Random geometry on the sphere

Understand

We introduce and study a universal model of random geometry in two dimensions.

• To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size $n$, for instance the class of all triangulations of the sphere with $n$ faces.
• We equip the vertex set of the graph with the usual graph distance rescaled by the factor $n^{-1/4}$.
• We then prove that the resulting random metric space converges in distribution as $n\to\infty$, in the Gromov-Hausdorff sense, toward a limiting random compact metric space called the Brownian map, which is universal in the sense that it does not depend on the class of graphs chosen initially.