In article <31FD037B.12A7@math.okstate.edu> David Ullrich <email@example.com> writes: >R E Sheskey wrote: >> >> Hi. I have an infinite collection of identical disks arranged >> randomly on the plane subject to two conditions: >> >> 1. No two of the disks overlap. >> 2. No disk can be added without violating #1. >> >> Question: what fraction of the plane is covered? > > By which we mean presumably the limit as R tends to >infinity of A(R), where A(R) is the fraction of the disc >about 0 of radius R that's covered by your collection of >discs. Or something like that... > There's no reason that A(R) should actually have >a limit as R tends to infinity. > [...] > ----------------------------------------------------------------
Sure, it would be easy to create a configuration of non-overlapping unit disks in the plane for which A(R) does not approach a limit as R -> oo.
But there is good reason to expect that there will be (with probability 1) a definite numerical limit that is approached, statistically.
Let D_R denote the disk of radius R about 0 in the plane.
Define B(R) as the average fraction of D_R covered, averaged over all configurations of unit disks within D_R satisfying 1. and 2.
(Intuitively, this could be made precise by considering one "try" to consist of the following: Pick points c_1, c_2,... at random from D_R, thinking of them as centers of unit disks -- rejecting any c_(n+1) which violates rule 1. Stop when there is no room for another disk. Repeating tries K times should, in the limit as K -> oo, lead to an average fractional coverage B(R). This limiting average should exist, I think, with probability 1.)
(A cleaner method might be to consider sequences of points c_1,...,c_n of D_R that are maximal with respect to the condition that all pairwise distances ||c_i - c_j|| < 1. Such sequences form a subset S (open) of D_R x ... x D_R (n factors), with a natural uniform measure on it inherited from R^(2n). Then it is straightforward to average the fraction of D_R associated to disks centered at c_1,...,c_n over the subset S, obtaining a number F(R,n). The rub is that the number n can vary. The correct definition for B(R) ought to be the weighted average w_1*F(R,1) + w_2*F(R,2) + ... (this infinite sum is actually finite), where w_n is the probability that n is the size of the maximal sequence c_1,...,c_n. But the w_i may be difficult to determine.)
Then I believe that B(R) will approach a limit as R -> oo.
The reason is that the "edge effects" -- due to the boundary of D_R -- should approach zero influence on B(R) as R -> oo.
Dr. Daniel Asimov Senior Research Scientist
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