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May 2022 Geodesic stars in random geometry
Jean-François Le Gall
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Ann. Probab. 50(3): 1013-1058 (May 2022). DOI: 10.1214/21-AOP1553

Abstract

A point of a metric space is called a geodesic star with m arms if it is the endpoint of m disjoint geodesics. For every m{1,2,3,4}, we prove that the set of all geodesic stars with m arms in the Brownian sphere has dimension 5m. This complements recent results of Miller and Qian, who proved that this dimension is smaller than or equal to 5m.

Funding Statement

The present work was supported by the ERC Advanced Grant 740943 GEOBROWN.

Acknowledgements

I thank Jason Miller and Wei Qian for keeping me informed of their ongoing work. I am also indebted to the referee for a careful reading of the manuscript and several useful remarks.

Citation

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Jean-François Le Gall. "Geodesic stars in random geometry." Ann. Probab. 50 (3) 1013 - 1058, May 2022. https://doi.org/10.1214/21-AOP1553

Information

Received: 1 February 2021; Revised: 1 August 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

Digital Object Identifier: 10.1214/21-AOP1553

Subjects:
Primary: 60D05
Secondary: 05C80

Keywords: Brownian snake , Brownian sphere , Geodesic star , Hausdorff dimension , hull

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 3 • May 2022
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