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). On an L× L box at a large inverse-temperature β the height of most sites concentrates on a single level h = ⌊ 1 4β logL⌋ 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 1 4β logL 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 L fluctuations from the side boundaries.