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Local limit of labeled trees and expected volume growth in a random quadrangulation

Philippe Chassaing·Bergfinnur Durhuus
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Abstract

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an 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 expected volume of the ball of radius $r$ around a marked point in the limit random surface is $Θ(r^4)$.

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Philippe Chassaing, Bergfinnur Durhuus. 2006-06-29. Local limit of labeled trees and expected volume growth in a random quadrangulation. https://doi.org/10.1214/009117905000000774

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