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

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