Wetting and layering for Solid-on-Solid I: Identification of the wetting point and critical behavior

We provide a complete description of the low temperature wetting transition for the two dimensional Solid-On-Solid model. More precisely we study the integer-valued field $(\phi(x))_{x\in \mathbb Z^2}$, associated associated to the energy functional $$V(\phi)=\beta \sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_{x}\left(h{\bf 1}_{\{\phi(x)=0\}}-\infty{\bf 1}_{\{\phi(x)<0\}} \right).$$ It is known since the pioneering work of Chalker (J. Phys. A {\bf 15} (1982) 481-485) that for every $\beta$, there exists $h_{w}(\beta)>0$ delimiting a transition between a delocalized phase ($h<h_{w}(\beta)$) where the proportion of points at level zero vanishes, and a localized phase ($h>h_{w}(\beta)$) where this proportion is positive. We prove in the present paper that for $\beta$ sufficiently large we have $$h_w(\beta)= \log \left(\frac{e^{4\beta}}{e^{4\beta}-1}\right).$$ Furthermore we provide a sharp asymptotic for the free energy at the vicinity of the critical point: We show that close to $h_w(\beta)$, the free energy is approximately piecewise affine and that the points of discontinuity for the derivative of the affine approximation forms a geometric sequence accumulating on the right of $h_w(\beta)$. This asymptotic behavior provides a strong evidence for the conjectured existence of countably many "layering transitions" at the vicinity of the critical point, corresponding to jumps for the typical height of the field.