This was previously asked on [MathSE](http://math.stackexchange.com/questions/1430446/nearest-neighbor-for-planar-poisson-is-normally-distributed), but was not answered.

Answering a [question](http://math.stackexchange.com/questions/1429831/finding-the-probability-density-for-a-poisson-process), I realized that the nearest point for a planar Poisson point process (with constant intensity $\lambda>0$) is normally distributed. 

Naturally, this property is just a coincidence, it is specific to dimension $2$, and in this sense it is similar to the conformal invariance of Brownian motion. But the latter property implies a lot of interesting facts; you can even prove theorems from complex analysis like little and big Picard, using the planar Brownian motion. 

Therefore, the question:
>Are there any interesting facts or properties following from the fact that the nearest neighbor distribution for a planar Poisson point process is normal?