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 space-filling SLE${}_{16/{\gamma }^2}$ curve on a quantum disk or on a $\gamma$-quantum wedge is a certain explicit conditioned two-dimensional 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.

arXiv comment69 pages, 10 figures; final version, to appear in AIHP

References in corpus (6)

• Uniqueness and universality of the Brownian map
• Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
• Two perspectives of the 2D unit area quantum sphere and their equivalence
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa >8$
• Gaussian multiplicative chaos through the lens of the 2D Gaussian free field

Cited by in corpus (2)

• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Mating of trees for critical Liouville quantum gravity