  We study the fluctuations of random surfaces on a two-dimensional discrete
torus. The random surfaces we consider are defined via a nearest-neighbor pair
potential which we require to be twice continuously differentiable on a
(possibly infinite) interval and infinity outside of this interval. No
convexity assumption is made and we include the case of the so-called hammock
potential, when the random surface is uniformly chosen from the set of all
surfaces satisfying a Lipschitz constraint. Our main result is that these
surfaces delocalize, having fluctuations whose variance is at least of order
$\log n$, where $n$ is the side length of the torus. We also show that the
expected maximum of such surfaces is of order at least $\log n$. The main tool
in our analysis is an adaptation to the lattice setting of an algorithm of
Richthammer, who developed a variant of a Mermin-Wagner-type argument
applicable to hard-core constraints. We rely also on the reflection positivity
of the random surface model. The result answers a question mentioned by
Brascamp, Lieb and Lebowitz 1975 on the hammock potential and a question of
Velenik 2006.
