Schnyder woods, SLE(16), and Liouville quantum gravity

Yiting Li, Xin Sun, Samuel S. Watson

Published 2017-05-10Version 1

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. We show that a uniformly sampled Schnyder-wood-decorated triangulation with $n$ vertices converges as $n\to\infty$ to the Liouville quantum gravity with parameter $1$, decorated with a triple of SLE$_{16}$'s of angle difference $2\pi/3$ in the imaginary geometry sense. Our convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via Liouville quantum gravity and imaginary geometry.