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

TL;DR

This paper establishes a connection between Schnyder woods, random walks, and Liouville quantum gravity, showing that scaled limits of Schnyder-wood-decorated triangulations converge to Brownian motions coupled with SLE$_{16}$ curves in the LQG framework.

Contribution

It demonstrates the continuum limit of Schnyder's embedding algorithm as a coupling of Brownian motions and SLE$_{16}$ curves within Liouville quantum gravity.

Findings

Convergence of Schnyder-wood-decorated triangulations to Brownian motions in the LQG setting.

Coupling of three SLE$_{16}$ curves with angle difference $2 extpi/3$ in imaginary geometry.

Description of the continuum limit of planar maps via LQG and SLE.

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.