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Schnyder woods, SLE(16), and Liouville quantum gravity
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abstract
In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood, to give a fundamental grid-embedding algorithm for planar maps. In the framework of mating of trees, a uniformly sampled Schnyder-wood-decorated triangulation can produce a triple of random walks. We show that these three walks converge in the scaling limit to three Brownian motions produced in the mating-of-trees framework by Liouville quantum gravity (LQG) with parameter $1$, decorated with a triple of SLE$_{16}$'s curves. These three SLE$_{16}$'s curves are coupled such that the angle difference between them is $2\pi/3$ in imaginary geometry. Our convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via LQG and SLE.
Forward citations
Cited by 6 Pith papers
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High-dimensional permutons: theory and applications
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Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity
Simulations of random planar maps and discrete Liouville quantum gravity contradict Watabiki's formula for the Hausdorff dimension and support the Ding-Gwynne formula for central charges in [-12.5, 0).
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Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators
A simple priority-based shelling heuristic produces well balanced Schnyder woods in practice, and experiments suggest these improve Schnyder drawing quality and cycle separator size.
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