The scaling limit of random simple triangulations and random simple quadrangulations

Let Mₙ be a simple triangulation of the sphere S², drawn uniformly at random from all such triangulations with n vertices. Endow Mₙ with the uniform probability measure on its vertices. After rescaling graph distance by (3/(4n))1/4, the resulting random measured metric space converges in distribution, in the Gromov–Hausdorff–Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of Mₙ. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. We establish similar results for simple quadrangulations.