September – The Universality of the Poisson Point Process (Breaking a Pile of Sand)

I thought that it would be nice to write about some purely mathematical subject today, so I decided to briefly discuss the famous Poisson process and its properties under a transformation.

It is a well-known fact that some point processes converge to the Poisson point process after a sufficient amount of right transformations. This is of course why the Poisson p.p appears everywhere in nature; for example, grain of sands in a sand castle – made by a child at a beach on a sunny day – will eventually spread by wind (or water) around the beach. The grains will be randomly displaced here and there forming something approximating a realization of a Poisson p.p., even though the original pile was more ordered.

Let's play the game with a pile of sand spreading to a flat area by random translations of the grains.

Consider a pile of sand at origo. Furthermore, let's assume that the pile is small enough (compared to unit distance) so that it is concentrated in a single point. Let's make a natural assumption that the amount of grains in the pile is Poisson distributed. Now our pile of sand is a Poisson point process concentrated in the origo.

Suppose that after a while, wind moves individual sand grains from the pile directing them with a uniformly distributed angle and to a random normally distributed distance between $[0, \infty)$. In other words, the probability that a sand grain is moved from the origo to a location $Y_j$ inside a circle $B(r_1,r_2)$ is $2\int_{r_1}^{r^2} \frac{1}{\sqrt{2 \pi}}e^{-y^2} dy.$

After a random translation of all points the Laplace functional of the distances of the points is

$\displaystyle \mathcal{L}_{\tau}(f)$ $\displaystyle = \mathbb{E} \exp \left[-\sum_i f(\vert Y_i\vert) \right] = \mathbb{E} \prod_i \exp \left[ -f(\vert Y_i\vert) \right]$    
  $\displaystyle = \mathbb{E}_n \left[ (2 \pi)^{-n/2} \int_0^{\infty} \dots \int_0...
...left[ -f(y_i) \right] \frac{2}{\sqrt{2 \pi}}e^{-y_i^2} dy_1 \dots d y_n \right]$    
  $\displaystyle = \mathbb{E}_n \left[(2 \pi)^{-n/2} \left( \int_{0}^{\infty} 2\exp \left[ -f(y) - y^2 \right] dy \right)^n\right]$    
  $\displaystyle = \mathbb{E}_n \exp \left[ \sum_{i= 0}^n \log \left( \int_{0}^{\infty} \exp \left[ -f(y) \right] \frac{2e^{-y^2}}{\sqrt{2 \pi}} dy \right) \right]$    

Evaluating the Poisson Laplace functional $\mathcal{L}(g)$ of the sand pile with $g = - \log \left( \int_{0}^{\infty} \exp \left[ -f(y) \right] \frac{e^{-y^2}}{\sqrt{2 \pi}} dy \right) $ we get

$\displaystyle \mathcal{L}(g) = \mathcal{L}_{\tau}(f)$ $\displaystyle = \exp \left[ -\int_{\mathbb{R}^2} 1 - e^{\log \left( \int_{0}^{\...
...t[ -f(y) \right] \frac{2e^{-y^2}}{\sqrt{2 \pi}} dy \right) }\Lambda(dx) \right]$    
  $\displaystyle = \exp \left[-\int_{\mathbb{R}^2} \left(1 - \int_{0}^{\infty} \exp [-f(y)] \frac{2e^{-y^2}}{\sqrt{2 \pi}} dy \right) \Lambda(dx) \right]$    
  $\displaystyle = \exp \left[-\int_{\mathbb{R}^2}\int_{0}^{\infty} \left(1 - \exp [-f(y)] \right) \frac{2e^{-y^2}}{\sqrt{2 \pi}}dy \Lambda(dx) \right]$    
  $\displaystyle = \exp \left[-\int_{0}^{\infty} \left(1 - \exp [-f(y)] \right) \sqrt{\frac{2}{\pi}}e^{-y^2} N dy \right], (1)$    

where $N$ denotes the expected number of points in the sand pile (Campbell's theorem). But the formula $(1)$ is the Laplace functional of the Poisson p.p. on $[0, \infty)$ with a density parameter $\sqrt{\frac{2}{\pi}}e^{-x^2} N.$ As the direction of the translation of a sand grain was uniformly distributed, we can conclude that the sand grains are Poisson distributed according to the Poisson p.p. with the two-dimensional multivariate Gaussian distribution as the density.

The result above applies to any translations as long as the translation is independently applied to every point (or grain); the Poisson point process remains Poisson in a transformation. Here we had to make the Poisson assumption also for the pile so that the result above could be derived. But indeed, we can show that non-Poisson processes approach Poisson p.p. when the points are translated in a certain manner. However, I leave this is out of the scope of this blog entry for now, and take the remark made here as a simple (possibly a bit weird) example of the universality of the Poisson point process.

Figure: $10000$ points randomly moved from the origo to the plane