Electronic Communications in Probability

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

Simon Buchholz, Jean-Dominique Deuschel, Noemi Kurt, and Florian Schweiger

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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

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

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

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Primary: 60G15: Gaussian processes
Secondary: 60G60: Random fields 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

membrane model random interface entropic repulsion Gaussian process

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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

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