The brownian cactus I. Scaling limits of discrete cactuses
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 340-373.

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 -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.

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 -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.

DOI: 10.1214/11-AIHP460
Classification: 60F17, 60D05
Keywords: random planar maps, scaling limit, brownian map, brownian cactus, Hausdorff dimension
@article{AIHPB_2013__49_2_340_0,
     author = {Curien, Nicolas and Le Gall, Jean-Fran\c{c}ois and Miermont, Gr\'egory},
     title = {The brownian cactus {I.} {Scaling} limits of discrete cactuses},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {340--373},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     doi = {10.1214/11-AIHP460},
     mrnumber = {3088373},
     zbl = {1275.60035},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP460/}
}
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
SP  - 340
EP  - 373
VL  - 49
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/11-AIHP460/
DO  - 10.1214/11-AIHP460
LA  - en
ID  - AIHPB_2013__49_2_340_0
ER  - 
%0 Journal Article
%A Curien, Nicolas
%A Le Gall, Jean-François
%A Miermont, Grégory
%T The brownian cactus I. Scaling limits of discrete cactuses
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 340-373
%V 49
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/11-AIHP460/
%R 10.1214/11-AIHP460
%G en
%F AIHPB_2013__49_2_340_0
Curien, Nicolas; Le Gall, Jean-François; Miermont, Grégory. The brownian cactus I. Scaling limits of discrete cactuses. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 340-373. doi : 10.1214/11-AIHP460. http://www.numdam.org/articles/10.1214/11-AIHP460/

[1] D. Aldous. The continuum random tree I. Ann. Probab. 19 (1991) 1-28. | MR | Zbl

[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. | MR | Zbl

[3] J. Bouttier, P. Di Francesco and E. Guitter. Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) R69. | MR | Zbl

[4] J. Bouttier and E. Guitter. Confluence of geodesic paths and separating loops in large planar quadrangulations. J. Stat. Mech. Theory Exp. (2009) P03001. | MR

[5] D. Burago, Y. Burago and S. Ivanov. A Course in Metric Geometry. Graduate Studies in Mathematics 33. Amer. Math. Soc., Boston, 2001. | MR | Zbl

[6] S. N. Evans. Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin, 2008. | MR | Zbl

[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. | MR | Zbl

[8] C. Favre and M. Jonsson. The Valuative Tree. Lecture Notes in Math. 1853. Springer, Berlin, 2004. | MR | Zbl

[9] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston, 2001. | MR | Zbl

[10] N. C. Jain and S. J. Taylor. Local asymptotic laws for Brownian motion. Ann. Probab. 1 (1973) 527-549. | MR | Zbl

[11] S. K. Lando and A. K. Zvonkin. Graphs on Surfaces and Their Applications. Encyclopedia of Mathematical Sciences 141. Springer, Berlin, 2004. | MR | Zbl

[12] J. F. Le Gall. Random trees and applications. Probab. Sur. 2 (2005) 245-311. | EuDML | MR | Zbl

[13] J. F. Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. 169 (2007) 621-670. | MR | Zbl

[14] J. F. Le Gall. Geodesics in large planar maps and in the Brownian map. Acta Math. 205 (2010) 287-360. | MR | Zbl

[15] J. F. Le Gall. Uniqueness and universality of the Brownian map. Preprint. Available at arXiv:1105.4842. | MR | Zbl

[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. | MR | Zbl

[17] J. F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007) 1642-1705. | MR | Zbl

[18] J. F. Marckert and A. Mokkadem. Limit of normalized quadrangulations. The Brownian map. Ann. Probab. 34 (2006) 2144-2202. | MR | Zbl

[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. | MR | Zbl

[20] G. Miermont. Invariance principles for spatial multitype Galton-Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 1128-1161. | EuDML | Numdam | MR | Zbl

[21] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Preprint. Available at arXiv:1104.1606. | MR | Zbl

[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. | EuDML | MR | Zbl

[23] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1991. | MR | Zbl

[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. | MR | Zbl

[25] W. T. Tutte. A census of planar maps. Canad. J. Math. 15 (1963) 249-271. | MR | Zbl

Cited by Sources: