Conformal welding of quantum disks

Abstract

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 ${SLE}_{\mathrm{\kappa }}({\mathit{\rho }}_{-};{\mathit{\rho }}_{+})$ 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 $W=\frac{{\mathit{\gamma }}^2}{2}$ quantum disk. Moreover, it serves as a key ingredient in our companion work, which proves an exact formula for ${SLE}_{\mathrm{\kappa }}({\mathit{\rho }}_{-};{\mathit{\rho }}_{+})$ using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.