The Annals of Probability

Random recursive triangulations of the disk via fragmentation theory

Nicolas Curien and Jean-François Le Gall

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We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension β* + 1, where β* = (√17 − 3)/2, and that it can be described as the geodesic lamination coded by a random continuous function which is Hölder continuous with exponent β* − ε, for every ε > 0. We also discuss recursive constructions of triangulations of the n-gon that give rise to the same continuous limit when n tends to infinity.

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Ann. Probab., Volume 39, Number 6 (2011), 2224-2270.

First available in Project Euclid: 17 November 2011

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Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C80: Random graphs [See also 60B20]

Triangulation of the disk noncrossing chords Hausdorff dimension geodesic lamination fragmentation process random recursive construction


Curien, Nicolas; Le Gall, Jean-François. Random recursive triangulations of the disk via fragmentation theory. Ann. Probab. 39 (2011), no. 6, 2224--2270. doi:10.1214/10-AOP608.

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  • [1] Aldous, D. (1994). Recursive self-similarity for random trees, random triangulations and Brownian excursion. Ann. Probab. 22 527–545.
  • [2] Aldous, D. (1994). Triangulating the circle, at random. Amer. Math. Monthly 101 223–233.
  • [3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [4] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge Univ. Press, Cambridge.
  • [5] Bertoin, J. and Gnedin, A. V. (2004). Asymptotic laws for nonconservative self-similar fragmentations. Electron. J. Probab. 9 575–593 (electronic).
  • [6] Bonahon, F. (2001). Geodesic laminations on surfaces. In Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998). Contemp. Math. 269 1–37. Amer. Math. Soc., Providence, RI.
  • [7] Brennan, M. D. and Durrett, R. (1986). Splitting intervals. Ann. Probab. 14 1024–1036.
  • [8] Brennan, M. D. and Durrett, R. (1987). Splitting intervals. II. Limit laws for lengths. Probab. Theory Related Fields 75 109–127.
  • [9] David, F., Dukes, W. M. B., Jonsson, T. and Stefánsson, S. Ö. (2009). Random tree growth by vertex splitting. J. Stat. Mech. P04009.
  • [10] David, F., Hagendorf, C. and Wiese, K. J. (2008). A growth model for RNA secondary structures. J. Stat. Mech. P04008.
  • [11] Devroye, L., Flajolet, P., Hurtado, F., Noy, M. and Steiger, W. (1999). Properties of random triangulations and trees. Discrete Comput. Geom. 22 105–117.
  • [12] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
  • [13] Evans, S. N. (2008). Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin.
  • [14] Le Gall, J.-F. and Paulin, F. (2008). Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 893–918.
  • [15] Liu, Q. (1997). Sur une équation fonctionnelle et ses applications: Une extension du théorème de Kesten-Stigum concernant des processus de branchement. Adv. in Appl. Probab. 29 353–373.
  • [16] Müller, M. (2003). Repliement d’hétéropolymères. Ph.D. thesis, Université Paris-Sud. Available at:
  • [17] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [18] Sleator, D. D., Tarjan, R. E. and Thurston, W. P. (1988). Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc. 1 647–681.