Liouville quantum gravity surfaces with boundary as matings of trees

TL;DR

This paper extends the mating of trees framework to Liouville quantum gravity surfaces with boundary, providing explicit descriptions of boundary length processes and conditional laws of quantum areas.

Contribution

It generalizes the mating of trees theorem to boundary cases of LQG surfaces, offering explicit Brownian motion characterizations and formulas for quantum area distributions.

Findings

Boundary length process is a conditioned correlated Brownian motion.

Explicit formula for quantum disk area given boundary length.

Extension of previous boundary case results to a broader parameter range.

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.