Electronic Journal of Probability

Some families of increasing planar maps

Marie Albenque and Jean-Francois Marckert

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Stack-triangulations appear as natural objects when one wants to define some families of increasing triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations.

Article information

Electron. J. Probab. Volume 13 (2008), paper no. 56, 1624-1671.

Accepted: 19 September 2008
First available in Project Euclid: 1 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60C05: Combinatorial probability
Secondary: 60F1

stackmaps triangulations Gromov-Hausdorff convergence continuum random tree

This work is licensed under a Creative Commons Attribution 3.0 License.


Albenque, Marie; Marckert, Jean-Francois. Some families of increasing planar maps. Electron. J. Probab. 13 (2008), paper no. 56, 1624--1671. doi:10.1214/EJP.v13-563. http://projecteuclid.org/euclid.ejp/1464819129.

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  • Aldous, David. The continuum random tree. II. An overview. Stochastic analysis (Durham, 1990), 23–70, London Math. Soc. Lecture Note Ser., 167, Cambridge Univ. Press, Cambridge, 1991.
  • Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289.
  • J.S. Andrade Jr, H.J. Herrmann, R.F.S. Andrade & L.R. da Silva (2005), Apollonian Networks: Simultaneously Scale-free, Small World, Euclidien, Space Filling, and with Matching graphs, Phys. Rev. Lett., 94, 018702.
  • Angel, O. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (2003), no. 5, 935–974.
  • Angel, Omer; Schramm, Oded. Uniform infinite planar triangulations. Comm. Math. Phys. 241 (2003), no. 2-3, 191–213.
  • F. Bergeron, P. Flajolet, B. Salvy. Varietes of increasing trees}, research report INRIA 1583, available at ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-1583.ps.gz
  • O. Bernardi & N. Bonichon (2007), Catalan intervals and realizers of triangulations., to appear in J. of Comb. Theo. - Series A, ArXiv: math.CO/0704.3731. Extended abstract FPSAC 2007.
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • O. Bodini, A. Darrasse, & M. Soria (2008), Distances in random Apollonian network structures, 20th Annual international Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2008, DMTCS Proceedings, to appear.
  • T. Böhme, M. Stiebitz & M. Voigt (1998), On Uniquely 4-Colorable Planar Graphs, url = citeseer.ist.psu.edu/110448.html
  • Bouttier, J.; Di Francesco, P.; Guitter, E. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004), no. 1, Research Paper 69, 27 pp. (electronic).
  • Broutin, N.; Devroye, L.; McLeish, E.; de la Salle, M. The height of increasing trees. Random Structures Algorithms 32 (2008), no. 4, 494–518.
  • Chassaing, Philippe; Durhuus, Bergfinnur. Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 (2006), no. 3, 879–917.
  • Chassaing, Philippe; Schaeffer, Gilles. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 (2004), no. 2, 161–212.
  • Cori, Robert; Vauquelin, Bernard. Planar maps are well labeled trees. Canad. J. Math. 33 (1981), no. 5, 1023–1042.
  • A. Darrasse & M. Soria, (2007), Degree distribution of RAN structures and Boltzmann generation for trees, International Conference on Analysis of Algorithms Antibes, June 2007,DIMACS Series in Discrete Mathematics and Theoretical Computer Science, p. 1-14.
  • Dong, Rui; Goldschmidt, Christina; Martin, James B. Coagulation-fragmentation duality, Poisson-Dirichlet distributions and random recursive trees. Ann. Appl. Probab. 16 (2006), no. 4, 1733–1750.
  • Duquesne, Thomas; Le Gall, Jean-François. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005), no. 4, 553–603.
  • Evans, Steven N.; Pitman, Jim; Winter, Anita. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 (2006), no. 1, 81–126.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Felsner, Stefan; Zickfeld, Florian. On the number of planar orientations with prescribed degrees. Electron. J. Combin. 15 (2008), no. 1, Research Paper 77, 41 pp.
