The shape of the $ (2+1)$d SOS surface above a wall

Caputo, Pietro; Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, Fabio Lucio

doi:10.1016/j.crma.2012.07.006

Probability Theory

The shape of the

(2 + 1)

d SOS surface above a wall
[La forme de lʼinterface SOS

(2 + 1)

-dimensionnelle au-dessus dʼun mur]

Présenté par : the Editorial Board

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

Comptes Rendus. Mathématique, Tome 350 (2012) no. 13-14, pp. 703-706.

Résumé
Abstract

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 $L \times L$ à basse température $T = 1 / β$ , la hauteur de la plupart des sommets se concentre au niveau $h = ⌊ \frac{1}{4 β} \log L ⌋$ , pour la plupart des valeurs de L. Pour une suite croissante de boîtes, lʼensemble de lignes de niveau de hauteur $(h, h - 1, \dots)$ admet une limite au sense de la distance de Hausdorff ssi la partie fractionnaire de $\frac{1}{4 β} \log L$ 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 $L^{1 / 3 + o (1)}$ .

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 $L \times L$ box at a large inverse-temperature β the height of most sites concentrates on a single level $h = ⌊ \frac{1}{4 β} \log L ⌋$ for most values of L. For a sequence of diverging boxes the ensemble of level lines of heights $(h, h - 1, \dots)$ has a scaling limit in Hausdorff distance iff the fractional parts of $\frac{1}{4 β} \log L$ 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 $L^{1 / 3 + o (1)}$ fluctuations from the side boundaries.

Reçu le : 2012-07-15
Accepté le : 2012-07-18
Publié le : 2012-07-01

DOI : 10.1016/j.crma.2012.07.006

Affiliations des auteurs :

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

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     title = {The shape of the $ (2+1)$d {SOS} surface above a wall},
     journal = {Comptes Rendus. Math\'ematique},
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Caputo, Pietro; Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, Fabio Lucio. The shape of the $ (2+1)$d SOS surface above a wall. Comptes Rendus. Mathématique, Tome 350 (2012) no. 13-14, pp. 703-706. doi : 10.1016/j.crma.2012.07.006. http://www.numdam.org/articles/10.1016/j.crma.2012.07.006/

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Cité par Sources :

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