Aggregations of randomly moving particles into clusters is studied by a wide variety of theoretical models, experiments, and numerical simulations. Theoretical studies are restricted by high complexity of the process, often requiring significant simplifications and assumptions on the parameters.

We introduce two lattice models, aggregations of l-dimensional boxes and aggregations of partitions with l parts, as simplified growth models, inspired by diffusion limited aggregation (DLA) process. We describe the parameter set of aggregations, compute characteristics of the random variable of the number of growth directions. Inspired by work of our colleagues from physics department, we discuss the asymptotical behavior of the proportions of the most frequent two- and three-dimensional self-aggregations.