Limit of normalized quadrangulations: The Brownian map

Abstract

Consider $q_n$ a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with $n$ faces. In this paper we show that, when $n$ goes to $+\infty$, $q_n$ suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of the Brownian map is defined. The weak convergences hold in these metric spaces.

Published at http://dx.doi.org/10.1214/009117906000000557 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Uniform Infinite Planar Triangulations
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity
• Uniform infinite planar triangulation and related time-reversed critical branching process
• Statistics of planar graphs viewed from a vertex: A study via labeled trees

Cited by in corpus (27)

• Uniqueness and universality of the Brownian map
• The topological structure of scaling limits of large planar maps
• Quantum Loewner Evolution
• The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection
• Invariance principles for random bipartite planar maps
• Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop
• Multi-critical behaviour of 4-dimensional tensor models up to order 6
• Scaling limits of random bipartite planar maps with a prescribed degree sequence
• Random non-crossing plane configurations: A conditioned Galton-Watson tree approach
• Quadrangulations with no pendant vertices
• The topology of scaling limits of positive genus random quadrangulations
• Statistics of geodesics in large quadrangulations
• Geodesic distances in Liouville quantum gravity
• Local convergence for permutations and local limits for uniform $\rho$-avoiding permutations with $|\rho |=3$
• Geodesics in Brownian surfaces (Brownian maps)
• Nesting statistics in the O(n) loop model on random planar maps
• On the diameter of random planar graphs
• Confluence of geodesic paths and separating loops in large planar quadrangulations
• Local convergence of large random triangulations coupled with an Ising model
• Trees with exponential height dependent weight
• Tree-decorated planar maps
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• Distance statistics in large toroidal maps
• The generating function of planar Eulerian orientations
• Scale-invariant random geometry from mating of trees: a numerical study
• Trees with power-like height dependent weight
• Distances on the ${\mathrm{CLE}}_4$, critical Liouville quantum gravity and $3/2$-stable maps