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Local limit of labeled trees and expected volume growth in a random quadrangulation. (English) Zbl 1102.60007

Well-labeled trees are studied by R. Cori and B. Vanquelin [Can. J. Math. 33, 1023–1042 (1981; Zbl 0415.05020)], who established a relationship between the number of well-labeled trees of size \(N\) and the number of quadrangulated planar maps with \(N\) faces. Given a probability measure on a space of planar maps (called also ensemble of planar maps), a relevant quantity of interest is the exponent \(\alpha\) that characterizes the expected volume of growth of the maps. Let \(B_r(\mathcal{M})\) be a ball of radius \(r\) around a marked point on the surface \(\mathcal{M}\) and let \(| B_r(\mathcal{M})|\) denote its volume, that is, the number of vertices in \(B_r(\mathcal{M})\). Then, \(\alpha\) is determined by the relation \(E(|B_r|)=\Theta(r^{\alpha})\) if this number \(\alpha\) exists. The expectation is taken with respect to the given probability measure and \(\Theta(r^{\alpha})\) denotes a function bounded from above and below by positive constant multiples of \(r^{\alpha}\) as \(r\) becomes large. In the present paper the authors exploit a bijective correspondence between planar quadrangulations and well-labeled trees to construct, using simple combinatorial arguments, a uniform probability measure \(\mu\) on the set of infinite well-labeled trees. They show how to identify well-labeled trees in the support of this measure with infinite quadrangulated planar surfaces. Then, viewing \(\mu\) as a measure on these surfaces, the authors prove that the exponent \(\alpha\) of the expected volume growth is equal to \(4\). To prove this result they show that the random surface, obtained in the limit, can be described in terms of a birth and death process and of a sequence of multitype Galton-Watson trees.

MSC:

60C05 Combinatorial probability
05C30 Enumeration in graph theory
05C05 Trees
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics

Citations:

Zbl 0415.05020
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