Invariance principles for labeled mobiles and bipartite planar maps
Résumé
A class of labeled trees, called mobiles, was introduced by Bouttier-di Francesco and Guitter in order to generalize the bijective studies of planar maps initiated by Cori-Vauquelin and Schaeffer. We prove an invariance principle for rescaled random mobiles associated with bipartite random planar maps under a Boltzmann distribution. We infer that the latter converge in a certain sense to the Brownian map introduced by Marckert and Mokkadem, which encompasses results of Chassaing and Schaeffer on quadrangulations (although in a slightly different context). These results are derived from a new invariance principle for a class of two-type Galton-Watson trees coupled with a spatial motion, which are shown to converge to the Brownian snake.