Let be a compact, negatively curved surface. From the (finite) set of all closed geodesics on of length at most , choose one, say, , at random, and let be the number of its self-intersections. It is known that there is a positive constant depending on the metric such that in probability as . The main results of this article concern the size of typical fluctuations of about . It is proved that if the metric has constant curvature , then typical fluctuations are of order ; in particular, as the random variables converge in distribution. In contrast, it is also proved that if the metric has variable negative curvature, then fluctuations of are of order ; in particular, the random variables 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.
"Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces." Duke Math. J. 163 (6) 1191 - 1261, 15 April 2014. https://doi.org/10.1215/00127094-2649425