## 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

Zbl 0415.05020
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### References:

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