Illinois Journal of Mathematics

Scaling limits for the uniform infinite quadrangulation

Jean-François Le Gall and Laurent Ménard

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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.

Article information

Illinois J. Math., Volume 54, Number 3 (2010), 1163-1203.

First available in Project Euclid: 3 May 2012

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 05C80: Random graphs [See also 60B20]


Le Gall, Jean-François; Ménard, Laurent. Scaling limits for the uniform infinite quadrangulation. Illinois J. Math. 54 (2010), no. 3, 1163--1203. doi:10.1215/ijm/1336049989.

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