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

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

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

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

Comptes Rendus. Mathématique, Volume 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 :

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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.07.006/

Bibliographie
Cité par

[1] K.S. Alexander Cube-root boundary fluctuations for droplets in random cluster models, Comm. Math. Phys., Volume 224 (2001) no. 3, pp. 733-781

[2] J. Bricmont; A. El Mellouki; J. Fröhlich Random surfaces in statistical mechanics: roughening, rounding, wetting, … , J. Stat. Phys., Volume 42 (1986) no. 5–6, pp. 743-798

[3] P. Caputo; E. Lubetzky; F. Martinelli; A. Sly; F.L. Toninelli Dynamics of $2 + 1$ dimensional SOS surfaces above a wall: slow mixing induced by entropic repulsion, 2012 (preprint. Available at) | arXiv

[4] I. Corwin; A. Hammond Brownian Gibbs property for Airy line ensembles, 2011 (preprint. Available at) | arXiv

[5] R. Dobrushin; R. Kotecký; S. Shlosman Wulff Construction. A Global Shape from Local Interaction, Transl. Math. Monogr., vol. 104, American Mathematical Society, Providence, RI, 1992

[6] P.L. Ferrari; H. Spohn Constrained Brownian motion: fluctuations away from circular and parabolic barriers, Ann. Probab., Volume 33 (2005) no. 4, pp. 1302-1325

[7] J. Fröhlich; T. Spencer Kosterlitz–Thouless transition in the two-dimensional plane rotator and Coulomb gas, Phys. Rev. Lett., Volume 46 (1981) no. 15, pp. 1006-1009

[8] J. Fröhlich; T. Spencer The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas, Comm. Math. Phys., Volume 81 (1981) no. 4, pp. 527-602

[9] J. Fröhlich; T. Spencer The Berežinskiĭ–Kosterlitz–Thouless transition (energy–entropy arguments and renormalization in defect gases), scaling and self-similarity in physics, Prog. Phys., Volume 7 (1983), pp. 29-138

[10] O. Hryniv; Y. Velenik Universality of critical behaviour in a class of recurrent random walks, Probab. Theory Related Fields, Volume 130 (2004) no. 2, pp. 222-258

[11] K. Johansson Discrete polynuclear growth and determinantal processes, Comm. Math. Phys., Volume 242 (2003) no. 1–2, pp. 277-329

[12] R.H. Schonmann; S.B. Shlosman Complete analyticity for 2D Ising completed, Comm. Math. Phys., Volume 170 (1995) no. 2, pp. 453-482

[13] R.H. Schonmann; S.B. Shlosman Constrained variational problem with applications to the Ising model, J. Stat. Phys., Volume 83 (1996) no. 5–6, pp. 867-905

[14] H.N.V. Temperley Statistical mechanics and the partition of numbers. II. The form of crystal surfaces, Proc. Cambridge Philos. Soc., Volume 48 (1952), pp. 683-697

[15] Y. Velenik Entropic repulsion of an interface in an external field, Probab. Theory Related Fields, Volume 129 (2004) no. 1, pp. 83-112

Cité par Sources :

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

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