Abstract
We define decorated α-stable trees which are informally obtained from an α-stable tree by blowing up its branchpoints into random metric spaces. This generalizes the α-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 α-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.
Funding Statement
The second author acknowledges support from the Icelandic Research Fund, Grant Number: 185233-051, and is grateful for the hospitality at Université Paris-Sud Orsay. The first author was a post-doctoral fellow at the University of British Columbia when most of the research that led to this paper was conducted.
Acknowledgments
We warmly thank the referee for the thorough reading of the manuscript and the helpful comments. We are grateful to Nicolas Curien for suggesting this collaboration, and to Eleanor Archer for discussing some earlier version of her project [3].
Citation
Delphin Sénizergues. Sigurdur Örn Stefánsson. Benedikt Stufler. "Decorated stable trees." Electron. J. Probab. 28 1 - 53, 2023. https://doi.org/10.1214/23-EJP1050
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