  Random planar maps are considered in the physics literature as the discrete
counterpart of random surfaces. It is conjectured that properly rescaled random
planar maps, when conditioned to have a large number of faces, should converge
to a limiting surface whose law does not depend, up to scaling factors, on
details of the class of maps that are sampled. Previous works on the topic,
starting with Chassaing and Schaeffer, have shown that the radius of a random
quadrangulation with $n$ faces, that is, the maximal graph distance on such a
quadrangulation to a fixed reference point, converges in distribution once
rescaled by $n^{1/4}$ to the diameter of the Brownian snake, up to a scaling
constant. Using a bijection due to Bouttier, Di Francesco and Guitter between
bipartite planar maps and a family of labeled trees, we show the corresponding
invariance principle for a class of random maps that follow a Boltzmann
distribution putting weight $q_k$ on faces of degree $2k$: the radius of such
maps, conditioned to have $n$ faces (or $n$ vertices) and under a criticality
assumption, converges in distribution once rescaled by $n^{1/4}$ to a scaled
version of the diameter of the Brownian snake. Convergence results for the
so-called profile of maps are also provided. The convergence of rescaled
bipartite maps to the Brownian map, in the sense introduced by Marckert and
Mokkadem, is also shown. The proofs of these results rely on a new invariance
principle for two-type spatial Galton--Watson trees.
