Critical prewetting in the 2d Ising model

In this paper we develop a detailed analysis of critical prewetting in the context of the two-dimensional Ising model. Namely, we consider a two-dimensional nearest-neighbor Ising model in a $2N\times N$ rectangular box with a boundary condition inducing the coexistence of the $+$ phase in the bulk and a layer of $-$ phase along the bottom wall. The presence of an external magnetic field of intensity $h=\lambda \mathrm{/}N$ (for some fixed $\lambda >0$) makes the layer of $-$ phase unstable. For any $\beta >{\beta }_{\mathrm{c}}$, we prove that, under a diffusing scaling by $N^{-2\mathrm{/}3}$ horizontally and $N^{-1\mathrm{/}3}$ vertically, the interface separating the layer of unstable phase from the bulk phase weakly converges to an explicit Ferrari-Spohn diffusion.