Typical height of the (2+1)-D Solid-on-Solid surface with pinning above a wall in the delocalized phase

We study the typical height of the (2+1)-dimensional solid-on-solid surface with pinning interacting with an impenetrable wall in the delocalization phase. More precisely, let ${\mathrm{\Lambda }}_N$ be a $N\times N$ box of ${\mathbb{Z}}^2$, and we consider a nonnegative integer-valued field $(\phi (x){)}_{x\in {\mathrm{\Lambda }}_N}$ with zero boundary conditions (i.e. $\phi {\mathrm{\mid }}_{{\mathrm{\Lambda }}_N^{\mathrm{\complement }}}=0$) associated with the energy functional $$\mathcal{V}(\phi )=\beta \sum _{x\sim y}\mathrm{\mid }\phi (x)-\phi (y)\mathrm{\mid }-\sum _{x}h{\mathbf{1}}_{\{\phi (x)=0\}},$$ where $\beta >0$ is the inverse temperature and $h\ge 0$ is the pinning parameter. Lacoin has shown that for sufficiently large $\beta$, there is a phase transition between delocalization and localization at the critical point $$h_w(\beta )=\log \left(\frac{e^{4\beta }}{e^{4\beta }-1}\right)\mathrm{.}$$ In this paper we show that for $\beta \ge 1$ and $h\in (0,h_w)$, the values of $\phi$ concentrate at the height $H=\lfloor (4\beta {)}^{-1}\log N\rfloor$ with constant order fluctuations. Moreover, at criticality $h=h_w$, we provide evidence for the conjectured typical height $H_w=\lfloor (6\beta {)}^{-1}\log N\rfloor$.