Self-intersections in combinatorial topology: statistical structure

TL;DR

This paper investigates the statistical distribution of self-intersection numbers of random curves on surfaces, demonstrating that for large word lengths, the distribution converges to a Gaussian, revealing a probabilistic structure in combinatorial topology.

Contribution

It establishes the asymptotic Gaussian distribution of self-intersection numbers for random classes of curves on surfaces, connecting topology with probability theory.

Findings

Self-intersection numbers follow a Gaussian distribution for large word lengths.

Distribution converges to normal as the number of generators increases.

Provides probabilistic insight into topological properties of curves.

Abstract

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of $m$ letters, then for large $m$ the distribution of the self-intersection number approaches the Gaussian distribution.