Comptes Rendus
Probability Theory
The shape of the (2+1)d SOS surface above a wall
Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 703-706.

We give a full description for the shape of the classical (2+1)d Solid-On-Solid model above a wall, introduced by Temperley (1952) [14]. On an L×L box at a large inverse-temperature β the height of most sites concentrates on a single level h=14βlogL for most values of L. For a sequence of diverging boxes the ensemble of level lines of heights (h,h1,) has a scaling limit in Hausdorff distance iff the fractional parts of 14βlogL converge to a noncritical value. The scaling limit is explicitly given by nested distinct loops formed via translates of Wulff shapes. Finally, the h-level lines feature L1/3+o(1) fluctuations from the side boundaries.

Nous donnons une description complète de la forme typique de lʼinterface SOS (2+1)-dimensionnelle au-dessus dʼun mur, introduite par Temperley (1952) [14]. Dans une boîte L×L à basse température T=1/β, la hauteur de la plupart des sommets se concentre au niveau h=14βlogL, pour la plupart des valeurs de L. Pour une suite croissante de boîtes, lʼensemble de lignes de niveau de hauteur (h,h1,) admet une limite au sense de la distance de Hausdorff ssi la partie fractionnaire de 14βlogL converge à une valeur non critique. La limite dʼéchelle est donnée explicitement par des boucles imbriquées, formées par des translations de formes de Wulff. Enfin, la distance entre le bord de la boîte et la ligne de niveau h a des fluctuations L1/3+o(1).

Published online:
DOI: 10.1016/j.crma.2012.07.006

Pietro Caputo 1; Eyal Lubetzky 2; Fabio Martinelli 1; Allan Sly 3; Fabio Lucio Toninelli 4

1 Università Roma Tre, Largo S. Murialdo 1, 00146 Roma, Italy
2 Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
3 UC Berkeley, Berkeley, CA 94720, USA
4 CNRS and ENS Lyon, laboratoire de physique, 46 allée dʼItalie, 69364 Lyon, France
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     title = {The shape of the $ (2+1)$d {SOS} surface above a wall},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {703--706},
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Pietro Caputo; Eyal Lubetzky; Fabio Martinelli; Allan Sly; Fabio Lucio Toninelli. The shape of the $ (2+1)$d SOS surface above a wall. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 703-706. doi : 10.1016/j.crma.2012.07.006.

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This work was supported by the European Research Council through the “Advanced Grant” PTRELSS 228032.

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