Random stable looptrees

Nicolas Curien (CNRS & University Paris 6)
Igor Kortchemski (DMA, Ecole Normale Supérieure)

Abstract


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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Pages: 1-35

Publication Date: November 11, 2014

DOI: 10.1214/EJP.v19-2732

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