The Scaling Limit of Random Simple Triangulations and Random Simple Quadrangulations

Let Mn be a simple triangulation of the sphere S, 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)), 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 ofMn. 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. The appearance of a winding number suggests that a discrete complex-analytic approach to the study of random triangulations may lead to further discoveries. Figure 1. The circle packing associated to a uniformly random simple triangulation of S with 10 vertices. Blue shaded circles form a shortest path between two uniformly random vertices (circles). Created using Ken Stephenson’s CirclePack program; the file for the above packing is included with the arXiv posting of this manuscript. Date: June 20, 2013 . 2010 Mathematics Subject Classification. 60F17,05C12,82B41.