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Liouville quantum gravity surfaces with boundary as matings of trees · Around

2019

Liouville quantum gravity surfaces with boundary as matings of trees

Ang, Morris, Gwynne, Ewain

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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).

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