Random stable looptrees

Abstract

We introduce a class of random compact metric spaces L(α) indexed by α\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 be informally be viewed as dual graphs of α-stable Lévy trees. We study their properties and prove in particular that the Hausdorff dimension of L(α) is almost surely equal to α. 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 3/2 is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations.

35 pages, 12 figures. Final version, Electron. J. Probab. 19 (2014), no. 108, 1-35

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• Scaling limits of random looptrees and bipartite plane maps with prescribed large faces
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• A line-breaking construction of the stable trees
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