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Random stable looptrees. (English) Zbl 1307.60061
Summary: We introduce a class of random compact metric spaces $$\mathscr{L}_{\alpha}$$ indexed by $$\alpha~\in(1,2)$$ and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can informally be viewed as dual graphs of $$\alpha$$-stable Lévy trees. We study their properties and prove in particular that the Hausdorff dimension of $$\mathscr{L}_{\alpha}$$ is almost surely equal to $$\alpha$$. We also show that stable looptrees are universal scaling limits, for the Gromov-Hausdorff topology, of various combinatorial models. In a companion paper, we prove that the stable looptree of parameter $$\frac{3}{2}$$ is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations.

##### MSC:
 60G52 Stable stochastic processes 60F17 Functional limit theorems; invariance principles 05C80 Random graphs (graph-theoretic aspects)
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