On the sphericity of scaling limits of random planar quadrangulations

TL;DR

This paper provides a new proof that the scaling limits of random planar quadrangulations are topologically equivalent to the 2-sphere, using an enhanced convergence concept to preserve surface topology.

Contribution

It introduces a reinforced Gromov-Hausdorff convergence method to establish the topological nature of scaling limits of random quadrangulations.

Findings

Scaling limits are homeomorphic to the 2-sphere

Introduces 1-regular convergence to preserve topology

Provides a new proof of Le Gall & Paulin's theorem

Abstract

We give a new proof of a theorem by Le Gall & 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.