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

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

2003-11-28

arxiv.org/abs/math/0311532v3

Annals of Probability 2006, Vol. 34, No. 3, 879-917

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000774 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000774

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 $\Theta(r^4)$.

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