A disc contains $n$ independent uniformly random points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.

For example, here are $20$ random points and $7$ clusters, with an average cluster size of $\frac{20}{7}$.

[![enter image description here][1]][1]

>What is the expectation of the average cluster size, as $n\to\infty$ ?

I made a [random point generator][2] that generates $20$ random points. The expectation of the average cluster size seems to be approximately $3$.

This question was [posted][3] on Math SE. This [answer][4] provides useful context (but does not answer the question).

This question was inspired by the Math SE question [Stars in the universe - probability of mutual nearest neighbors][5].


  [1]: https://i.sstatic.net/PXovh.png
  [2]: https://www.desmos.com/calculator/v4ziiyaonx?lang=zh-CN
  [3]: https://math.stackexchange.com/q/4845153/398708
  [4]: https://math.stackexchange.com/a/4845173/398708
  [5]: https://math.stackexchange.com/questions/271497/stars-in-the-universe-probability-of-mutual-nearest-neighbors