Open Access
2023 Conformal welding of quantum disks
Morris Ang, Nina Holden, Xin Sun
Author Affiliations +
Electron. J. Probab. 28: 1-50 (2023). DOI: 10.1214/23-EJP943


Two-pointed quantum disks with a weight parameter W>0 are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal SLEκ(ρ;ρ+) curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees theory. Our results can be used to give unified proofs of the mating-of-trees theorems for the quantum disk and the quantum sphere, in addition to a mating-of-trees description of the weight W=γ22 quantum disk. Moreover, it serves as a key ingredient in our companion work, which proves an exact formula for SLEκ(ρ;ρ+) using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.

Funding Statement

M. A. was supported by NSF grant DMS-1712862. N. H. was supported by Dr. Max Rossler, the Walter Haefner Foundation, and the ETH Zurich Foundation. X. S. was supported by a Junior Fellow award from the Simons Foundation, and NSF Grant DMS-1811092 and DMS-2027986.


We thank Yilin Wang for inspiring discussions on the conformal welding of quantum disks, and Guillaume Baverez, Ewain Gwynne and Jason Miller for helpful comments on the first version of the paper.


Download Citation

Morris Ang. Nina Holden. Xin Sun. "Conformal welding of quantum disks." Electron. J. Probab. 28 1 - 50, 2023.


Received: 15 August 2021; Accepted: 31 March 2023; Published: 2023
First available in Project Euclid: 12 April 2023

MathSciNet: MR4574484
zbMATH: 1517.60105
Digital Object Identifier: 10.1214/23-EJP943

Primary: 60G60 , 60J67

Keywords: Liouville quantum gravity , Schramm-Loewner evolution

Vol.28 • 2023
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