On the central role of the scale invariant Poisson processes on (0,infty)

TL;DR

This paper explores the fundamental role of scale invariant Poisson processes on (0,∞) in unifying various discrete structures across number theory, combinatorics, and genetics through their continuous limit properties.

Contribution

It highlights the central importance of scale invariant Poisson processes as unifying objects in probability theory and their connections to diverse discrete phenomena.

Findings

Unify diverse discrete structures via continuous limits

Identify the central role of scale invariant Poisson processes

Connect Poisson processes to number theory, combinatorics, and genetics

Abstract

The scale invariant Poisson processes on (0,infty) play a central but mildly disguised role in number theory, combinatorics, and genetics. They give the continuous limits which underly and unify diverse discrete structures, including the prime factorization of a uniformly chosen integer, the factorization of polynomials over finite fields, the decomposition into cycles of random permutations, the decomposition into components of random mappings, and the Ewens sampling formula. They deserve attention as one of the fundamental and central objects of probability theory.