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2014 Random stable looptrees
Nicolas Curien, Igor Kortchemski
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Electron. J. Probab. 19: 1-35 (2014). DOI: 10.1214/EJP.v19-2732


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 be 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.


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Nicolas Curien. Igor Kortchemski. "Random stable looptrees." Electron. J. Probab. 19 1 - 35, 2014.


Accepted: 11 November 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1307.60061
MathSciNet: MR3286462
Digital Object Identifier: 10.1214/EJP.v19-2732

Primary: 60F17 , 60G52
Secondary: 05C80

Keywords: Random metric spaces , Random non-crossing configurations , Stable processes


Vol.19 • 2014
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