Annals of Probability

The dual tree of a recursive triangulation of the disk

Nicolas Broutin and Henning Sulzbach

Full-text: Open access


In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011) 2224–2270] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process $\mathscr{M}$. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov–Hausdorff sense to a limit real tree $\mathscr{T}$, which is encoded by $\mathscr{M}$. This confirms a conjecture of Curien and Le Gall.

Article information

Ann. Probab., Volume 43, Number 2 (2015), 738-781.

First available in Project Euclid: 2 February 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60F17: Functional limit theorems; invariance principles 05C05: Trees
Secondary: 11Y16: Algorithms; complexity [See also 68Q25] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16: Asymptotic enumeration

Real tree Gromov–Hausdorff convergence functional limit theorem contraction method


Broutin, Nicolas; Sulzbach, Henning. The dual tree of a recursive triangulation of the disk. Ann. Probab. 43 (2015), no. 2, 738--781. doi:10.1214/13-AOP894.

Export citation


  • [1] Addario-Berry, L., Broutin, N., Goldschmidt, C. and Miermont, G. (2013). The scaling limit of the minimum spanning tree of the complete graph. Preprint. Available at arXiv:1301.1664.
  • [2] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
  • [3] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990) (M. T. Barlow and N. H. Bingham, eds.) 23–70. Cambridge Univ. Press, Cambridge.
  • [4] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
  • [5] Aldous, D. (1994). Recursive self-similarity for random trees, random triangulations and Brownian excursion. Ann. Probab. 22 527–545.
  • [6] Aldous, D. (1994). Triangulating the circle, at random. Amer. Math. Monthly 101 223–233.
  • [7] Bai, Z.-D., Hwang, H.-K., Liang, W.-Q. and Tsai, T.-H. (2001). Limit theorems for the number of maxima in random samples from planar regions. Electron. J. Probab. 6 41 pp. (electronic).
  • [8] Baryshnikov, Y. and Gnedin, A. (2001). Counting intervals in the packing process. Ann. Appl. Probab. 11 863–877.
  • [9] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, Cambridge.
  • [10] Bertoin, J. and Gnedin, A. V. (2004). Asymptotic laws for nonconservative self-similar fragmentations. Electron. J. Probab. 9 575–593.
  • [11] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [12] Broutin, N., Neininger, R. and Sulzbach, H. (2012). Partial match queries in random quadtrees. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA) (Y. Rabani, ed.) 1056–1065.
  • [13] Broutin, N., Neininger, R. and Sulzbach, H. (2013). A limit process for partial match queries in random quadtrees and $2$-d trees. Ann. Appl. Probab. 23 2560–2603.
  • [14] Chern, H.-H. and Hwang, H.-K. (2003). Partial match queries in random quadtrees. SIAM J. Comput. 32 904–915 (electronic).
  • [15] Coffman, E. G. Jr., Mallows, C. L. and Poonen, B. (1994). Parking arcs on the circle with applications to one-dimensional communication networks. Ann. Appl. Probab. 4 1098–1111.
  • [16] Curien, N. and Joseph, A. (2011). Partial match queries in two-dimensional quadtrees: A probabilistic approach. Adv. in Appl. Probab. 43 178–194.
  • [17] Curien, N. and Kortchemski, I. (2014). Random noncrossing plane configurations: A conditioned Galton–Watson tree approach. Random Structures Algorithms. To appear.
  • [18] Curien, N. and Le Gall, J.-F. (2011). Random recursive triangulations of the disk via fragmentation theory. Ann. Probab. 39 2224–2270.
  • [19] Curien, N. and Werner, W. (2013). The Markovian hyperbolic triangulation. J. Eur. Math. Soc. (JEMS) 15 1309–1341.
  • [20] David, F., Hagendorf, C. and Wiese, K. J. (2008). A growth model for rna secondary structures. J. Stat. Mech. Theory Exp. 2008 P04008.
  • [21] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147.
  • [22] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
  • [23] Evans, S. N. (2008). Probability and Real Trees. Lecture Notes in Math. 1920. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 623, 2005. Springer, Berlin.
  • [24] Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester.
  • [25] Falconer, K. J. (1986). The Geometry of Fractal Sets. Cambridge Tracts in Mathematics 85. Cambridge Univ. Press, Cambridge.
  • [26] Flajolet, P., Gonnet, G., Puech, C. and Robson, J. M. (1993). Analytic variations on quadtrees. Algorithmica 10 473–500.
  • [27] Flajolet, P. and Puech, C. (1986). Partial match retrieval of multidimensional data. J. Assoc. Comput. Mach. 33 371–407.
  • [28] Flajolet, P. and Sedgewick, R. (1995). Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theoret. Comput. Sci. 144 101–124.
  • [29] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge Univ. Press, Cambridge.
  • [30] Gromov, M. (1999). Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics 152. Birkhäuser, Boston, MA.
  • [31] Haas, B. and Miermont, G. (2004). The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 57–97 (electronic).
  • [32] Haas, B. and Miermont, G. (2012). Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 2589–2666.
  • [33] Knuth, D. E. (1973). The Art of Computer Programming: Sorting and Searching. Addison-Wesley, Reading, MA.
  • [34] Kortchemski, I. (2014). Random stable laminations of the disk. Ann. Probab. To appear.
  • [35] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
  • [36] Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213–252.
  • [37] 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.
  • [38] Marckert, J.-F. and Panholzer, A. (2002). Noncrossing trees are almost conditioned Galton–Watson trees. Random Structures Algorithms 20 115–125.
  • [39] Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378–418.
  • [40] Neininger, R. and Sulzbach, H. (2014). On a functional contraction method. Ann. Probab. To appear.
  • [41] Nörlund, N. E. (1924). Vorlesungen über Differenzenrechnung. Springer, Berlin.
  • [42] Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. in Appl. Probab. 27 770–799.
  • [43] Ragab, M. and Roesler, U. (2014). The Quicksort process. Stochastic Process. Appl. 124 1036–1054.
  • [44] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • [45] Rösler, U. (1991). A limit theorem for “Quicksort.” RAIRO Inform. Théor. Appl. 25 85–100.
  • [46] Sedgewick, R. and Flajolet, P. (1996). An Introduction to the Analysis of Algorithm. Addison-Wesley, Reading, MA.