The Annals of Probability

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

Philippe Chassaing and 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(r4).

Article information

Source
Ann. Probab., Volume 34, Number 3 (2006), 879-917.

Dates
First available in Project Euclid: 27 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1151418487

Digital Object Identifier
doi:10.1214/009117905000000774

Mathematical Reviews number (MathSciNet)
MR2243873

Zentralblatt MATH identifier
1102.60007

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 05C30: Enumeration in graph theory 05C05: Trees 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Random surface quadrangulation expected volume growth well-labeled trees Galton–Watson trees birth and death process quantum gravity

Citation

Chassaing, Philippe; Durhuus, Bergfinnur. Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab. 34 (2006), no. 3, 879--917. doi:10.1214/009117905000000774. https://projecteuclid.org/euclid.aop/1151418487


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