Suppose one sprinkles points uniformly at random on the infinite Euclidean plane, with some density $\rho$ per unit area. View the points as disks of radius zero. Now the radii $r$ of all disks grows simultaneously at the same (slow) rate, with the disks pushing one another when they come into contact, with never two disk interiors overlapping. Each disk-disk contact pushes both disks apart along the line connecting their centers. More complex clusters move the disks by summing the disk-disk force-separation vectors, although I want the disks to move only when pushed, i.e., they never gain momentum to drift without pushing. Three time-snapshots for a region containing $8$ points might look like this (just a hand simulation):
Here the disks do not yet touch at $t=1$ (blue), but at $t=2$ (tan), $\lbrace 1,2 \rbrace$ and $\lbrace 7,8 \rbrace$ are in contact, and by $t=3$ (pink), $\lbrace 1,2,3,4 \rbrace$ and $\lbrace 5,6,7,8 \rbrace$ are touching.

Although the model is not fully specified, perhaps analogous processes have been studied in the literature? Essentially I want to know:

Q. What happens? Is there some limit configuration as $r$ grows?

Intuitively, it feels like all empty space would be squeezed out. Perhaps the behavior for points initially confined to a finite region $R$, as that region grows larger, would hint at the situation for the infinite plane? I'd appreciate pointers or observations—Thanks!

Update. Here is one of the many images from the Lubachevsky, Graham, Stillinger work to which Henry Cohn points, "Spontaneous Patterns in Disk Packings":


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    $\begingroup$ This resembles greatly the way that "jammed" packings are prepared; see my answer to one of your previous questions mathoverflow.net/questions/71397/… Of course in those models they typically simulate "soft" disks and balls with say a harmonic repulsion when two particles overlap, and they work with a finite region with periodic boundary conditions, but I expect that much of the behavior will be the same. $\endgroup$ – j.c. Jun 14 '13 at 23:39
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    $\begingroup$ I imagine someone must have simulated this, but I'm not sure who or where. I think jc's probably right about the behavior. Incidentally, are you familiar with the Lubachevsky-Stillinger algorithm? (For example, see princeton.edu/~fhs/lubachevsky.htm.) It's reminiscent of your question, but the particles have momentum and bounce off each other. It's a different model and I don't see any obvious implications for your question, but it's similar in spirit. $\endgroup$ – Henry Cohn Jun 15 '13 at 2:36
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    $\begingroup$ Probably an unimportant side question, but how does one put point uniformly at random on the infinite plain (since there is no uniform probability measure)? $\endgroup$ – Dirk Jun 15 '13 at 4:19
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    $\begingroup$ @Dirk: Joseph was probably referring to en.wikipedia.org/wiki/Point_process#Poisson_point_process . $\endgroup$ – Ori Gurel-Gurevich Jun 15 '13 at 5:22
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    $\begingroup$ After some more thought I think the pictures on the page linked to by Henry Cohn should be a good representation for what happens. For packings in two dimensions, a strong tendency to crystallize has been observed if the particle radii are monodisperse, so in the 2D simulations I mentioned the particle radii are often set to be different, since the primary interest is in studying disordered packings. For 3D particles there is apparently less of a tendency to crystallize and so I expect that the result of your model will be disordered packings. $\endgroup$ – j.c. Jun 15 '13 at 7:18

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