The brownian cactus I. Scaling limits of discrete cactuses

Nicolas Curien; Jean-François Le Gall; Grégory Miermont

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 340-373
  • ISSN: 0246-0203

Abstract

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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 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.

How to cite

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Curien, Nicolas, Le Gall, Jean-François, and Miermont, Grégory. "The brownian cactus I. Scaling limits of discrete cactuses." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 340-373. <http://eudml.org/doc/272049>.

@article{Curien2013,
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.},
author = {Curien, Nicolas, Le Gall, Jean-François, Miermont, Grégory},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random planar maps; scaling limit; brownian map; brownian cactus; Hausdorff dimension; Brownian map; Brownian cactus},
language = {eng},
number = {2},
pages = {340-373},
publisher = {Gauthier-Villars},
title = {The brownian cactus I. Scaling limits of discrete cactuses},
url = {http://eudml.org/doc/272049},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Curien, Nicolas
AU - Le Gall, Jean-François
AU - Miermont, Grégory
TI - The brownian cactus I. Scaling limits of discrete cactuses
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 340
EP - 373
AB - 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.
LA - eng
KW - random planar maps; scaling limit; brownian map; brownian cactus; Hausdorff dimension; Brownian map; Brownian cactus
UR - http://eudml.org/doc/272049
ER -

References

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  1. [1] D. Aldous. The continuum random tree I. Ann. Probab.19 (1991) 1–28. Zbl0722.60013MR1085326
  2. [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. Zbl1096.82500MR1465433
  3. [3] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) R69. Zbl1060.05045MR2097335
  4. [4] J. Bouttier and E. Guitter. Confluence of geodesic paths and separating loops in large planar quadrangulations. J. Stat. Mech. Theory Exp. (2009) P03001. Zbl1179.82069MR2495866
  5. [5] D. Burago, Y. Burago and S. Ivanov. A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Boston, 2001. Zbl0981.51016MR1835418
  6. [6] S. N. Evans. Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin, 2008. Zbl1139.60006MR2351587
  7. [7] S. N. Evans, J. W. Pitman and A. Winter. Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields134 (2006) 81–126. Zbl1086.60050MR2221786
  8. [8] C. Favre and M. Jonsson. The Valuative Tree. Lecture Notes in Math. 1853. Springer, Berlin, 2004. Zbl1064.14024MR2097722
  9. [9] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston, 2001. Zbl0953.53002MR1699320
  10. [10] N. C. Jain and S. J. Taylor. Local asymptotic laws for Brownian motion. Ann. Probab.1 (1973) 527–549. Zbl0261.60053MR365732
  11. [11] S. K. Lando and A. K. Zvonkin. Graphs on Surfaces and Their Applications. Encyclopedia of Mathematical Sciences 141. Springer, Berlin, 2004. Zbl1040.05001MR2036721
  12. [12] J. F. Le Gall. Random trees and applications. Probab. Sur.2 (2005) 245–311. Zbl1189.60161MR2203728
  13. [13] J. F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math.169 (2007) 621–670. Zbl1132.60013MR2336042
  14. [14] J. F. Le Gall. Geodesics in large planar maps and in the Brownian map. Acta Math.205 (2010) 287–360. Zbl1214.53036MR2746349
  15. [15] J. F. Le Gall. Uniqueness and universality of the Brownian map. Preprint. Available at arXiv:1105.4842. Zbl1282.60014MR3112934
  16. [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. Zbl1166.60006MR2438999
  17. [17] J. F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab.35 (2007) 1642–1705. Zbl1208.05135MR2349571
  18. [18] J. F. Marckert and A. Mokkadem. Limit of normalized quadrangulations. The Brownian map. Ann. Probab. 34 (2006) 2144–2202. Zbl1117.60038MR2294979
  19. [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. Zbl1195.60049MR2509622
  20. [20] G. Miermont. Invariance principles for spatial multitype Galton–Watson trees. Ann. Inst. Henri Poincaré Probab. Stat.44 (2008) 1128–1161. Zbl1178.60058MR2469338
  21. [21] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Preprint. Available at arXiv:1104.1606. Zbl1278.60124MR3070569
  22. [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. Zbl1190.60024MR2375600
  23. [23] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1991. Zbl0917.60006MR1083357
  24. [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. Zbl1131.60088MR2334202
  25. [25] W. T. Tutte. A census of planar maps. Canad. J. Math.15 (1963) 249–271. Zbl0115.17305MR146823

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