Self-intersections of Random Geodesics on Negatively Curved Surfaces

We study the fluctuations of self-intersection counts of random geodesic segments of length t on a compact, negatively curved surface in the limit of large t. If the initial direction vector of the geodesic is chosen according to the Liouville measure, then it is not difficult to show that the number N(t) of self-intersections by time t grows like κt2, where κ = κM is a positive constant depending on the surface M . We show that (for a smooth modification of N(t)) the fluctuations are of size t, and the limit distribution is a weak limit of Gaussian quadratic forms. We also show that the fluctuations of localized self-intersection counts (that is, only self-intersections in a fixed subset of M are counted) are typically of size t3/2, and the limit distribution is Gaussian. 1. FLUCTUATIONS OF SELF-INTERSECTION COUNTS Choose a point x and a direction θ at random on a compact, negatively curved surface M , and let γ(t) = γ(t;x, θ) be the (unit speed) geodesic ray in direction θ started at x. For large t the number N(t) = N(γ[0, t]) of transversal1 self-intersections of the geodesic segment γ[0, t] will be of order t; in fact, if κM = (2π|M |)−1, where |M | denotes the surface area of M , then (1) lim t→∞ N(t)/t = κM/2 with probability 1. See section 3 below for the (easy) proof. Furthermore, the empirical distribution of the self-intersection points converges to the uniform distribution on the surface. Similar results hold for a randomly chosen closed geodesic [8]: If from among all closed geodesics of length ≤ L one is chosen at random, then the number of self-intersections, normalized by L, will, with probability approaching one as L→∞, be close to κg . These results have been extended [12] to the number and distribution of self-intersections at angles in fixed intervals [α, β]. Closed geodesics with no self-intersections have long been of interest in geometry — see, for instance, [3, 2] — and recently, M. Mirzakhani [10] found the asymptotic growth rate of the number of simple closed geodesics of length ≤ t as t → ∞. That this number is not 0 shows (in view of the Law of Large Numbers (1) ) that there is substantial variation in the random variable N(t). The primary objective of this paper is to investigate the fluctuations (second-order asymptotics) of the self-intersection numbers. One’s first guess might be that these are of order t, and this is indeed the case; however, lest this seem too obvious we add that if one counts only self-intersections Date: July 1, 2009. 1991 Mathematics Subject Classification. Primary 58F17, secondary 53C22, 37D20.