Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy
[Mouillage et stratification pour le modèle SOS II : transitions de niveau, états de Gibbs et régularité de l’énergie libre]
Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1-62.

Nous considérons le modèle « Solid-On-Solid » (SOS) incluant une interaction avec une paroi. Il s’agit du modèle de mécanique statistique associé au champ à valeurs entières (ϕ(x)) x 2 et à la fonctionnelle d’énergie

V(ϕ)=β xy |ϕ(x)-ϕ(y)|- x h1 {ϕ(x)=0} -1 {ϕ(x)<0} .

Nous démontrons que pour des valeurs de β suffisamment grandes, il existe une suite décroissante (h n * (β)) n0 , satisfaisant lim n h n * (β)=h w (β), et telle que : (A) l’énergie libre associée au système est infiniment dérivable sur {h n * } n1 h w (β), et n’admet pas de dérivée aux points {h n * } n1  ; (B) pour tout entier n0, pour les valeurs de h dans l’intervalle (h n+1 * ,h n * ) (avec la convention h 0 * =), il existe une unique mesure de Gibbs correspondant à une hauteur de localisation n, alors qu’aux points de non-dérivabilité il y a multiplicité des états de Gibbs, en particulier il en existe deux correspondant aux hauteurs de localisation n-1 et n respectivement. La valeur h n * marque donc une transition de niveau entre la hauteur n et la hauteur n-1. Ces résultats et ceux prouvés dans [28] fournissent une description complète des transitions de niveau et de la transition de mouillage pour le modèle SOS.

We consider the Solid-on-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field (ϕ(x)) x 2 , and the energy functional

V(ϕ)=β xy |ϕ(x)-ϕ(y)|- x h1 {ϕ(x)=0} -1 {ϕ(x)<0} .

We prove that for β sufficiently large, there exists a decreasing sequence (h n * (β)) n0 , satisfying lim n h n * (β)=h w (β), and such that: (A) The free energy associated with the system is infinitely differentiable on {h n * } n1 h w (β), and not differentiable on {h n * } n1 . (B) For each n0 within the interval (h n+1 * ,h n * ) (with the convention h 0 * =), there exists a unique translation invariant Gibbs state which is localized around height n, while at a point of non-differentiability, at least two ergodic Gibbs states coexist. The respective typical heights of these two Gibbs states are n-1 and n. The value h n * corresponds thus to a first order layering transition from level n to level n-1. These results combined with those obtained in [28] provide a complete description of the wetting and layering transition for SOS.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.110
Classification : 60K35, 60K37, 82B27, 82B44
Keywords: Solid-on-Solid, wetting, layering transitions, Gibbs states
Mot clés : Modèle SOS, mouillage, transitions de niveau, état de Gibbs
Hubert Lacoin 1

1 IMPA, Institudo de Matemática Pura e Aplicada Estrada Dona Castorina 110, Rio de Janeiro, CEP-22460-320, Brasil
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Hubert Lacoin. Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1-62. doi : 10.5802/jep.110. https://jep.centre-mersenne.org/articles/10.5802/jep.110/

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