Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces

Abstract

Let $\Upsilon$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $\Upsilon$ of length at most $L$, choose one, say, ${\gamma }_L$, at random, and let $N\left({\gamma }_L\right)$ be the number of its self-intersections. It is known that there is a positive constant $\kappa$ depending on the metric such that $N\left({\gamma }_L\right)/L^2\to \kappa$ in probability as $L\to \infty$. The main results of this article concern the size of typical fluctuations of $N\left({\gamma }_L\right)$ about $\kappa L^2$. It is proved that if the metric has constant curvature $-1$, then typical fluctuations are of order $L$; in particular, as $L\to \infty$ the random variables $\left(N\right({\gamma }_L)-\kappa L^2)/L$ converge in distribution. In contrast, it is also proved that if the metric has variable negative curvature, then fluctuations of $N\left({\gamma }_L\right)$ are of order $L^{3/2}$; in particular, the random variables $\left(N\right({\gamma }_L)-\kappa L^2)/L^{3/2}$ converge in distribution to a Gaussian distribution with positive variance. Similar results are proved for generic geodesics, that is, geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure.