Invariance principles for random bipartite planar maps
Résumé
It is conjectured in the Physics literature 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 & Schaeffer, have shown that the radius of a random quadrangulation with $n$ faces 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 \&\ 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: 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, as well as convergence of rescaled bipartite maps to the Brownian map, as introduced by Marckert & Mokkadem. The proofs of these results rely on a new invariance principle for two-type spatial Galton-Watson trees.