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May 2022 Simple conformal loop ensembles on Liouville quantum gravity
Jason Miller, Scott Sheffield, Wendelin Werner
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Ann. Probab. 50(3): 905-949 (May 2022). DOI: 10.1214/21-AOP1550

Abstract

We show that, when one draws a simple conformal loop ensemble (CLEκ for κ(8/3,4)) on an independent κ-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, (4/κ)-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be cos(4π/κ) which can be interpreted as a “density” of CLE loops in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces.

Some consequences of this result are the following: (i) It provides a construction of a CLE on LQG as a patchwork/welding of quantum disks. (ii) It allows us to construct the “natural quantum measure” that lives in a CLE carpet. (iii) It enables us to derive some new properties and formulas for SLE processes and CLE themselves (without LQG) such as the exact distribution of the trunk of the general SLEκ(κ6) processes.

The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of O(N) models on planar maps with large faces and CLE on LQG. Indeed, our Lévy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps, such as in recent work of Bertoin, Budd, Curien and Kortchemski.

The case of nonsimple CLEs on LQG is studied in another paper.

Funding Statement

JM was supported by ERC Starting Grant 804166 (SPRS). SS was supported by NSF award DMS-1712862. WW was supported by the SNF Grant 175505.

Acknowledgments

WW acknowledges the hospitality of the University of Cambridge, where part of this work has been written. We also thank an anonymous referee as well as Nina Holden and Matthis Lehmkühler for very useful comments.

Citation

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Jason Miller. Scott Sheffield. Wendelin Werner. "Simple conformal loop ensembles on Liouville quantum gravity." Ann. Probab. 50 (3) 905 - 949, May 2022. https://doi.org/10.1214/21-AOP1550

Information

Received: 1 March 2020; Revised: 1 May 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

Digital Object Identifier: 10.1214/21-AOP1550

Subjects:
Primary: 60J67 , 60K35
Secondary: 60G52 , 60G60 , 60J80 , 82B27 , 82B41

Keywords: Conformal loop ensembles , growth-fragmentation trees , Liouville quantum gravity , Schramm–Loewner evolutions

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 3 • May 2022
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