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 concentra
tion 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.
In this paper we present a multilayer particle deposition model on a random tree. We derive the time dependent densities of the first and second layer analytically and show that in all trees the limiting density of the first layer exceeds the density
in the second layer. We also provide a procedure to calculate higher layer densities and prove that random trees have a higher limiting density in the first layer than regular trees. Finally, we compare densities between the first and second layer and between regular and random trees.
