Topics in Random Conformal Geometry: SLE Bubble Measures, Conformal Weldings of Liouville Quantum Gravity Surfaces, and Applications

In this dissertation, we showed that the SLE bubble measure recently constructed by Zhan arises naturally from the conformal welding of two Liouville quantum gravity disks. The proof of the main results relies on 1) a ``quantum version'' of the limiting construction of the SLE bubble, 2) the conformal welding between quantum triangles and quantum disks due to Ang, Sun and Yu, and 3) the uniform embedding techniques of Ang, Holden and Sun. As a by-product of our proof, we obtained a decomposition formula of the SLE bubble measure. Furthermore, we provided two applications of our conformal welding results. First, we computed the moments of the conformal radius of the SLE bubble on the upper half plane conditioning on surrounding i. The second application concerns the bulk-boundary correlation function in the Liouville Conformal Field Theory (LCFT). Within probabilistic frameworks, we derived a formula linking the bulk-boundary correlation function in the LCFT to the joint law of left & right quantum boundary lengths and the quantum area of the two-pointed quantum disk. This relation will be used by Ang, Remy, Sun and Zhu in a concurrent work to verify the formula of two-pointed bulk-boundary correlation function in physics predicted by Hosomichi (2001).