We give a new proof of a theorem by Le Gall and Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces.
"On the sphericity of scaling limits of random planar quadrangulations." Electron. Commun. Probab. 13 248 - 257, 2008. https://doi.org/10.1214/ECP.v13-1368