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(r4).
Philippe Chassaing. Bergfinnur Durhuus. "Local limit of labeled trees and expected volume growth in a random quadrangulation." Ann. Probab. 34 (3) 879 - 917, May 2006. https://doi.org/10.1214/009117905000000774