  We study in detail a one-dimensional lattice model of a continuum, conserved
field (mass) that is transferred deterministically between neighbouring random
sites. The model falls in a wider class of lattice models capturing the joint
effect of random advection and diffusion and encompassing as specific cases,
some models studied in the literature, like the Kang-Redner,
Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup
comes from a straightforward interpretation as advection of particles in
one-dimensional turbulence, but it is also related to a problem of
synchronization of dynamical systems driven by common noise. For finite
lattices, we study both the coalescence of an initially spread field
(interpreted as roughening), and the statistical steady-state properties. We
distinguish two main size-dependent regimes, depending on the strength of the
diffusion term and on the lattice size. Using numerical simulations and
mean-field approach, we study the statistics of the field. For weak diffusion,
we unveil a characteristic hierarchical structure of the field. We also connect
the model and the iterated function systems concept.
