Two-pointed quantum disks with a weight parameter 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 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 quantum disk. Moreover, it serves as a key ingredient in our companion work, which proves an exact formula for using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.
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.
"Conformal welding of quantum disks." Electron. J. Probab. 28 1 - 50, 2023. https://doi.org/10.1214/23-EJP943