The limiting law of the Discrete Gaussian level-lines

Consider the $(2+1)$D Discrete Gaussian (ZGFF, integer-valued Gaussian free field) model in an $L\times L$ box above a hard floor. Bricmont, El-Mellouki and Fr\"ohlich (1986) established that, at low enough temperature, this random surface exhibits entropic repulsion: the floor propels the average height to be poly-logarithmic in $L$. The second author, Martinelli and Sly (2016) showed that, for all but exceptional values of $L$, the surface has a plateau whose height concentrates on an explicit integer $H(L)$, and fills nearly the full square. It was conjectured there that the boundary of this plateau -- the top level-line of the surface -- should have random fluctuations of $L^{1/3+o(1)}$. We confirm this conjecture of [LMS16] and further recover the limiting law of the top level-line: there exists an explicit sequence $N=L^{1-o(1)}$ such that the distance of the top level-line from $I$, the interval of length $N^{2/3}$ centered along the side boundary, converges, after rescaling it by $N^{1/3}$ and the width of the interval by $N^{2/3}$, to a Ferrari--Spohn diffusion. In particular, the level-line fluctuations at, say, the center of $I$, have a limit law involving the Airy function rescaled by $N^{1/3}$. This gives the first example of one of the $(2+1)$D $|\nabla \phi|^p$ models (approximating 3D Ising and crystal formation) where a Ferrari--Spohn limit law of its level-lines is confirmed (ZGFF is the case $p=2$). More generally, we find the joint limit law of any finite number of top level-lines: rescaling their distances from the side boundary, each by its $(N_n^{2/3},N_n^{1/3})$, yields a product of Ferrari--Spohn laws. These new results extend to the full universality class of $|\nabla\phi|^p$ models for any fixed $p>1$.