Self-Intersections of Random Geodesics on Negatively Curved Surfaces

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.