A planar map is a discretization of the two-sphere into a
finite number of polygons. Planar maps appear in physics for example
as models of random surfaces in two dimensional quantum gravity and as
Feynman diagrams in matrix models. I will discuss a model of random
planar maps which is defined by assigning non-negative Boltzmann
weights to each polygon where the weight depends only on its degree. I
will explain the phase diagram of the model and how it can be
understood by considering the phase diagram of a model of random trees
called simply generated trees. The main tool is the Bouttier-Di
Francesco-Guitter bijection between planar maps and a class of labeled
trees called mobiles. By throwing away labels one can, via another
bijection, relate the mobiles to the model of simply generated trees.
A novel result is that 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.
