Random stable looptrees

We introduce a class of random compact metric spaces Lα indexed by α ∈ (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 ofLα is almost surely equal to α. We also show that stable looptrees are universal scaling limits, for the Gromov–Hausdorff topol-ogy, of various combinatorial models. In a companion paper, we prove that the stable looptree of parameter 32 is the scaling limit of cluster boundaries in critical site-percolation on large random triangulations. Figure 1: An α = 1.1 stable tree, and its associated looptreeL1.1, embedded non isomet-rically in the plane (this embedding ofL1.1 contains intersecting loops, even though they are disjoint in the metric space)