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
     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},
     zbl = {1275.60035},
     mrnumber = {3088373},
     language = {en},
     url = {}
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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.

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