Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limits of random outerplanar maps with independent link-weights

Benedikt Stufler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The scaling limit of large simple outerplanar maps was established by Caraceni using a bijection due to Bonichon, Gavoille and Hanusse. The present paper introduces a new bijection between outerplanar maps and trees decorated with ordered sequences of edge-rooted dissections of polygons. We apply this decomposition in order to provide a new, short proof of the scaling limit that also applies to the general setting of first-passage percolation. We obtain sharp tail-bounds for the diameter and recover the asymptotic enumeration formula for outerplanar maps. Our methods also enable us to treat subclasses such as bipartite outerplanar maps.

Résumé

La limite d’échelle des cartes planaires extérieures simples a été établie par Caraceni via une bijection de Bonichon, Gavoille et Hanusse. Dans ce papier, nous construisons une nouvelle bijection entre les cartes planaires extérieures, et les arbres décorés par des suites ordonnées de dissections de polygones enracinées. Nous utilisons cette décomposition pour obtenir une nouvelle preuve, plus courte, du résultat de Caraceni, qui s’étend en outre au cadre général de la percolation de premier passage. Nous obtenons des bornes précises sur la queue de distribution du diamètre des cartes planaires extérieures, et retrouvons les formules d’énumération asymptotiques de ces cartes. Nos méthodes nous permettent également de traiter le cas de sous-classes comme les cartes planaires extérieures biparties.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 900-915.

Dates
Received: 28 May 2015
Revised: 28 December 2015
Accepted: 13 January 2016
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897750

Digital Object Identifier
doi:10.1214/16-AIHP741

Mathematical Reviews number (MathSciNet)
MR3634279

Zentralblatt MATH identifier
1367.60031

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60C05: Combinatorial probability
Secondary: 05C80: Random graphs [See also 60B20]

Keywords
Continuum random tree Scaling limits Random graphs Outerplanar maps

Citation

Stufler, Benedikt. Scaling limits of random outerplanar maps with independent link-weights. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 900--915. doi:10.1214/16-AIHP741. https://projecteuclid.org/euclid.aihp/1491897750


Export citation

References

  • [1] C. Abraham. Rescaled bipartite planar maps converge to the Brownian map. Ann. Inst. Henri Poincaré Probab. Stat. 52 (2) (2016) 575–595.
  • [2] L. Addario-Berry, L. Devroye and S. Janson. Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees. Ann. Probab. 41 (2) (2013) 1072–1087.
  • [3] D. Aldous. The continuum random tree. I. Ann. Probab. 19 (1) (1991) 1–28.
  • [4] D. Aldous. The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990) 23–70. London Math. Soc. Lecture Note Ser. 167. Cambridge Univ. Press, Cambridge, 1991.
  • [5] D. Aldous. The continuum random tree. III. Ann. Probab. 21 (1) (1993) 248–289.
  • [6] J. Ambjorn and T. Budd. Multi-point functions of weighted cubic maps, 2014. Available at http://arxiv.org/abs/1408.3040.
  • [7] N. Bernasconi, K. Panagiotou and A. Steger. On properties of random dissections and triangulations. Combinatorica 30 (6) (2010) 627–654.
  • [8] J. Bettinelli. Scaling limit of random planar quadrangulations with a boundary. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2) (2015) 432–477.
  • [9] N. Bonichon, C. Gavoille and N. Hanusse. Canonical decomposition of outerplanar maps and application to enumeration, coding and generation. J. Graph Algorithms Appl. 9 (2) (2005) 185–204 (electronic).
  • [10] N. Broutin and P. Flajolet. The distribution of height and diameter in random non-plane binary trees. Random Structures Algorithms 41 (2) (2012) 215–252.
  • [11] A. Caraceni. The scaling limit of random outerplanar maps. Ann. Inst. Henri Poincaré Probab. Stat. To appear.
  • [12] N. Curien, B. Haas and I. Kortchemski. The CRT is the scaling limit of random dissections. Random Structures Algorithms 47 (2) (2015) 304–327.
  • [13] N. Curien and J.-F. Le Gall. Scaling limits for the peeling process on random maps. Ann. Inst. Henri Poincaré Probab. Stat. To appear.
  • [14] N. Curien and J.-F. Le Gall. First-passage percolation and local modifications of distances in random triangulations, 2015. Available at http://arxiv.org/abs/1511.04264.
  • [15] R. Diestel. Graph Theory, 4th edition. Graduate Texts in Mathematics 173. Springer, Heidelberg, 2010.
  • [16] P. Duchon, P. Flajolet, G. Louchard and G. Schaeffer. Random sampling from Boltzmann principles. In Automata, Languages and Programming 501–513. Lecture Notes in Comput. Sci. 2380. Springer, Berlin, 2002.
  • [17] P. Duchon, P. Flajolet, G. Louchard and G. Schaeffer. Boltzmann samplers for the random generation of combinatorial structures. Combin. Probab. Comput. 13 (4–5) (2004) 577–625.
  • [18] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge Univ. Press, Cambridge, 2009.
  • [19] B. Haas and G. Miermont. Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees. Ann. Probab. 40 (6) (2012) 2589–2666.
  • [20] S. Janson. Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9 (2012) 103–252.
  • [21] S. Janson and S. Ö. Stefánsson. Scaling limits of random planar maps with a unique large face. Ann. Probab. 43 (3) (2015) 1045–1081.
  • [22] A. Joyal. Une théorie combinatoire des séries formelles. Adv. in Math. 42 (1) (1981) 1–82.
  • [23] K. Panagiotou and B. Stufler. Scaling limits of random Pólya trees, 2015. Available at http://arxiv.org/abs/1502.07180.
  • [24] K. Panagiotou, B. Stufler and K. Weller. Scaling limits of random graphs from subcritical classes. Ann. Probab. To appear.
  • [25] B. Stufler. The continuum random tree is the scaling limit of unlabelled unrooted trees, 2014. Available at http://arxiv.org/abs/1412.6333.
  • [26] G. Szekeres. Distribution of labelled trees by diameter. In Combinatorial Mathematics, X (Adelaide, 1982) 392–397. Lecture Notes in Math. 1036. Springer, Berlin, 1983.
  • [27] M. Wang and Height and diameter of Brownian tree. Electron. Commun. Probab. 20 (2015) no. 88, 15.