Random stable looptrees

TL;DR

This paper introduces stable looptrees, a new class of random metric spaces formed by loops connected along a tree, and demonstrates their properties and universality as scaling limits of various models.

Contribution

It defines stable looptrees as dual graphs of lpha-stable Lévy trees and establishes their Hausdorff dimension and universality as scaling limits.

Findings

Hausdorff dimension of L(lpha) is almost surely lpha

Stable looptrees are universal scaling limits for various combinatorial models

Stable looptree of parameter 3/2 is the limit of cluster boundaries in critical percolation

Abstract

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, and can be informally be viewed as dual graphs of \alpha-stable L\'evy trees. We study their properties and prove 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. 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.