Non-Gaussian Surface Pinned by a Weak Potential

We consider a model of a two-dimensional interface of the SOS type, with finite-range, even, strictly convex, twice continuously differentiable interactions. We prove that, under an arbitrarily weak potential favouring zero-height, the surface has finite mean square heights. We consider the cases of both square well and $\delta$ potentials. These results extend previous results for the case of nearest-neighbours Gaussian interactions in \cite{DMRR} and \cite{BB}. We also obtain estimates on the tail of the height distribution implying, for example, existence of exponential moments. In the case of the $\delta$ potential, we prove a spectral gap estimate for linear functionals. We finally prove exponential decay of the two-point function (1) for strong $\delta$-pinning and the above interactions, and (2) for arbitrarily weak $\delta$-pinning, but with finite-range Gaussian interactions.