Schnyder woods, \(\mathrm{SLE}_{16}\), and Liouville quantum gravity
The authors consider a 3-spanning-tree decomposition of a simple triangulation, known as the Schnyder wood. In the framework of mating of trees, a uniformly sampled Schnyder-wood-decorated triangulation can produce a triple of random walks. The authors 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 SLE16's curves. These three SLE16's curves are coupled such that the angle difference between them is \(\frac{2\pi}{3}\) in imaginary geometry. \N\NThe convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via LQG and SLE.
scientific article; zbMATH DE number 7876026
scientific article