  We present classes of models in which particles are dropped on an arbitrary
fixed finite connected graph, obeying adhesion rules with screening. We prove
that there is an invariant distribution for the resulting height profile, and
Gaussian concentration for functions depending on the paths of the profiles. As
a corollary we obtain a law of large numbers for the maximum height. This
describes the asymptotic speed with which the maximal height increases. The
results incorporate the case of independent particle droppings but extend to
droppings according to a driving Markov chain, and to droppings with possible
deposition below the top layer up to a fixed finite depth, obeying a
non-nullness condition for the screening rule. The proof is based on an
analysis of the Markov chain on height-profiles using coupling methods. We
construct a finite communicating set of configurations of profiles to which the
chain keeps returning.
