Answering a recent question, 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?



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