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Imagine I perform Random Sequential Adsorption (RSA) of discs of some radius $r$ on $[0, 1]^2$, eventually covering the surface to some density $Q \leq 0.543$ with $N$ total discs (where $\approx 0.543$ is the approximate jamming limit).

Now imagine I randomly select some disc $c_i \in C$, and "glue" the remaining $(N-1)$ discs in place on the surface. If I can freely move my disc $c_i$, what mean maximum displacement can I hope to be able to achieve from its initial position provided that I cannot intersect any of the "glued" discs?

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This is not an knowledgeable answer, so forgive me if this is irrelevant.

The paper, "Random sequential adsorption," by Jens Feder (Elsevier link), seems to indicate that "the size distribution for non-overlapping holes" is known, which, at least superficially, seems to relate to the quantity you want to compute...?

Abstract. By placing at random disks onto a surface, but adsorbing only those that do not overlap previously adsorbed disks, one will finally reach the jamming limit beyond which no more disks can be adsorbed. We find, using computer simulations, that the coverage at the jamming limit is θ = 0·547 ± 0·002 for disks and θ = 0·562 ± 0·002 for aligned squares. For the jammed state we have evaluated the pair correlation function for the disks and the size distribution for non-overlapping holes. [...]


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@Joseph O'Rourke I apologize, but I can't access the paper for the next day or so. But do you know what they mean by "holes"? Do they mean to refer to the distribution for the disc size that can fit in the gaps between discs after some RSA process? – A.T. Apr 21 '13 at 16:38

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