Random geometry on the sphere

TL;DR

This paper introduces the Brownian map as a universal scaling limit for random planar graphs on the sphere, revealing its geometric properties and connections to quantum gravity.

Contribution

It proves the convergence of various classes of random planar graphs to the Brownian map, establishing its universality and detailed geometric structure.

Findings

The Brownian map is the universal limit for large random planar graphs.

It has Hausdorff dimension 4 and is homeomorphic to the sphere.

The paper describes the structure of geodesics in the Brownian map.

Abstract

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. The Brownian map is homeomorphic to the sphere, but its Hausdorff dimension is equal to $4$. We obtain detailed information about the structure of geodesics in the Brownian map. We also present the infinite-volume variant of the Brownian map called the Brownian plane, which arises as the scaling limit of the uniform infinite planar quadrangulation. Finally, we discuss certain open problems. This study is motivated in part by the use of random geometry in the physical theory of two-dimensional quantum gravity.