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Random stable looptrees

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We introduce a class of random compact metric spaces $L_\alpha$ 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, can be informally be viewed as dual graphs of $\alpha$ -stable Lévy trees and are coded by a spectrally positive $\alpha$-stable Lévy process. We study their properties and see in particular that the Hausdorff dimension of $L_\alpha$ 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. We finally see that the stable looptree of parameter $3/2$ is closely related to the scaling limits of cluster boundaries in critical site-percolation on large random triangulations. Based on joint works with Nicolas Curien

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