The Brownian cactus I. Scaling limits of discrete cactuses

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 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. © 2013 Association des Publications de l'Institut Henri Poincaré.