## Journal of the Mathematical Society of Japan

### Scaling limits for weakly pinned Gaussian random fields under the presence of two possible candidates

#### Abstract

We study the scaling limit and prove the law of large numbers for weakly pinned Gaussian random fields under the critical situation that two possible candidates of the limits exist at the level of large deviation principle. This paper extends the results of [3], [7] for one dimensional fields to higher dimensions: $d\ge3$, at least if the strength of pinning is sufficiently large.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1359-1412.

Dates
First available in Project Euclid: 27 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1445951154

Digital Object Identifier
doi:10.2969/jmsj/06741359

Mathematical Reviews number (MathSciNet)
MR3417501

Zentralblatt MATH identifier
1334.60205

#### Citation

BOLTHAUSEN, Erwin; CHIYONOBU, Taizo; FUNAKI, Tadahisa. Scaling limits for weakly pinned Gaussian random fields under the presence of two possible candidates. J. Math. Soc. Japan 67 (2015), no. 4, 1359--1412. doi:10.2969/jmsj/06741359. https://projecteuclid.org/euclid.jmsj/1445951154

#### References

• R. A. Adams and J. Fournier, Sobolev spaces, Second edition, Elsevier/Academic Press, Amsterdam, 2003, xiv+305 pp.
• G. Ben Arous and J.-D. Deuschel, The construction of the $d+1$-dimensional Gaussian droplet, Commun. Math. Phys., 179 (1996), 467–488.
• E. Bolthausen, T. Funaki and T. Otobe, Concentration under scaling limits for weakly pinned Gaussian random walks, Probab. Theory Relat. Fields, 143 (2009), 441–480.
• E. Bolthausen and D. Ioffe, Harmonic crystal on the wall: a microscopic approach, Commun. Math. Phys., 187 (1997), 523–566.
• J.-D. Deuschel, G. Giacomin and D. Ioffe, Large deviations and concentration properties for $\nabla\varphi$ interface models, Probab. Theory Relat. Fields, 117 (2000), 49–111.
• T. Funaki, Stochastic Interface Models, In: Lectures on Probability Theory and Statistics, Ecole d'Eté de Probabilités de Saint-Flour XXXIII, 2003, (ed. J. Picard), Lect. Notes in Math., 1869, Springer, 2005, 103–274.
• T. Funaki and T. Otobe, Scaling limits for weakly pinned random walks with two large deviation minimizers, J. Math. Soc. Japan, 62 (2010), 1005–1041.
• T. Funaki and H. Sakagawa, Large deviations for $\nabla\varphi$ interface model and derivation of free boundary problems, In: Proceedings of Shonan/Kyoto meetings “Stochastic Analysis on Large Scale Interacting Systems” 2002, (eds. T. Funaki and H. Osada), Adv. Stud. Pure Math., 39, Math. Soc. Japan, 2004, 173–211.
• T. Funaki and K. Toukairin, Dynamic approach to a stochastic domination: The FKG and Brascamp-Lieb inequalities, Proc. Amer. Math. Soc., 135 (2007), 1915–1922.
• V. A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal., 1 (1992), 61–82.
• G. F. Lawler, Intersections of Random Walks, Birkhäuser, 1996, 220 pp.
• M. Ledoux, Isoperimetry and Gaussian Analysis, In: Lectures on Probability Theory and Statistics, Ecole d'Eté de Probabilités de Saint-Flour XXIV, 1994, Lect. Notes in Math., 1648, Springer, 1996, 165–294.