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 $\frac1{4\beta}\log n+O(1)$; 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 $\lfloor \frac1{6\beta}\log n+\frac13\rfloor$, confirming a conjecture of Lacoin.