  A minimal surface in a random environment (MSRE) is a surface which minimizes
the sum of its elastic energy and its environment potential energy, subject to
prescribed boundary conditions. Apart from their intrinsic interest, such
surfaces are further motivated by connections with disordered spin systems and
first-passage percolation models. We wish to study the geometry of
d-dimensional minimal surfaces in a (d+n)-dimensional random environment.
Specializing to a model that we term harmonic MSRE, in an ``independent''
random environment, we rigorously establish bounds on the geometric and
energetic fluctuations of the minimal surface, as well as a scaling relation
that ties together these two types of fluctuations. In particular, we prove,
for all values of $n$, that the surfaces are delocalized in dimensions $d\le 4$
and localized in dimensions $d\ge 5$. Moreover, the surface delocalizes with
power-law fluctuations when $d\le 3$ and sub-power-law fluctuations when $d=4$.
Our localization results apply also to harmonic minimal surfaces in a periodic
random environment.
