MPS-RR 2003-36
December 2003

Statistical Hausdorff Dimension of Labelled Trees and Quadrangulations

by: Philippe Chassaing , Bergfinnur Durhuus

Abstract

Exploiting a bijective correspondence between planar quadrangulations and so-called well labelled trees, we define a random ensemble of infinite surfaces, as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton Watson trees. As a consequence, we find that the volume of the ball of radius $r$ around a marked point in the limit random surface is $\Theta(r^{4})$, leading to statistical Hausdorff dimension $4$ for this random ensemble of infinite surfaces.

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This paper has now been published in Annals of Probability (to appear). Published as "Local limit of labelled trees and expected volume growth in a random quadrangulation".