Electronic Journal of Probability

The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection

Jérémie Bettinelli, Emmanuel Jacob, and Grégory Miermont

Full-text: Open access

Abstract

We prove that a uniform rooted plane map with n edges converges in distribution after asuitable normalization to the Brownian map for the Gromov–Hausdorff topology. A recent bijection due to Ambjørn and Budd allows to derive this result by a direct coupling with a uniform random quadrangulation with n faces.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 74, 16 pp.

Dates
Accepted: 19 August 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065716

Digital Object Identifier
doi:10.1214/EJP.v19-3213

Mathematical Reviews number (MathSciNet)
MR3256874

Zentralblatt MATH identifier
1320.60088

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60C05: Combinatorial probability

Keywords
random maps scaling limits Brownian map Gromov-Hausdorff topology random metric spaces bijections

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

Citation

Bettinelli, Jérémie; Jacob, Emmanuel; Miermont, Grégory. The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection. Electron. J. Probab. 19 (2014), paper no. 74, 16 pp. doi:10.1214/EJP.v19-3213. https://projecteuclid.org/euclid.ejp/1465065716


Export citation

References

  • Ambjørn, Jan; Budd, Timothy G. Trees and spatial topology change in causal dynamical triangulations. J. Phys. A 46 (2013), no. 31, 315201, 33 pp.
  • Addario-Berry, Louigi; Albenque, Marie. The scaling limit of random simple triangulations and random simple quadrangulations. Preprint, arXiv:1306.5227, 2013.
  • Abraham, Céline. Rescaled bipartite planar maps converge to the Brownian map. In preparation, 2013.
  • Ambjørn, Jan; Durhuus, Bergfinnur; Jonsson, Thordur. Quantum geometry. A statistical field theory approach. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1997. xiv+363 pp. ISBN: 0-521-46167-7
  • Angel, Omer; Schramm, Oded. Uniform infinite planar triangulations. Comm. Math. Phys. 241 (2003), no. 2-3, 191–213.
  • Bouttier, J.; Di Francesco, P.; Guitter, E. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004), no. 1, Research Paper 69, 27 pp.
  • Bouttier, Jérémie; Fusy, Éric; Guitter, Emmanuel. On the two-point function of general planar maps and hypermaps. Preprint, arXiv:1312.0502, 2013.
  • Bouttier, Jérémie; Guitter, Emmanuel. The three-point function of planar quadrangulations. J. Stat. Mech. Theory Exp., 7:P07020, 39, 2008.
  • 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.
  • Gromov, Misha. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original [ (85e:53051)]. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. ISBN: 0-8176-3898-9
  • Gao, Zhicheng; Wormald, Nicholas C. The distribution of the maximum vertex degree in random planar maps. J. Combin. Theory Ser. A 89 (2000), no. 2, 201–230.
  • 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. Geodesics in large planar maps and in the Brownian map. Acta Math. 205 (2010), no. 2, 287–360.
  • Le Gall, Jean-François. Uniqueness and universality of the Brownian map. Ann. Probab. 41 (2013), no. 4, 2880–2960.
  • Le Gall, Jean-François; Miermont, Grégory. Scaling limits of random trees and planar maps. Probability and statistical physics in two and more dimensions, 155–211, Clay Math. Proc., 15, Amer. Math. Soc., Providence, RI, 2012.
  • Le Gall, Jean-François; Weill, Mathilde. Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 4, 455–489.
  • Marckert, Jean-François. The rotation correspondence is asymptotically a dilatation. Random Structures Algorithms 24 (2004), no. 2, 118–132.
  • Miermont, Grégory. An invariance principle for random planar maps. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 39–57, Discrete Math. Theor. Comput. Sci. Proc., AG, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2006.
  • Miermont, Grégory. Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 725–781.
  • Miermont, Grégory. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 (2013), no. 2, 319–401.
  • Marckert, Jean-François; Mokkadem, Abdelkader. Limit of normalized quadrangulations: the Brownian map. Ann. Probab. 34 (2006), no. 6, 2144–2202.
  • Marckert, Jean-François; Miermont, Grégory. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007), no. 5, 1642–1705.
  • Poulalhon, Dominique; Schaeffer, Gilles. Optimal coding and sampling of triangulations. Algorithmica 46 (2006), no. 3-4, 505–527.
  • Schaeffer, Gilles. Conjugaison d'arbres et cartes combinatoires aléatoires. PhD thesis, Université Bordeaux I, 1998.