The Brownian Cactus I. Scaling limits of discrete cactuses

Nicolas Curien (1) , Jean-François Le Gall (2) , Grégory Miermont (3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 ICAN - Unité de Recherche sur les Maladies Cardiovasculaires, du Métabolisme et de la Nutrition = Research Unit on Cardiovascular and Metabolic Diseases
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Domaines: Probabilités [math.PR]

Nicolas Curien, Jean-François Le Gall, Grégory Miermont. The Brownian Cactus I. Scaling limits of discrete cactuses. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2011, 49 (2), pp.340-373. ⟨10.1214/11-AIHP460⟩. ⟨hal-00567333⟩