The brownian cactus I. Scaling limits of discrete cactuses

Nicolas Curien; Jean-François Le Gall; Grégory Miermont

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 340-373
  • ISSN: 0246-0203

Abstract

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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 , 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.

How to cite

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Curien, Nicolas, Le Gall, Jean-François, and Miermont, Grégory. "The brownian cactus I. Scaling limits of discrete cactuses." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 340-373. <http://eudml.org/doc/272049>.

@article{Curien2013,
abstract = {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.},
author = {Curien, Nicolas, Le Gall, Jean-François, Miermont, Grégory},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random planar maps; scaling limit; brownian map; brownian cactus; Hausdorff dimension; Brownian map; Brownian cactus},
language = {eng},
number = {2},
pages = {340-373},
publisher = {Gauthier-Villars},
title = {The brownian cactus I. Scaling limits of discrete cactuses},
url = {http://eudml.org/doc/272049},
volume = {49},
year = {2013},
}

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
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 340
EP - 373
AB - 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.
LA - eng
KW - random planar maps; scaling limit; brownian map; brownian cactus; Hausdorff dimension; Brownian map; Brownian cactus
UR - http://eudml.org/doc/272049
ER -

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