Scaling limits for the uniform infinite quadrangulation

TL;DR

This paper investigates the asymptotic geometric properties of the uniform infinite quadrangulation, revealing its scaling limits and volume growth behavior through advanced probabilistic tools and bijections.

Contribution

It introduces a detailed analysis of the scaling limits of the uniform infinite quadrangulation using Brownian snakes and extends Schaeffer's bijection for this purpose.

Findings

Scaling limit of the contour functions described by eternal conditioned Brownian snakes

Asymptotics for volume of large balls in the quadrangulation

Establishment of the scaling limit for the quadrangulation via bijection

Abstract

The uniform infinite planar quadrangulation is an infinite random graph embedded in the plane, which is the local limit of uniformly distributed finite quadrangulations with a fixed number of faces. We study asymptotic properties of this random graph. In particular, we investigate scaling limits of the profile of distances from the distinguished point called the root, and we get asymptotics for the volume of large balls. As a key technical tool, we first describe the scaling limit of the contour functions of the uniform infinite well-labeled tree, in terms of a pair of eternal conditioned Brownian snakes. Scaling limits for the uniform infinite quadrangulation can then be derived thanks to an extended version of Schaeffer's bijection between well-labeled trees and rooted quadrangulations.