The Brownian map

What is this?

This picture represents a random triangulation of the sphere, chosen uniformly among all the triangulations with 30000 vertices. The sampling of the random triangulation was done using edge-flips, and the embedding was computed using the GraphPlot3D function of Mathematica. An interesting feature of random planar maps is that the graph distances between vertices in a uniform triangulation of size n are typically of order n1/4. Moreover, after renormalizing distances by n1/4 a uniform random triangulation converges in distribution to a random metric space called the Brownian map. The picture above can be seen as an approximation of the Brownian map.

Picture created by Thomas Budzinski.