The shape of the (2+1)d SOS surface above a wall

We give a full description for the shape of the classical (2+1)d Solid-On-Solid model above a wall, introduced by Temperley (1952) [14]. On an L×L box at a large inverse-temperature β the height of most sites concentrates on a single level h=⌊14βlogL⌋ for most values of L. For a sequence of diverging boxes the ensemble of level lines of heights (h,h−1,…) has a scaling limit in Hausdorff distance iff the fractional parts of 14βlogL converge to a noncritical value. The scaling limit is explicitly given by nested distinct loops formed via translates of Wulff shapes. Finally, the h-level lines feature L1/3+o(1) fluctuations from the side boundaries.