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.