The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space , one can associate an -tree called the continuous cactus of . 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 , on peut associer un -arbre appelé cactus continu de . 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.
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/
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