Critical wetting in the (2+1)D Solid-On-Solid model

In this note, we study the low temperature $(2+1)$D SOS interface above a hard floor with critical pinning potential ${\lambda }_w=\log (\frac{1}{1-e^{-4\beta }})$. At $\lambda <{\lambda }_w$ entropic repulsion causes the surface to delocalize and be rigid at height $\frac{1}{4\beta }\log n+O(1)$; at $\lambda >{\lambda }_w$ it is localized at some $O(1)$ height. We show that at $\lambda ={\lambda }_w$, there is delocalization, with rigidity now at height $\lfloor \frac{1}{6\beta }\log n+\frac{1}{3}\rfloor$, confirming a conjecture of Lacoin.