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

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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 $\mathbb{R}$-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.


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

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

First available in Project Euclid: 16 April 2013

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Zentralblatt MATH identifier

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

Random planar maps Scaling limit Brownian map Brownian cactus Hausdorff dimension


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.

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