  We study how the typical gradient and typical height of a random surface are
modified by the addition of quenched disorder in the form of a random
independent external field. The results provide quantitative estimates, sharp
up to multiplicative constants, in the following cases.
  It is shown that for real-valued disordered random surfaces of the $\nabla
\phi$ type with a uniformly convex interaction potential: (i) The gradient of
the surface delocalizes in dimensions $1\le d\le 2$ and localizes in dimensions
$d\ge3$. (ii) The surface delocalizes in dimensions $1\le d\le 4$ and localizes
in dimensions $d\ge 5$.
  It is further shown that for the integer-valued disordered Gaussian free
field: (i) The gradient of the surface delocalizes in dimensions $d=1,2$ and
localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions
$d=1,2$. (iii) The surface localizes in dimensions $d\ge 3$ at weak disorder
strength. The behavior in dimensions $d\ge 3$ at strong disorder is left open.
  The proofs rely on several tools: explicit identities satisfied by the
expectation of the random surface, the Efron--Stein concentration inequality, a
coupling argument for Langevin dynamics (originally due to Funaki and Spohn)
and the Nash--Aronson estimate.
