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September 2022 Isoperimetric inequalities in the Brownian plane
Armand Riera
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Ann. Probab. 50(5): 2013-2055 (September 2022). DOI: 10.1214/22-AOP1576

Abstract

We consider the model of the Brownian plane, which is a pointed noncompact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar triangulation or the uniform infinite planar quadrangulation and is conjectured to be the universal scaling limit of many others random planar lattices. We establish sharp bounds on the probability of having a short cycle separating the ball of radius r centered at the distinguished point from infinity. Then we prove a strong version of the spatial Markov property of the Brownian plane. Combining our study of short cycles with this strong spatial Markov property we obtain sharp isoperimetric bounds for the Brownian plane.

Funding Statement

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

Acknowledgements

I warmly thank Jean-François Le Gall for all his suggestions and for carefully reading earlier versions of this manuscript. We also thank the referee for several useful comments.

Citation

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Armand Riera. "Isoperimetric inequalities in the Brownian plane." Ann. Probab. 50 (5) 2013 - 2055, September 2022. https://doi.org/10.1214/22-AOP1576

Information

Received: 1 May 2021; Revised: 1 February 2022; Published: September 2022
First available in Project Euclid: 24 August 2022

Digital Object Identifier: 10.1214/22-AOP1576

Subjects:
Primary: 60D05
Secondary: 60J65

Keywords: Brownian plane , Brownian snake , Isoperimetric inequality , separating cycles

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 5 • September 2022
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