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

Lacoin, Hubert

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

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

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 $(h_{n+1}^{*},h_n^{*})$ (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 ${\left(\varphi \left(x\right)\right)}_{x\in {\mathbb{Z}}^2}$, and the energy functional

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

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 $(h_{n+1}^{*},h_n^{*})$ (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.

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

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