Self-Intersections of Random Geodesics on Negatively Curved Surfaces

TL;DR

This paper investigates the statistical fluctuations of self-intersection counts of random geodesics on negatively curved surfaces, revealing different fluctuation scales and limit distributions for global and localized counts as the geodesic length grows.

Contribution

It provides a detailed analysis of the fluctuation behavior and limit distributions of self-intersection counts for random geodesics on negatively curved surfaces, including localized counts.

Findings

Global self-intersection counts grow like rac14;t^2 with Gaussian quadratic form fluctuations.

Localized counts grow like rac14;t^{3/2} with Gaussian fluctuations.

Limit distributions depend on whether counts are global or localized.

Abstract

We study the fluctuations of self-intersection counts of random geodesic segments of length $t$ on a compact, negatively curved surface in the limit of large $t$. If the initial direction vector of the geodesic is chosen according to the \emph{Liouville measure}, then it is not difficult to show that the number $N (t)$ of self-intersections by time $t$ grows like $\kappa t^{2}$, where $\kappa =\kappa_{M}$ is a positive constant depending on the surface $M$. We show that (for a smooth modification of $N (t)$) the fluctuations are of size $t$, and the limit distribution is a weak limit of Gaussian quadratic forms. We also show that the fluctuations of \emph{localized} self-intersection counts (that is, only self-intersections in a fixed subset of $M$ are counted) are typically of size $t^{3/2}$, and the limit distribution is Gaussian.