Electronic Journal of Probability

Asymptotics for Rooted Bipartite Planar Maps and Scaling Limits of Two-Type Spatial Trees

Mathilde Weill

Full-text: Open access

Abstract

We prove some asymptotic results for the radius and the profile of large random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted bipartite planar maps and certain two-type trees with positive labels, we derive our results from a conditional limit theorem for two-type spatial trees. Finally we apply our estimates to separating vertices of bipartite planar maps: with probability close to one when n tends to infinity, a random $2k$-angulation with n faces has a separating vertex whose removal disconnects the map into two components each with size greater that $n^{1/2 - \varepsilon}$.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 31, 862-925.

Dates
Accepted: 13 June 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818502

Digital Object Identifier
doi:10.1214/EJP.v12-425

Mathematical Reviews number (MathSciNet)
MR2318414

Zentralblatt MATH identifier
1127.05096

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Planar maps two-type Galton-Watson trees Conditioned Brownian snake

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Weill, Mathilde. Asymptotics for Rooted Bipartite Planar Maps and Scaling Limits of Two-Type Spatial Trees. Electron. J. Probab. 12 (2007), paper no. 31, 862--925. doi:10.1214/EJP.v12-425. https://projecteuclid.org/euclid.ejp/1464818502


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References

  • Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289. (94c:60015)
  • Banderier, Cyril; Flajolet, Philippe; Schaeffer, Gilles; Soria, Michèle. Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Analysis of algorithms (Krynica Morska, 2000). Random Structures Algorithms 19 (2001), no. 3-4, 194–246. (2002k:05012)
  • 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). (2005i:05087)
  • Chassaing, Philippe; Schaeffer, Gilles. Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 (2004), no. 2, 161–212. (2004k:60016)
  • Cori, Robert; Vauquelin, Bernard. Planar maps are well labeled trees. Canad. J. Math. 33 (1981), no. 5, 1023–1042. (83c:05070)
  • Drmota, Michael; Gittenberger, Bernhard. On the profile of random trees. Random Structures Algorithms 10 (1997), no. 4, 421–451. (99c:05176)
  • Duquesne, Thomas. A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31 (2003), no. 2, 996–1027. (2004a:60076)
  • Duquesne, Thomas; Le Gall, Jean-François. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005), no. 4, 553–603. (2006d:60123)
  • Janson, Svante; Marckert, Jean-François. Convergence of discrete snakes. J. Theoret. Probab. 18 (2005), no. 3, 615–647. (2006g:60126)
  • Le Gall, Jean-François. Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1999. x+163 pp. ISBN: 3-7643-6126-3 (2001g:60211)
  • Le Gall, Jean-François. Random trees and applications. Probab. Surv. 2 (2005), 245–311 (electronic). (Review)
  • Le Gall, Jean-François. A conditional limit theorem for tree-indexed random walk. Stochastic Process. Appl. 116 (2006), no. 4, 539–567. (2007g:60098)
  • c Le Gall, J.F. (2006) The topological structure of scaling limits of large planar maps. arXiv: math.PR/0607567.
  • Le Gall, Jean-Francois; Le Jan, Yves. Branching processes in Lévy processes: the exploration process. Ann. Probab. 26 (1998), no. 1, 213–252. (99d:60096)
  • Le Gall, Jean-François; Weill, Mathilde. Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 4, 455–489.
  • Marckert, J.F., Miermont, G. (2006) Invariance principles for random bipartite planar maps. Ann. Probab., to appear.
  • Marckert, Jean-François; Mokkadem, Abdelkader. States spaces of the snake and its tour–-convergence of the discrete snake. J. Theoret. Probab. 16 (2003), no. 4, 1015–1046 (2004). (2005c:60095)
  • Marckert, Jean-François; Mokkadem, Abdelkader. Limit of normalized quadrangulations: the Brownian map. Ann. Probab. 34 (2006), no. 6, 2144–2202.
  • Schaeffer G. (1998) Conjugaison d'arbres et cartes aleatoires, These, Universite de Bordeaux I.