Random geometry on the sphere

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.

To appear in the Proceedings of ICM 2014, Seoul

References in corpus (9)

• Uniqueness and universality of the Brownian map
• The topological structure of scaling limits of large planar maps
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• Local structure of random quadrangulations
• The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection
• The three-point function of planar quadrangulations
• Quadrangulations with no pendant vertices
• The scaling limit of random simple triangulations and random simple quadrangulations
• A glimpse of the conformal structure of random planar maps

Cited by in corpus (14)

• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• The Tensor Theory Space
• Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa >8$
• Random planar maps & growth-fragmentations
• Local convergence of large random triangulations coupled with an Ising model
• Geodesics in the Brownian map: Strong confluence and geometric structure
• Plane bipolar orientations and quadrant walks
• A combinatorial criterion for macroscopic circles in planar triangulations
• Discrete analytic functions on non-uniform lattices without global geometric control
• A phase transition creates the geometry of the continuum from discrete space
• On the structure of random graphs with constant $r$-balls
• Introduction to the Liouville quantum gravity metric
• The generating function of planar Eulerian orientations
• Tridiagonalized GUE matrices are a matrix model for labeled mobiles