Abstract
Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of m letters, then for large m the distribution of the self-intersection number approaches the Gaussian distribution. The theorem was strongly suggested by a computer experiment with four million curves producing a very nearly Gaussian distribution.
Similar content being viewed by others
References
Billingsley, P.: Probability and Measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1995). A Wiley-Interscience Publication
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999). A Wiley-Interscience Publication
Birman, J.S., Series, C.: An algorithm for simple curves on surfaces. J. Lond. Math. Soc. (2) 29(2), 331–342 (1984)
Chas, M.: Experimental results in combinatorial topology. Manuscript
Chas, M.: Combinatorial Lie bialgebras of curves on surfaces. Topology 43(3), 543–568 (2004)
Chas, M., Phillips, A.: Self-intersection numbers of curves in the doubly punctured plane. arXiv:1001.4568
Chas, M., Phillips, A.: Self-intersection numbers of curves on the punctured torus. Exp. Math. 19(2), 129–148 (2010)
Cohen, M., Lustig, M.: Paths of geodesics and geometric intersection numbers. I. In: Combinatorial Group Theory and Topology, Alta, Utah, 1984. Annals of Mathematics Studies, vol. 111, pp. 479–500. Princeton University Press, Princeton (1987)
Denker, M., Keller, G.: On U-statistics and v. Mises’ statistics for weakly dependent processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 64(4), 505–522 (1983)
Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19, 293–325 (1948)
Iosifescu, M.: On U-statistics and von Mises statistics for a special class of Markov chains. J. Stat. Plan. Inference 30(3), 395–400 (1992)
Lalley, S.P.: Self-intersections of random geodesics on negatively curved surfaces. arXiv:0907.0259
Lalley, S.P.: Self-intersections of closed geodesics on a negatively curved surface: statistical regularities. In: Convergence in Ergodic Theory and Probability, Columbus, OH, 1993. Ohio State University Mathematical Research Institute Publications, vol. 5, pp. 263–272. de Gruyter, Berlin (1996)
Mirzakhani, M.: Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. Math. (2) 168(1), 97–125 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSF grants DMS-0805755 and DMS-0757277.
Rights and permissions
About this article
Cite this article
Chas, M., Lalley, S.P. Self-intersections in combinatorial topology: statistical structure. Invent. math. 188, 429–463 (2012). https://doi.org/10.1007/s00222-011-0350-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-011-0350-7