Decorated stable trees

TL;DR

This paper introduces decorated alpha-stable trees, extending stable trees by replacing branchpoints with random metric spaces, and explores their properties, constructions, and convergence from discrete models.

Contribution

It defines decorated alpha-stable trees, provides multiple constructions, analyzes their geometric properties, and proves convergence from discrete models.

Findings

Decorated alpha-stable trees generalize stable trees with complex branchpoint structures.

The paper establishes invariance principles linking discrete and continuous decorated trees.

Examples include applications in random trees and planar maps.

Abstract

We define decorated $\alpha$-stable trees which are informally obtained from an $\alpha$-stable tree by blowing up its branchpoints into random metric spaces. This generalizes the $\alpha$-stable looptrees of Curien and Kortchemski, where those metric spaces are just deterministic circles. We provide different constructions for these objects, which allows us to understand some of their geometric properties, including compactness, Hausdorff dimension and self-similarity in distribution. We prove an invariance principle which states that under some conditions, analogous discrete objects, random decorated discrete trees, converge in the scaling limit to decorated $\alpha$-stable trees. We mention a few examples where those objects appear in the context of random trees and planar maps, and we expect them to naturally arise in many more cases.