%0 Journal Article
%T The Brownian Cactus I. Scaling limits of discrete cactuses
%A Nicolas Curien
%A Jean-Fran£¿ois Le Gall
%A Gr¨¦gory Miermont
%J Mathematics
%D 2011
%I arXiv
%X 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.
%U http://arxiv.org/abs/1102.4177v1