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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.

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