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
to
or
.
An essential innovation is that random surface models with nonpair
interactions of long- and infinite-range are included in the analysis. More
specifically, we cover at least: uniformly random graph homomorphisms from
to a
-regular tree
for any
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 conjectures of Sheffield and of Menz and Tassy. 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.