Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The Brownian cactus I. Scaling limits of discrete cactuses

Nicolas Curien, Jean-François Le Gall, and Grégory Miermont

Full-text: Open access

Abstract

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\mathbb{R}$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.

Résumé

Le cactus d’un graphe pointé est un certain arbre discret associé à ce graphe. De façon similaire, à tout espace métrique géodésique pointé $E$, on peut associer un $\mathbb{R}$-arbre appelé cactus continu de $E$. Sous des hypothèses générales, nous montrons que le cactus de cartes planaires aléatoires – dont la loi est déterminée par des poids de Boltzmann, et qui sont conditionnées à avoir un grand nombre fixé de sommets – converge en loi vers un espace limite appelé cactus brownien, au sens de la topologie de Gromov–Hausdorff. De plus, le cactus brownien peut être interprété comme le cactus continu de la carte brownienne.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 2 (2013), 340-373.

Dates
First available in Project Euclid: 16 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1366117650

Digital Object Identifier
doi:10.1214/11-AIHP460

Mathematical Reviews number (MathSciNet)
MR3088373

Zentralblatt MATH identifier
1275.60035

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

Keywords
Random planar maps Scaling limit Brownian map Brownian cactus Hausdorff dimension

Citation

Curien, Nicolas; Le Gall, Jean-François; Miermont, Grégory. The Brownian cactus I. Scaling limits of discrete cactuses. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 2, 340--373. doi:10.1214/11-AIHP460. https://projecteuclid.org/euclid.aihp/1366117650


Export citation

References

  • [1] D. Aldous. The continuum random tree I. Ann. Probab. 19 (1991) 1–28.
  • [2] J. Ambjørn, B. Durhuus and T. Jonsson. Quantum Geometry. A Statistical Field Theory Approach. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 1997.
  • [3] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) R69.
  • [4] J. Bouttier and E. Guitter. Confluence of geodesic paths and separating loops in large planar quadrangulations. J. Stat. Mech. Theory Exp. (2009) P03001.
  • [5] D. Burago, Y. Burago and S. Ivanov. A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Boston, 2001.
  • [6] S. N. Evans. Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin, 2008.
  • [7] S. N. Evans, J. W. Pitman and A. Winter. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 (2006) 81–126.
  • [8] C. Favre and M. Jonsson. The Valuative Tree. Lecture Notes in Math. 1853. Springer, Berlin, 2004.
  • [9] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston, 2001.
  • [10] N. C. Jain and S. J. Taylor. Local asymptotic laws for Brownian motion. Ann. Probab. 1 (1973) 527–549.
  • [11] S. K. Lando and A. K. Zvonkin. Graphs on Surfaces and Their Applications. Encyclopedia of Mathematical Sciences 141. Springer, Berlin, 2004.
  • [12] J. F. Le Gall. Random trees and applications. Probab. Sur. 2 (2005) 245–311.
  • [13] J. F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (2007) 621–670.
  • [14] J. F. Le Gall. Geodesics in large planar maps and in the Brownian map. Acta Math. 205 (2010) 287–360.
  • [15] J. F. Le Gall. Uniqueness and universality of the Brownian map. Preprint. Available at arXiv:1105.4842.
  • [16] J. F. Le Gall and F. Paulin. Scaling limits of bipartite planar maps are homeomorphic to the $2$-sphere. Geomet. Funct. Anal. 18 (2008) 893–918.
  • [17] J. F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007) 1642–1705.
  • [18] J. F. Marckert and A. Mokkadem. Limit of normalized quadrangulations. The Brownian map. Ann. Probab. 34 (2006) 2144–2202.
  • [19] G. Miermont. An invariance principle for random planar maps. In Fourth Colloquium on Mathematics and Computer Science, Algorithms, Trees, Combinatorics and Probabilities 39–57 (electronic). Discrete Math. Theor. Comput. Sci. Proc., AG. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2006.
  • [20] G. Miermont. Invariance principles for spatial multitype Galton–Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 1128–1161.
  • [21] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Preprint. Available at arXiv:1104.1606.
  • [22] G. Miermont and M. Weill. Radius and profile of random planar maps with faces of arbitrary degrees. Electron. J. Probab. 13 (2008) 79–106.
  • [23] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1991.
  • [24] O. Schramm. Conformally invariant scaling limits: An overview and a collection of problems. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. I 513–543. European Math. Soc., Zürich, 2007.
  • [25] W. T. Tutte. A census of planar maps. Canad. J. Math. 15 (1963) 249–271.