Open Access
2023 The SLE loop via conformal welding of quantum disks
Morris Ang, Nina Holden, Xin Sun
Electron. J. Probab. 28: 1-20 (2023). DOI: 10.1214/23-EJP914


We prove that the SLEκ loop measure arises naturally from the conformal welding of two γ-Liouville quantum gravity (LQG) disks for γ2=κ(0,4). The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE83 loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.

Funding Statement

M.A. was supported by the Simons Foundation as a Junior Fellow at the Simons Society of Fellows, and partially supported by NSF grant DMS-1712862. N.H. was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation, along with grant 175505 of the Swiss National Science Foundation. X.S. was supported by the Simons Foundation as a Junior Fellow at the Simons Society of Fellows, and by the NSF grant DMS-2027986 and the Career award 2046514.


We are in debt to Yilin Wang for her important insight on SLE loop. In our opinion, her contribution to Theorem 1.1 is as much as ours. We are also grateful to Steffen Rohde, Scott Sheffield, and Dapeng Zhan for helpful discussions. We thank two anonymous referees for helpful comments on an earlier version of this paper.


Download Citation

Morris Ang. Nina Holden. Xin Sun. "The SLE loop via conformal welding of quantum disks." Electron. J. Probab. 28 1 - 20, 2023.


Received: 24 May 2022; Accepted: 1 February 2023; Published: 2023
First available in Project Euclid: 21 February 2023

MathSciNet: MR4550760
zbMATH: 1517.60106
Digital Object Identifier: 10.1214/23-EJP914

Primary: 60G60 , 60J67

Keywords: Liouville quantum gravity , Schramm-Loewner evolution

Vol.28 • 2023
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