Sigurdur Stefánsson: Condensation in planar maps

Random planar maps are defined by assigning non–negative Boltzmann weights to each face of a planar map and the weight of a face depends only on its degree. I will explain the Bouttier-Di Francesco-Guitter bijection between the planar maps and a class of labeled trees called mobiles. By throwing away labels one can essentially reduce the model of random maps to the model of simply generated trees. For certain choices of Boltzmann weights a unique large face, having degree proportional to the total number of edges in the maps, appears with high probability when the maps are large. This corresponds to a recently studied phenomenon of condensation in simply generated trees where a vertex having degree proportional to the size of the trees appears. In this case the planar maps, with a properly rescaled graph metric, are shown to converge in distribution towards Aldous’ Brownian tree in the Gromov-Hausdorff topology. This was a joint work with Svante Janson. (arXiv:1212.5072)