The scaling limit of random simple triangulations and random simple quadrangulations

Understand

Let $M_n$ be a simple triangulation of the sphere $S^2$, drawn uniformly at random from all such triangulations with n vertices.

• Endow $M_n$ with the uniform probability measure on its vertices.
• After rescaling graph distance on $V(M_n)$ 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 $M_n$.