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 Λ_N be a N×N box of Z², and we consider a nonnegative integer-valued field (ϕ(x))_x∈ΛN with zero boundary conditions (i.e. ϕ|Λ_N^∁=0) associated with the energy functional V(ϕ)=β∑x∼y|ϕ(x)−ϕ(y)|−∑xh1_{ϕ(x)=0}, where β>0 is the inverse temperature and h≥0 is the pinning parameter. Lacoin has shown that for sufficiently large β, there is a phase transition between delocalization and localization at the critical point h_w(β)=log [Formula presented]. In this paper we show that for β≥1 and h∈(0,h_w), the values of ϕ concentrate at the height H=⌊(4β)⁻¹logN⌋ with constant order fluctuations. Moreover, at criticality h=h_w, we provide evidence for the conjectured typical height H_w=⌊(6β)⁻¹logN⌋.