  Height functions of growing random surfaces are often conjectured to be
superconcentrated, meaning that their variances grow sublinearly in time. This
article introduces a new concept, called subroughness, meaning that there exist
two distinct points such that the expected squared difference between the
heights at these points grows sublinearly in time. The main result of the paper
is that superconcentration is equivalent to subroughness in a class of growing
random surfaces. The result is applied to establish superconcentration in a
variant of the restricted solid-on-solid (RSOS) model and in a variant of the
ballistic deposition model.
