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

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We give a full description for the shape of the classical (2+1) 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) L 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) L 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.