The Brownian map is the scaling limit of uniform random plane quadrangulations
The central object of study in this work is quadrangulation. A map is called quadrangulation if all faces have degree four, where the degree of a face is the number of edges that are incident to it (an edge which is incident to only one face has to be counted twice in the computation of the degree of this face). The author proves that uniform random quadrangulations of the sphere with \(n\) faces, endowed with the usual graph distance and renormalized by \(n-1/4\), converge as \(n\to\infty\) in distribution for the Gromov-Hausdorff topology to a limiting metric space. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.
scientific article; zbMATH DE number 6199563
scientific article