Limiting shape and cube-root fluctuations of (2+1)-dimensional SOS
We give a full description for the shape of the classical (2+1)-dimensional Solid-On-Solid model above a wall, introduced by Temperley (1952). On an $L \times L$ box at a large inverse-temperature $\beta$ the height of most sites concentrates on a single level $h = \lfloor \frac14 \beta\log L \rfloor$ for most values of $L$. For a sequence of diverging boxes the ensemble of level lines of heights $(h,h-1,\ldots)$ has a scaling limit in Hausdorff distance iff the fractional parts of $\frac14\beta\log L$ 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 $L^{1/3+o(1)}$ fluctuations from the side boundaries.
Based on joint works with Pietro Caputo, Fabio Martinelli, Allan Sly and Fabio Toninelli.