# Nearest neighbor for planar Poisson is normally distributed

Answering a 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?

• This is not really an answer, but a possibly related reference. Terry Soo and co-authors have some papers where they study "Poisson thinning". That is, they construct for fixed $\mu<\lambda$ a deterministic measurable map which sends an instance of a Poisson process with rate $\lambda$ to an instance of a Poisson process with rate $\mu$. This is all about "extracting randomness" from PPs; maybe the fact that you mention is used somewhere. (NB: of course non-deterministic Poisson thinning is simple) – Anthony Quas Sep 20 '15 at 6:13
• Normal random variables can be negative, while distances cannot... – Michael Sep 20 '15 at 17:41
• The $e^{-x^2}$ function is associated with the Gaussian density. In your link, the $e^{-x^2}$ is associated with a CDF function. I do not think it is correct to call your distance "normally distributed," even if you overlook the non-negative issue. – Michael Sep 20 '15 at 17:49
• @Michael, maybe I was not very careful writing this. I don't claim that the distance is normally distributed, but rather that the neighbor is. The neighbor is a two dimensional vector, whose length has Rayleigh distribution with cdf $1-e^{-ax^2}$. But the vector does have normal distribution. I copied the paragraph from my MathSE post in order to clarify. – zhoraster Sep 20 '15 at 18:01