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

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 $2π/3$ in imaginary geometry. Our convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via LQG and SLE.