TITLE: Tetris-like falling sticky disks
QUESTION [51 upvotes]: Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, if the disk center reaches $y=0$, the disk stops
with its center resting on the $x$-axis.
Here is an example of 1000 disks falling at uniformly random
$x$-locations within $[-50,50]$:

         

There are many questions one could ask about this (to me)
beautiful and intriguing structure (e.g., about its contact graph),
but to be specific,
let me concentrate on one quantity: the maximum height
$h_{\max}$ as a function of the number of disks $n$ and
the $x$-range $R$. (In the above example, $R=100$ and $h_{\max}=94.9$.)
It appears that $h_{\max}$ grows linearly, with $h_{\max} \approx n \frac{10}{R}$.
Here is plot, where each point is an average of ten trials:

                   

Two questions:

Q1. Is there a simple explanation of the growth of $h_{\max}$?
Q2. Has this process, or something close to it, been studied before?


Ultimately I am interested in determining packing densities of
randomly jostled shapes, as explored in the earlier MO question
"Average degree of contact graph for balls in a box."
Sticky disks are a very simple model along these lines.

Update (3Mar16).
An article by Ivan Corwin on KPZ universality has just appeared
(AMS Notices PDF),
including this figure to illustrate the "random ballistic" model:

     


Users ansobol and Nechaev and Jeremy Voltz 
previously pointed to the relevance of KPZ universality.

REPLY [8 votes]: That the growth is asymptotically linear is clear. But if you keep the width $2R$ of the strip (over which the centers are chosen uniformly) fixed, then the growth speed $c_R$ is not strictly proportional to $R^{-1}$. This is clear when thinking to the small width case: if $R<1$, then there is only one branch in the tree because two consecutive disks always touch. If $d$ is the distance between their centers, then they will arrange with a height gap equal to $\sqrt{4-d^2}$. Because the distribution of $d$ has density 
$$\frac{2R-d}{2R^2},$$
we find that the growth average speed
$$\frac1n\sum_1^nd_j$$
tends to the expectation of  $\sqrt{4-d^2}$:
$$c_R=\int_0^{2R}\sqrt{4-x^2}\frac{2R-x}{2R^2}dx.$$
Calculus gives
$$c_R=\frac2R\sin^{-1}R+2\sqrt{1-R^2}+\frac4{3R^2}\left((1-R^2)^{3/2}-1\right).$$
As expected, $c_R\rightarrow2$ as $R\rightarrow0$. On the other hand, $c_1=\pi-\frac43$.
Presumably, the OP is interested with $\gamma=\lim_{R\rightarrow+\infty}Rc_R$.

REPLY [6 votes]: Regarding the question

Has this process, or something close to it, been studied before?

I was recently made aware of an intriguing approach of Bob Macpherson and his post-doc Ben Schweinhart at IAS for investigating Brownian trees via computational topology in this pre-print. Since the investigation is topological, it will not yield answers to your question about maximum height, but it does capture other interesting global statistics.
The central idea is to let a diffusion-limited aggregation process run for a while and generate a configuration similar to that of your picture, but lacking any loops whatsoever. Essentially, each time a disc falling from infinity creates a cycle, it is discarded. However, this discarding step is not necessary for subsequent analysis and so would also apply to your (considerably loopier) situation. 
In any case, once the process has run its course and generated a space consisting of the union of balls, you start growing the radii of these balls (allowing overlap of course). As you increase these radii, the homology of the space changes: some loops form, others get filled in and so forth. It is possible to unambiguously associate a (birth, death) interval to each such loop via the theory of persistent homology.