Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy
Journal de l'École polytechnique — Mathématiques, Volume 7  (2020), p. 1-62

We consider the Solid-on-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field ${\left(\phi \left(x\right)\right)}_{x\in {ℤ}^{2}}$, and the energy functional

$V\left(\phi \right)=\beta \sum _{x\sim y}|\phi \left(x\right)-\phi \left(y\right)|-\sum _{x}\left(h{\mathbf{1}}_{\left\{\phi \left(x\right)=0\right\}}-\infty {\mathbf{1}}_{\left\{\phi \left(x\right)<0\right\}}\right).$

We prove that for $\beta$ sufficiently large, there exists a decreasing sequence ${\left({h}_{n}^{*}\left(\beta \right)\right)}_{n\ge 0}$, satisfying ${lim}_{n\to \infty }{h}_{n}^{*}\left(\beta \right)={h}_{w}\left(\beta \right)$, and such that: $\left(A\right)$ The free energy associated with the system is infinitely differentiable on $ℝ\setminus \left({\left\{{h}_{n}^{*}\right\}}_{n\ge 1}\cup {h}_{w}\left(\beta \right)\right)$, and not differentiable on ${\left\{{h}_{n}^{*}\right\}}_{n\ge 1}$. $\left(B\right)$ For each $n\ge 0$ within the interval $\left({h}_{n+1}^{*},{h}_{n}^{*}\right)$ (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.

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

$V\left(\phi \right)=\beta \sum _{x\sim y}|\phi \left(x\right)-\phi \left(y\right)|-\sum _{x}\left(h{\mathbf{1}}_{\left\{\phi \left(x\right)=0\right\}}-\infty {\mathbf{1}}_{\left\{\phi \left(x\right)<0\right\}}\right).$

Nous démontrons que pour des valeurs de $\beta$ suffisamment grandes, il existe une suite décroissante ${\left({h}_{n}^{*}\left(\beta \right)\right)}_{n\ge 0}$, satisfaisant ${lim}_{n\to \infty }{h}_{n}^{*}\left(\beta \right)={h}_{w}\left(\beta \right)$, et telle que : $\left(A\right)$ l’énergie libre associée au système est infiniment dérivable sur $ℝ\setminus \left({\left\{{h}_{n}^{*}\right\}}_{n\ge 1}\cup {h}_{w}\left(\beta \right)\right)$, et n’admet pas de dérivée aux points ${\left\{{h}_{n}^{*}\right\}}_{n\ge 1}$ ; $\left(B\right)$ pour tout entier $n\ge 0$, pour les valeurs de $h$ dans l’intervalle $\left({h}_{n+1}^{*},{h}_{n}^{*}\right)$ (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.

Accepted : 2019-10-15
Published online : 2019-11-08
DOI : https://doi.org/10.5802/jep.110
Classification:  60K35,  60K37,  82B27,  82B44
Keywords: Solid-on-Solid, wetting, layering transitions, Gibbs states
@article{JEP_2020__7__1_0,
author = {Hubert Lacoin},
title = {Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy},
journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
publisher = {\'Ecole polytechnique},
volume = {7},
year = {2020},
pages = {1-62},
doi = {10.5802/jep.110},
zbl = {07128376},
language = {en},
url = {https://jep.centre-mersenne.org/item/JEP_2020__7__1_0}
}

Lacoin, Hubert. Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy. Journal de l'École polytechnique — Mathématiques, Volume 7 (2020) , pp. 1-62. doi : 10.5802/jep.110. https://jep.centre-mersenne.org/item/JEP_2020__7__1_0/

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