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

TL;DR

This paper characterizes the shape and fluctuations of the (2+1)D SOS surface above a wall, revealing a precise description of level lines, their scaling limits, and fluctuation behavior at large scales.

Contribution

It provides a full description of the surface shape, level line scaling limits, and fluctuation properties for the classical SOS model above a wall.

Findings

Most sites concentrate on a single height level.

Level lines form nested loops with explicit Wulff shape structure.

Level line fluctuations are of order L^{1/3+o(1)}.

Abstract

We give a full description for the shape of the classical (2+1)\Dim 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 $h = \lfloor (1/4\beta)\log L\rfloor$ for most values of $L$. For a sequence of diverging boxes the ensemble of level lines of heights $(h,h-1,...)$ has a scaling limit in Hausdorff distance iff the fractional parts of $(1/4\beta)\log L$ 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.