Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy

Hubert Lacoin

doi:10.5802/jep.110

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]

Hubert Lacoin ¹

¹ IMPA, Institudo de Matemática Pura e Aplicada Estrada Dona Castorina 110, Rio de Janeiro, CEP-22460-320, Brasil

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1-62.

Résumé
Abstract

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 ${(ϕ (x))}_{x \in ℤ^{2}}$ et à la fonctionnelle d’énergie

V (ϕ) = β \sum_{x \sim y} | ϕ (x) - ϕ (y) | - \sum_{x} (h 1_{{ϕ (x) = 0}} - \infty 1_{{ϕ (x) < 0}}) .

Nous démontrons que pour des valeurs de $β$ suffisamment grandes, il existe une suite décroissante ${(h_{n}^{*} (β))}_{n \geq 0}$ , satisfaisant ${lim}_{n \to \infty} h_{n}^{*} (β) = h_{w} (β)$ , et telle que : $(A)$ l’énergie libre associée au système est infiniment dérivable sur $ℝ ∖ ({h_{n}^{*}}_{n \geq 1} \cup h_{w} (β))$ , et n’admet pas de dérivée aux points ${h_{n}^{*}}_{n \geq 1}$ ; $(B)$ pour tout entier $n \geq 0$ , pour les valeurs de $h$ dans l’intervalle $(h_{n + 1}^{*}, h_{n}^{*})$ (avec la convention $h_{0}^{*} = \infty$ ), 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 ${(ϕ (x))}_{x \in ℤ^{2}}$ , and the energy functional

V (ϕ) = β \sum_{x \sim y} | ϕ (x) - ϕ (y) | - \sum_{x} (h 1_{{ϕ (x) = 0}} - \infty 1_{{ϕ (x) < 0}}) .

We prove that for $β$ sufficiently large, there exists a decreasing sequence ${(h_{n}^{*} (β))}_{n \geq 0}$ , satisfying ${lim}_{n \to \infty} h_{n}^{*} (β) = h_{w} (β)$ , and such that: $(A)$ The free energy associated with the system is infinitely differentiable on $ℝ ∖ ({h_{n}^{*}}_{n \geq 1} \cup h_{w} (β))$ , and not differentiable on ${h_{n}^{*}}_{n \geq 1}$ . $(B)$ For each $n \geq 0$ within the interval $(h_{n + 1}^{*}, h_{n}^{*})$ (with the convention $h_{0}^{*} = \infty$ ), 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 : 2018-09-18
Accepté le : 2019-10-15
Publié le : 2019-11-08

MR Zbl

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

Affiliations des auteurs :

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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     title = {Wetting and layering for {Solid-on-Solid} {II:} {Layering} transitions, {Gibbs} states, and regularity of the free energy},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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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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