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

We consider the Solid-On-Solid model interacting with a wall, which is the statistical mechanics model associated with the integer-valued field $(\varphi (x){)}_{x\in {\mathbb{Z}}^2}$, and the energy functional $$V(\varphi )=\beta \sum _{x\sim y}\mathrm{\mid }\varphi (x)-\varphi (y)\mathrm{\mid }-\sum _x(h{\mathbf{1}}_{\{\varphi (x)=0\}}-\mathrm{\infty }{\mathbf{1}}_{\{\varphi (x)<0\}})\mathrm{.}$$ We prove that for $\beta$ sufficiently large, there exists a decreasing sequence $(h_n^{\ast }(\beta ){)}_{n\ge 0}$, satisfying ${\lim }_{n\to \mathrm{\infty }}{h}_n^{\ast }(\beta )=h_w(\beta ),$ and such that: $(A)$ The free energy associated with the system is infinitely differentiable on $\mathbb R \setminus \left({h^*n}{n\ge 1}\cup h_w(\beta)\right)$, and not differentiable on $\{h_n^{\ast }{\}}_{n\ge 1}$. $(B)$ For each $n\ge 0$ within the interval $(h_{n+1}^{\ast },h_n^{\ast })$ (with the convention $h_0^{\ast }=\mathrm{\infty }$), 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 state coexist. The respective typical heights of these two Gibbs states are $n-1$ and $n$. The value $h_n^{\ast }$ corresponds thus to a first order layering transition from level $n$ to level $n-1$. These results combined with those obtained in [23] provide a complete description of the wetting and layering transition for SOS.