The Eventual Gaussian Distribution for Self-Intersection Numbers on Closed Surfaces

TL;DR

This paper proves that for large classes of loops on closed surfaces, the distribution of their self-intersection numbers tends to a Gaussian, extending previous results to closed surfaces using Markov chains and the central limit theorem.

Contribution

It generalizes Lalley and Chas's result by establishing a Gaussian distribution for self-intersection numbers on closed surfaces using probabilistic methods.

Findings

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

Markov chains effectively model the exponential mixing of geodesic flow.

The result extends previous work to include closed surfaces.

Abstract

Oriented loops on an orientable surface are, up to equivalence by free homotopy, in one-to-one correspondence with the conjugacy classes of the surface's fundamental group. These conjugacy classes can be expressed (not uniquely in the case of closed surfaces) as a cyclic word of minimal length in terms of the fundamental group's generators. The self-intersection number of a conjugacy class is the minimal number of transverse self-intersections of representatives of the class. By using Markov chains to encapsulate the exponential mixing of the geodesic flow and achieve sufficient independence, we can use a form of the central limit theorem to describe the statistical nature of the self-intersection number. For a class chosen at random among all classes of length n, the distribution of the self intersection number approaches a Gaussian when n is large. This theorem generalizes the result of Steven Lalley and Moira Chas to include the case of closed surfaces.