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

Answering a [question](https://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. 

Indeed, it is easy to see that if $R_x$ measures the distance to nearest neighbor $n_x$ of some point $x$, then
$$
P(R_x>r) = P(\text{no points in a ball of radius }r\text{ around }x)  = e^{-\lambda\pi r^2},
$$
so $R_x$ has the Rayleigh distribution with parameter $\sigma = 1/(\lambda\pi\sqrt{2})$. And since the direction to nearest neighbor is obviously uniformly distributed, the vector $n_x - x$ is distributed as a pair of independent centered normal variables with variance $\sigma^2$. It is not hard to see that this is also true if we speak about the nearest neighbor of a point taken from the Poisson point process.

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?