  We study central limit theorems for a totally asymmetric, one-dimensional
interacting random system. The models we work with are the
Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H
process represents a particle configuration on the line, or a 1-dimensional
interface on the plane which moves in one fixed direction through random local
jumps. The stick model is the process of local slopes of the A-D-H process, and
has a conserved quantity. The results describe the fluctuations of these
systems around the deterministic evolution to which the random system converges
under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean
fluctuations on the scale of the classical central limit theorem. In the
scaling limit these fluctuations obey deterministic equations with random
initial conditions given by the initial fluctuations. Of particular interest is
the effect of macroscopic shocks, which play a dominant role because dynamical
noise is suppressed on the scale we are working.
