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Harmonic Pinnacles in the Discrete Gaussian Model

Author(s): Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan M.

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Abstract: The 2D Discrete Gaussian model gives each height function a probability proportional to , where is the inverse-temperature and sums over nearest-neighbor bonds. We consider the model at large fixed , where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an box with 0 boundary conditions concentrates on two integers M, M + 1 with . The key is a large deviation estimate for the height at the origin in , dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where . This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5-6):743-798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order . Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.
Publication Date: Jun-2016
Electronic Publication Date: 17-May-2016
Citation: Lubetzky, Eyal, Martinelli, Fabio, Sly, Allan. (2016). Harmonic Pinnacles in the Discrete Gaussian Model. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 344 (673 - 717). doi:10.1007/s00220-016-2628-5
DOI: doi:10.1007/s00220-016-2628-5
ISSN: 0010-3616
EISSN: 1432-0916
Pages: 673 - 717
Language: English
Type of Material: Journal Article
Version: Author's manuscript

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