Random stable looptrees

We introduce a class of random compact metric spaces ℒ_α indexed by α ∊ 2 (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 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.