughening, rounding, wetting. J. Stat. Phys. 42 (1986), 743-798. Math. Review 87i:82008
  5. D. Brydges, J. Fr"ohlich and T. Spencer. The random walk representation of classical spin systems and correlation inequalities. Comm. Math. Phys. 83 (1982), 123-150. Math. Review 83i:82032
  6. F. Dunlop, J. Magnen, V. Rivasseau and P. Roche. Pinning of an interface by a weak potential. J. Stat. Phys. 66 (1992), 71-98. Math. Review 93a:82007
  7. P. Ferrari and S. Grynberg. No phase transition for Gaussian fields with bounded spins. J. Stat. Phys. 130 (2008), 195-202. Math. Review 2009b:82026
  8. L. Flatto. The multiple range of two-dimensional recurrent walk. Ann. Probab. 4 (1976), 229-248. Math Review 55 #4388
  9. T. Funaki. Stochastic interface models, In: Lectures on Probability Theory and Statistics, Ecole d'Et'e de Probabilit'es de Saint -Flour XXXIII-2003 (ed. J. Picard), 103-274. Lect. Notes Math. 1869 Springer, 2005. Math Review 2007b:60236
  10. J. Ginibre. General formulation of Griffith's inequalities. Comm. Math. Phys. 16 (1970), 310-328. Math Review 42 #4148
  11. D.-S. Lee. Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces. Phys. Rev. Lett. 95 (2005), 150601.
  12. J. Pitt. Multiple points of transient random walks. Proc. Amer. Math. Soc. 43 (1974), 195-199. Math Review 52 #6880
  13. H. Sakagawa. Entropic repulsion for the high dimensional Gaussian lattice field between two walls. J. Stat. Phys. 124 (2006), 1255-1274. Math Review 2007h:82024
  14. H. Sakagawa. Bounds on the mass for the high dimensional Gaussian lattice field between two hard walls. J. Stat. Phys. 129 (2007), 537-553. Math Review 2009a:82027
  15. O. Schramm and S. Sheffield, Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 (2009), 21-137. MR2486487
  16. Y. Velenik. Localization and delocalization of random interfaces. Probab. Surv. 3 (2006), 112-169. Math Review 2007f:82038
  17. Y. Velenik. private communication.
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489