The scaling limit of random simple triangulations and random simple quadrangulations

TL;DR

This paper proves that uniformly random simple triangulations and quadrangulations of the sphere, when properly rescaled, converge to the Brownian map, a universal limiting metric space, using a novel vertex labelling technique.

Contribution

It introduces a new vertex labelling method to analyze the metric structure of random simple triangulations and quadrangulations, establishing their convergence to the Brownian map.

Findings

Random simple triangulations converge to the Brownian map after rescaling.

A vertex labelling function effectively captures distances to a distinguished point.

Similar convergence results are established for simple quadrangulations.

Abstract

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 $M_n$. 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.