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

Let $\mathrm{{\rm Y}}$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $\mathrm{{\rm Y}}$ of length $\le L$, 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 $N({\gamma }_L)\mathrm{/}{L}^2\to \kappa$ in probability as $L\to \mathrm{\infty }$. 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, $(N({\gamma }_L)-\kappa L^2)\mathrm{/}L$ 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 $L^{3\mathrm{/}2}$, 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.