Work

Random stable looptrees

Year: 2014

Type: article

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.

Source: Electronic Journal of Probability

DOI: 10.1214/ejp.v19-2732

Authors: Nicolas Curien, Igor Kortchemski

Institutions: Centre National de la Recherche Scientifique, Université Pierre-et-Marie-Curie, École Normale Supérieure - PSL, Département de mathématiques et applications

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $675

Funder: Agence Nationale de la Recherche