  The motivation for this article is to derive strict convexity of the surface
tension for Lipschitz random surfaces, that is, for models of random Lipschitz
functions from $\mathbb Z^d$ to $\mathbb Z$ or $\mathbb R$. An essential
innovation is that random surface models with long- and infinite-range
interactions are included in the analysis. More specifically, we cover at
least: uniformly random graph homomorphisms from $\mathbb Z^d$ to a $k$-regular
tree for any $k\geq 2$ and Lipschitz potentials which satisfy the FKG lattice
condition. The latter includes perturbations of dimer- and six-vertex models
and of Lipschitz simply attractive potentials introduced by Sheffield. The main
result is that we prove strict convexity of the surface tension -- which
implies uniqueness for the limiting macroscopic profile -- if the model of
interest is monotone in the boundary conditions. This solves a conjecture of
Menz and Tassy, and answers a question posed by Sheffield. Auxiliary to this,
we prove several results which may be of independent interest, and which do not
rely on the model being monotone. This includes existence and topological
properties of the specific free energy, as well as a characterization of its
minimizers. We also prove a general large deviations principle which describes
both the macroscopic profile and the local statistics of the height functions.
This work is inspired by, but independent of, Random Surfaces by Sheffield.
