Consider the (2+1)D Discrete Gaussian (integer-valued Gaussian free field) on an LxL box above a floor. Bricmont, El-Mellouki and Fröhlich (1986) proved at low temperature there is entropic repulsion: the floor propels the average height to be poly-logarithmic in L. A more detailed picture was established by Lubetzky, Martinelli, Sly (2016), showing the height function looks like a plateau.

We answer some conjectures about the boundary of this plateau, i.e. the top level-line, as L diverges. The global scaling limit is either the full box or a translation of Wulff shapes, depending on the sequence of L. In either case, along the flat sides of the limit shape, the top level-line converges to a Ferrari–Spohn diffusion on intervals of length N^{2/3} after scaling by (N^{2/3}, N^{1/3}), for an explicit N = L^{1-o(1)}. In particular, for “most” choices of L, the fluctuations are o(L^{1/3}). Moreover, the joint law of any finite number of top level-lines, appropriately rescaled, is independent Ferrari–Spohn diffusions. These new results extend to the full universality class of grad-phi models for any fixed p>1. Joint works with Eyal Lubetzky.