  • P. Flajolet, P. Dumas & V. Puyhaubert, (2006). Some exactly solvable models of urn process theory. Discrete Mathematics and Computer Science, vol. AG, 59-118.
  • F. Gillet, (2003), …tude d'algorithmes stochastiques et arbres. PhD thesis manuscript.
  • Gittenberger, Bernhard. A note on: "State spaces of the snake and its tour–-convergence of the discrete snake" [J. Theoret. Probab. 16 (2003), no. 4, 1015–1046; ] by J.-F. Marckert and A. Mokkadem. J. Theoret. Probab. 16 (2003), no. 4, 1063–1067 (2004).
  • Graham, Ronald L.; Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Allan R.; Yan, Catherine H. Apollonian circle packings: number theory. J. Number Theory 100 (2003), no. 1, 1–45.
  • Janson, Svante. Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 (2006), no. 3, 417–452.
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
  • Krikun, M. A. A uniformly distributed infinite planar triangulation and a related branching process. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 307 (2004), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10, 141–174, 282–283; translation in J. Math. Sci. (N. Y.) 131 (2005), no. 2, 5520–5537
  • Kurtz, Thomas; Lyons, Russell; Pemantle, Robin; Peres, Yuval. A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes. Classical and modern branching processes (Minneapolis, MN, 1994), 181–185, IMA Vol. Math. Appl., 84, Springer, New York, 1997.
  • Le Gall, Jean-François. Random trees and applications. Probab. Surv. 2 (2005), 245–311 (electronic).
  • Le Gall, Jean-François. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (2007), no. 3, 621–670.
  • Le Gall, Jean-François; Weill, Mathilde. Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 4, 455–489.
  • Lyons, Russell; Pemantle, Robin; Peres, Yuval. Conceptual proofs of $Llog L$ criteria for mean behavior of branching processes. Ann. Probab. 23 (1995), no. 3, 1125–1138.
  • Mahmoud, Hosam M.; Neininger, Ralph. Distribution of distances in random binary search trees. Ann. Appl. Probab. 13 (2003), no. 1, 253–276.
  • Marckert, Jean-François. The lineage process in Galton-Watson trees and globally centered discrete snakes. Ann. Appl. Probab. 18 (2008), no. 1, 209–244. (Review)
  • Marckert, Jean-François; Miermont, Grégory. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007), no. 5, 1642–1705. (Review)
  • Marckert, Jean-François; Mokkadem, Abdelkader. The depth first processes of Galton-Watson trees converge to the same Brownian excursion. Ann. Probab. 31 (2003), no. 3, 1655–1678.
  • Marckert, Jean-François; Mokkadem, Abdelkader. Limit of normalized quadrangulations: the Brownian map. Ann. Probab. 34 (2006), no. 6, 2144–2202.
  • G. Miermont, (2006). An invariance principle for random planar maps, Fourth Colloquium in Mathematics and Computer Sciences CMCS'06, DMTCS Proceedings AG, 39–58, Nancy.
  • Miermont, Grégory; Weill, Mathilde. Radius and profile of random planar maps with faces of arbitrary degrees. Electron. J. Probab. 13 (2008), no. 4, 79–106. (Review)
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • G. Schaeffer, (1998). Conjugaison d'arbres et cartes combinatoires aléatoires., PhD Thesis, Université Bordeaux I.
  • Weill, Mathilde. Asymptotics for rooted bipartite planar maps and scaling limits of two-type spatial trees. Electron. J. Probab. 12 (2007), no. 31, 887–925 (electronic). (Review)
  • Z. Zhang, L. Rong, F. Comellas, (2006). High-dimensional random Apollonian networks, Physica A, Volume 364, p. 610-618.
  • Z. Zhang, L. Chen, S. Zhou, L. Fang, J. Guan, Y. Zhang, (2007). Exact analytical solution of average path length for Apollonian networks, arXiv:0706.3491.
  • T. Zhou, G. Yan & B.H. Wang (2005). Maximal planar networks with large clustering oefficient and power-law degree distribution, Phys. Rev. E 71, 046141.