Eur. Phys. J. Special Topics 216, 3-20 (2013)
https://doi.org/10.1140/epjst/e2013-01724-4

Review

Statistical universality and the method of Poissonian randomizations

Holon Institute of Technology, PO Box 305, Holon 58102, Israel

^a e-mail: eliazar@post.tau.ac.il

Received: 30 November 2012
Revised: 6 December 2012
Published online: 31 January 2013

Abstract

In this paper we demonstrate the remarkable effectiveness of Poissonian randomizations in the generation of statistical universality. We do so via a highly versatile spatio-statistical model in which points are randomly scattered, according to a Poisson process, across a general metric space. The points have general independent and identically distributed random physical characteristics. A probe is positioned in space, and is affected by the points. The effect of a given point on the probe is a function of the physical characteristic of the point and the distance of the point from the probe. We determine the classes of Poissonian randomizations – i.e., the spatial Poissonian scatterings of the points – that render the effects of the points invariant with respect to the physical characteristics of the points. These Poissonian randomizations have intrinsic power-law structures, yield statistical robustness, and generate universal statistics including Lévy distributions and extreme-value distributions. In effect, our results establish how “fractal” spatial geometries lead to statistical universality.