The Brownian Cactus I. Scaling limits of discrete cactuses
  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. 
Nicolas Curien (DMA), Jean-Fran\c{c}ois Le Gall (LM-Orsay), Gr\'egory   Miermont (LM-Orsay)
math.PR