Diffusion-limited aggregation is different in that you consider ballistic rather than diffusive motion: randomness enters only through x-coordinates of the falling disks.
Have a look at the paper "Ballistic deposition patterns beneath a growing KPZ interface" by Konstantin Khanin et al (http://arxiv.org/abs/1006.4576; I happen to be one of the authors, and will ask my coauthors who are more versed in statistical physics to join the discussion). In particular, it contains some references to the existing literature on ballistic random growth.
People are usually interested in fluctuations of the upper envelope of the growing cluster, because for many such models it falls into the ``KPZ universality class'' (meaning that upon a proper rescaling its continuous limit converges to a kind of Airy process). In particular the behavior of $h_{\mathrm{max}}$ is a superposition of two phenomena: the obviously linear scaling of the mean height and the scatter of local heights around that mean, which is described by the Tracy-Widom law.
