Liouville quantum gravity surfaces with boundary as matings of trees
Abstract
For $\gamma \in (0,2)$, the quantum disk and $\gamma$quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a spacefilling SLE$_{16/\gamma^2}$ curve on a quantum disk or on a $\gamma$quantum wedge is a certain explicit conditioned twodimensional Brownian motion with correlation $\cos(\pi\gamma^2/4)$. This extends the mating of trees theorem of Duplantier, Miller, and Sheffield (2014) to the case of quantum surfaces with boundary (the disk case for $\gamma \in (\sqrt 2 , 2)$ was previously treated by Duplantier, Miller, Sheffield using different methods). As an application, we give an explicit formula for the conditional law of the LQG area of a quantum disk given its boundary length by computing the law of the corresponding functional of the correlated Brownian motion.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 DOI:
 10.48550/arXiv.1903.09120
 arXiv:
 arXiv:1903.09120
 Bibcode:
 2019arXiv190309120A
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Complex Variables;
 60J67;
 60D05;
 60J65
 EPrint:
 69 pages, 10 figures