## Electronic Communications in Probability

### Probability to be positive for the membrane model in dimensions 2 and 3

#### Abstract

We consider the membrane model on a box $V_{N}\subset \mathbb{Z} ^{n}$ of size $(2N+1)^{n}$ with zero boundary condition in the subcritical dimensions $n=2$ and $n=3$. We show optimal estimates for the probability that the field is positive in a subset $D_{N}$ of $V_{N}$. In particular we obtain for $D_{N}=V_{N}$ that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area $N^{n-1}$.

#### Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 44, 14 pp.

Dates
Received: 5 November 2018
Accepted: 23 May 2019
First available in Project Euclid: 5 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1562292105

Digital Object Identifier
doi:10.1214/19-ECP245

#### Citation

Buchholz, Simon; Deuschel, Jean-Dominique; Kurt, Noemi; Schweiger, Florian. Probability to be positive for the membrane model in dimensions 2 and 3. Electron. Commun. Probab. 24 (2019), paper no. 44, 14 pp. doi:10.1214/19-ECP245. https://projecteuclid.org/euclid.ecp/1562292105

#### References

• [1] Vladimir I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998.
• [2] Erwin Bolthausen, Jean-Dominique Deuschel, and Giambattista Giacomin, Entropic repulsion and the maximum of the two-dimensional harmonic crystal, Ann. Probab. 29 (2001), no. 4, 1670–1692.
• [3] Erwin Bolthausen, Jean-Dominique Deuschel, and Ofer Zeitouni, Entropic repulsion of the lattice free field, Comm. Math. Phys. 170 (1995), no. 2, 417–443.
• [4] Francesco Caravenna and Loïc Chaumont, An invariance principle for random walk bridges conditioned to stay positive, Electron. J. Probab. 18 (2013), no. 60, 32.
• [5] Francesco Caravenna and Jean-Dominique Deuschel, Pinning and wetting transition for $(1+1)$-dimensional fields with Laplacian interaction, Ann. Probab. 36 (2008), no. 6, 2388–2433.
• [6] Alessandra Cipriani, Biltu Dan, and Rajat Subhra Hazra, The scaling limit of the membrane model, 2018, arXiv:1801.05663.
• [7] Amir Dembo, Jian Ding, and Fuchang Gao, Persistence of iterated partial sums, Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 3, 873–884.
• [8] Denis Denisov and Vitali Wachtel, Exit times for integrated random walks, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 1, 167–193.
• [9] Jean-Dominique Deuschel, Entropic repulsion of the lattice free field. II. The $0$-boundary case, Comm. Math. Phys. 181 (1996), no. 3, 647–665.
• [10] Jean-Dominique Deuschel and Giambattista Giacomin, Entropic repulsion for the free field: pathwise characterization in $d\geq 3$, Comm. Math. Phys. 206 (1999), no. 2, 447–462.
• [11] Richard M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis 1 (1967), 290–330.
• [12] Giambattista Giacomin, Aspects of statistical mechanics of random surfaces., Notes of the lectures given at the IHP, 2001.
• [13] Christin Hiergeist and Reinhard Lipowsky, Local contacts of membranes and strings, Physica A: Statistical Mechanics and its Applications 244 (1997), no. 1, 164 – 175.
• [14] Noemi Kurt, Entropic repulsion for a class of Gaussian interface models in high dimensions, Stochastic Process. Appl. 117 (2007), no. 1, 23–34.
• [15] Noemi Kurt, Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension, Ann. Probab. 37 (2009), no. 2, 687–725.
• [16] Rafał Latała and Dariusz Matlak, Royen’s proof of the Gaussian correlation inequality, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2169, Springer, Cham, 2017, pp. 265–275.
• [17] Joel L. Lebowitz and Christian Maes, The effect of an external field on an interface, entropic repulsion, J. Statist. Phys. 46 (1987), no. 1-2, 39–49.
• [18] Wenbo V. Li and Qi-Man Shao, Lower tail probabilities for Gaussian processes, Ann. Probab. 32 (2004), no. 1A, 216–242.
• [19] Reinhard Lipowsky, Generic interactions of flexible membranes, Structure and Dynamics of Membranes (R. Lipowsky and E. Sackmann, eds.), Handbook of Biological Physics, vol. 1, North-Holland, 1995, pp. 521–602.
• [20] Stefan Müller and Florian Schweiger, Estimates for the Green’s Function of the Discrete Bilaplacian in Dimensions 2 and 3, Vietnam J. Math. 47 (2019), no. 1, 133–181.
• [21] Thomas Royen, A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, Far East J. Theor. Stat. 48 (2014), no. 2, 139–145.
• [22] Juan Jesús Ruiz-Lorenzo, Rodolfo Cuerno, Esteban Moro, and Angel Sánchez, Phase transition in tensionless surfaces, Biophysical Chemistry 115 (2005), no. 2, 187–193, BIFI 2004 International Conference Biology after the Genoma: A Physical View.
• [23] Hironobu Sakagawa, Entropic repulsion for a Gaussian lattice field with certain finite range interaction, J. Math. Phys. 44 (2003), no. 7, 2939–2951.
• [24] Hironobu Sakagawa, On the probability that Laplacian interface models stay positive in subcritical dimensions, Stochastic analysis on large scale interacting systems, RIMS Kôkyûroku Bessatsu, B59, Res. Inst. Math. Sci. (RIMS), Kyoto, 2016, pp. 273–288.
• [25] Florian Schweiger, Discrete Green’s functions and statistical physics of membranes, Master’s thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 2016.
• [26] Michel Talagrand, Majorizing measures: the generic chaining, Ann. Probab. 24 (1996), no. 3, 1049–1103.
• [27] Yvan Velenik, Localization and delocalization of random interfaces, Probab. Surv. 3 (2006), 112–169.