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September 2022 Stable shredded spheres and causal random maps with large faces
Jakob Björnberg, Nicolas Curien, Sigurdur Örn Stefánsson
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Ann. Probab. 50(5): 2056-2084 (September 2022). DOI: 10.1214/22-AOP1579

Abstract

We introduce a new familiy of random compact metric spaces Sα for α(1,2), which we call stable shredded spheres. They are constructed from excursions of α-stable Lévy processes on [0,1] possessing no negative jumps. Informally, viewing the graph of the Lévy excursion in the plane, each jump of the process is “cut open” and replaced by a circle, and then all points on the graph at equal height, which are not separated by a jump, are identified. We show that the shredded spheres arise as scaling limits of models of causal random planar maps with large faces introduced by Di Francesco and Guitter. We also establish that their Hausdorff dimension is almost surely equal to α. Point identification in the shredded spheres is intimately connected to the presence of decrease points in stable spectrally positive Lévy processes, as studied by Bertoin in the 1990s.

Funding Statement

The first author acknowledges support from Vetenskapsrådet, Grants 2015-0519 and 2019-04185 and is grateful for the hospitality at Université Paris-Sud Orsay. The second author acknowledges supports from ERC “GeoBrown” as well as the grant ANR-14-CE25-0014 “ANR GRAAL”. The third author acknowledges support from the Icelandic Research Fund, Grant Number: 185233-051, and is grateful for the hospitality at Université Paris-Sud Orsay and at Chalmers.

Acknowledgments

We thank the two anonymous referees for crucial remarks that helped clarifying the paper.

Citation

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Jakob Björnberg. Nicolas Curien. Sigurdur Örn Stefánsson. "Stable shredded spheres and causal random maps with large faces." Ann. Probab. 50 (5) 2056 - 2084, September 2022. https://doi.org/10.1214/22-AOP1579

Information

Received: 1 July 2021; Published: September 2022
First available in Project Euclid: 24 August 2022

Digital Object Identifier: 10.1214/22-AOP1579

Subjects:
Primary: 05C80 , 60F17 , 60G51

Keywords: Gromov–Hausdorff convergence , Hausdorff dimension , Random planar map , Scaling limit , stable distribution

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 5 • September 2022
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