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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.) (English. French summary) Zbl 07128376
Summary: We consider the Solid-on-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field $$(\phi (x))_{x\in \mathbb{Z}^2}$$, and the energy functional $V(\phi )=\beta \sum _{x\sim y}|\phi (x)-\phi (y)|-\sum _x\left(h\mathbf{1}_{\{ \phi (x)=0\} }-\infty \mathbf{1}_{\{ \phi (x)<0\} } \right).$ We prove that for $$\beta$$ sufficiently large, there exists a decreasing sequence $$(h^*_n(\beta ))_{n\ge 0}$$, satisfying $$\lim_{n\rightarrow \infty }h^*_n(\beta )=h_w(\beta)$$, and such that: $$(A)$$ The free energy associated with the system is infinitely differentiable on $$\mathbb{R} \smallsetminus \left(\{ h^*_n\} _{n\ge 1}\cup h_w(\beta )\right)$$, and not differentiable on $$\{ h^*_n\} _{n\ge 1}. (B)$$ 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 Part I [the author, Commun. Math. Phys. 362, No. 3, 1007–1048 (2018; Zbl 1398.82023)] provide a complete description of the wetting and layering transition for SOS.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K37 Processes in random environments 82B27 Critical phenomena in equilibrium statistical mechanics 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
##### Keywords:
solid-on-solid; wetting; layering transitions; Gibbs states
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