This comes from Keith Lynch, by way of the usenet group geometry.puzzles.
If a large number of points are distributed uniformly evenly over a region,
what is the probability that a given point's nearest neighbor's nearest neighbor
is that same given point again ?
I don't know the answer,
results. several trials.
with D > 1 the deviation is a significant, so the numbers below are my eyeball averaging of several runs.
it makes sense that higher dimension means the probability drops
(because each point can have more equidistant neighbors),
but i'm surprised the values don't drop off faster.
Peter Ralph is getting similar but not identical results with a derived solution,
but that solution itself involves simulation to calculate an integral, so there may be room for discrepency.
.. more to come!