2011/11/30 by Steven P. Lalley
Mathematics · #Constant (computer programming) #Constant curvature #Curvature #Distribution (mathematics) #Gaussian curvature #Geodesic #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric (unit) #Random variable #Stochastic processes and statistical mechanics #Tangent #math.DS #math.GT #msc:37D40
paper · pdf · doi:10.1215/00127094-2649425
published as Duke Math. J. 163, no. 6 (2014), 1191-1261
arxiv created 2013/04/19 · openalex publication_date 2014/04/11 · arxiv updated 2015/01/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let ϒ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on ϒ of length at most L, choose one, say, γL, at random, and let N(γL) be the number of its self-intersections. It is known that there is a positive constant κ depending on the metric such that N(γL)/L2→κ in probability as L→∞. The main results of this article concern the size of typical fluctuations of N(γL) about κL2. It is proved that if the metric has constant curvature −1, then typical fluctuations are of order L; in particular, as L→∞ the random variables (N(γL)−κL2)/L converge in distribution. In contrast, it is also proved that if the metric has variable negative curvature, then fluctuations of N(γL) are of order L3/2; in particular, the random variables (N(γL)−κL2)/L3/2 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.