### Metadata

### Abstract

We show that the tessellation of a compact, negatively curved surface induced by a long random geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.

### References

[AF82] Cross section map for the geodesic flow on the modular surface, Conference in modern analysis and probability (New Haven, Conn., 1982) (Contemporary Mathematics) Volume 26 (1982), pp. 9-24 | Article | Zbl 0552.58026

[Bil68] Convergence of probability measures, John Wiley & Sons, 1968 | Zbl 0172.21201

[Bow72] The equidistribution of closed geodesics, Am. J. Math., Volume 94 (1972), pp. 413-423 | Article | MR 315742 | Zbl 0249.53033

[Bow73] Symbolic dynamics for hyperbolic flows, Am. J. Math., Volume 95 (1973), pp. 429-460 | Article | MR 339281 | Zbl 0282.58009

[Bow75] Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Volume 470, Springer, 1975 | MR 442989 | Zbl 0308.28010

[BS79] Markov maps associated with Fuchsian groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 50 (1979), pp. 153-170 | Article | Numdam | Zbl 0439.30033

[BW72] Expansive one-parameter flows, J. Differ. Equations, Volume 12 (1972), pp. 180-193 | Article | MR 341451 | Zbl 0242.54041

[Cal53] A general ergodic theorem, Ann. Math., Volume 58 (1953), pp. 182-191 | Article | MR 55415 | Zbl 0052.11903

[Cal03] An explicit expression for the distribution of the number of sides of the typical Poisson–Voronoi cell, Adv. Appl. Probab., Volume 35 (2003) no. 4, pp. 863-870 | Article | MR 2014258 | Zbl 1038.60008

[CFF02] Processes with long memory: regenerative construction and perfect simulation, Ann. Appl. Probab., Volume 12 (2002) no. 3, pp. 921-943 | MR 1925446 | Zbl 1016.60061

[Dol04] Limit theorems for partially hyperbolic systems, Trans. Am. Math. Soc., Volume 356 (2004) no. 4, pp. 1637-1689 | Article | MR 2034323 | Zbl 1031.37031

[DZ98] Large deviations techniques and applications, Applications of Mathematics, Volume 38, Springer, 1998 | MR 1619036 | Zbl 0896.60013

[EJM91] Multiple intersections on negatively curved surfaces, J. Differ. Geom., Volume 33 (1991) no. 1, pp. 253-261 | Article | MR 1085143 | Zbl 0722.53043

[Kin93] Poisson processes, Oxford Studies in Probability; Oxford Science Publications, Volume 3, Clarendon Press; Oxford University Press, 1993 | MR 1207584 | Zbl 0771.60001

[KM99] Logarithm laws for flows on homogeneous spaces, Invent. Math., Volume 138 (1999) no. 3, pp. 451-494 | Article | MR 1719827 | Zbl 0934.22016

[Lal86] Regenerative representation for one-dimensional Gibbs states, Ann. Probab., Volume 14 (1986) no. 4, pp. 1262-1271 | Article | MR 866347 | Zbl 0612.60093

[Lal87] Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math., Volume 8 (1987) no. 2, pp. 154-193 | Article | MR 886923 | Zbl 0637.58013

[Lal89] Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J., Volume 58 (1989) no. 3, pp. 795-821 | MR 1016446 | Zbl 0732.53035

[Lal96] Self-intersections of closed geodesics on a negatively curved surface: statistical regularities, Convergence in ergodic theory and probability (Columbus, OH, 1993) (Ohio State University Mathematical Research Institute Publications) Volume 5, Walter de Gruyter, 1996, pp. 263-272 | MR 1412610 | Zbl 0868.58066

[Lal14] Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces, Duke Math. J., Volume 163 (2014) no. 6, pp. 1191-1261 | Article | MR 3192528 | Zbl 1328.37033

[LC60] An approximation theorem for the Poisson binomial distribution, Pac. J. Math., Volume 10 (1960), pp. 1181-1197 | Article | MR 142174 | Zbl 0118.33601

[LNP19] Kleinian Schottky groups, Patterson–Sullivan measures, and Fourier decay, with an appendix on stationarity of Patterson–Sullivan measures (2019) (https://arxiv.org/abs/1902.01103, to appear in Duke Mathematical Journal)

[Mau06] Dynamical Borel–Cantelli lemma for hyperbolic spaces, Isr. J. Math., Volume 152 (2006), pp. 143-155 | Article | MR 2214457 | Zbl 1129.53057

[Mil64a] Random polygons determined by random lines in a plane, Proc. Natl. Acad. Sci. USA, Volume 52 (1964), pp. 901-907 | Article | MR 168000

[Mil64b] Random polygons determined by random lines in a plane. II, Proc. Natl. Acad. Sci. USA, Volume 52 (1964), pp. 1157-1160 | Article | MR 169258 | Zbl 0133.16001

[PPS15] Equilibrium states in negative curvature, Astérisque, Volume 373, Société Mathématique de France, 2015 | Zbl 1347.37001

[Rat73] Markov partitions for Anosov flows on $n$-dimensional manifolds, Isr. J. Math., Volume 15 (1973), pp. 92-114 | Article | MR 339282 | Zbl 0269.58010

[San04] Integral geometry and geometric probability, Cambridge Mathematical Library, Cambridge University Press, 2004 (with a foreword by Mark Kac) | Zbl 1116.53050

[Ser81] Symbolic dynamics for geodesic flows, Acta Math., Volume 146 (1981) no. 1-2, pp. 103-128 | Article | MR 594628 | Zbl 0488.58016

[Ser85] The modular surface and continued fractions, J. Lond. Math. Soc., Volume 31 (1985) no. 1, pp. 69-80 | Article | MR 810563 | Zbl 0545.30001

[SKM87] Stochastic geometry and its applications, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1987 (With a foreword by D. G. Kendall.) | Zbl 0622.60019

[Sul82] Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., Volume 149 (1982) no. 3, p. 3-4 | MR 688349 | Zbl 0517.58028

[SY94] Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, Volume 1, International Press, 1994 (Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso.) | Zbl 0830.53001